@htl("""
<style>
@import url("https://mmogib.github.io/math102/custom.css");
</style>
""")
TableOfContents(title="MATH102")
cm"""
<div style="color:red;font-size:16pt;">
[Syllabus](https://www.dropbox.com/scl/fi/blnlbqsaxakwoexafg4uq/Math102SyllabusT231.pdf?rlkey=fzy43rthbvpb9lnq0seiu4qb4&raw=1)
</div>
"""
begin
cm"""
<div class="img-container">
$(Resource("https://www.dropbox.com/s/cat9ots4ausfzyc/qrcode_itempool.com_kfupm.png?raw=1",:width=>300))
</div>
"""
@htl("")
end
5.2: Area
Objectives
Use sigma notation to write and evaluate a sum.
Understand the concept of area.
Approximate the area of a plane region.
Find the area of a plane region using limits.
Sigma Notation
md"""
# 5.2: Area
__Objectives__
> 1. Use sigma notation to write and evaluate a sum.
> 2. Understand the concept of area.
> 3. Approximate the area of a plane region.
> 4. Find the area of a plane region using limits.
## Sigma Notation
"""
Sigma Notation
The sum of
where
cm"""
$(beginBlock("Sigma Notation",""))
The sum of ``n`` terms ``a_1, a_2, \cdots, a_n`` is written as
```math
\sum_{i=1}^n a_i = a_1+ a_2+ \cdots+ a_n
```
where ``i`` is the __index of summation__, ``a_i`` is the th __``i``th term__ of the sum, and the upper and lower bounds of summation are ``n`` and ``1``.
$(endBlock())
"""
Summation Properties
Theorem
cm"""
$(beginBlock("Summation Properties",""))
```math
\begin{array}{lcl}
\displaystyle\sum_{i=1}^n k a_i &=& \displaystyle k\sum_{i=1}^n a_i \\
\\
\displaystyle\sum_{i=1}^n (a_i\pm b_i) &=& \displaystyle\sum_{i=1}^n a_i\pm\sum_{i=1}^n b_i \\
\\
\end{array}
```
$(endBlock())
$(beginTheorem("Summation Formulas"))
```math
\displaystyle
\begin{array}{ll}
(1) & \displaystyle\sum_{i=1}^n c = cn, \quad c \text{ is a constant} \\
\\
(2) & \displaystyle\sum_{i=1}^n i = \frac{n(n+1)}{2} \\
\\
(3) &\displaystyle \sum_{i=1}^n i^2 = \frac{n(n+1)(2n+1)}{6} \\
\\
(4) & \displaystyle\sum_{i=1}^n i^3 = \left[\frac{n(n+1)}{2}\right]^2 \\
\\
\end{array}
```
$(endTheorem())
Example 1:
Evaluating a Sum
cm"""
$(example("Example 1","Evaluating a Sum"))
Evaluate ``\displaystyle \sum_{i=1}^n\frac{i+1}{n}`` for ``n=10, 100, 1000`` and ``10,000``.
"""
Area
In Euclidean geometry, the simplest type of plane region is a rectangle. Although people often say that the formula for the area of a rectangle is
it is actually more proper to say that this is the definition of the area of a rectangle.
For a triangle
md"""
## Area
In __Euclidean geometry__, the simplest type of plane region is a rectangle. Although people often say that the *formula* for the area of a rectangle is
```math
A = bh
```
it is actually more proper to say that this is the *definition* of the __area of a rectangle__.
For a triangle ``A=\frac{1}{2}bh``
$(Resource("https://www.dropbox.com/s/sfsg0d4ha1m2gc6/triangle_area.jpg?raw=1", :width=>300))
"""
The Area of a Plane Region
Example
Use five rectangles to find two approximations of the area of the region lying between the graph of
and the
md"""
## The Area of a Plane Region
__Example__
Use __five__ rectangles to find two approximations of the area of the region lying between the graph of
```math
f(x)=5-x^2
```
and the $x$-axis between $x=0$ and $x=2$.
"""
f (generic function with 1 method)n =
begin
# # plot(f;xlim=(-2π,2π), xticks=(-2π:(π/2):2π,["$c π" for c in -2:0.5:2]))
# # recs= [rect(sample(p,Δx),Δx,p,f) for p in partition]
# pp1=plot(xx1,f.(xx1);legend=nothing)
# pp1 = plot!(xx1, f.(xx1), fillrange=zero, fillalpha=0.35, c=:blue, framestyle=:origin, label=nothing)
end
end
# @bind showConnc Radio(["show" => "✅", "hide" => "❌"], default="hide")
# (showConnc == "show") ? md"""
# $$A=\lim_{n\to \infty} R_n =\lim_{n\to \infty} L_n =\frac{22}{3}$$
# """ : ""
Finding Area by the Limit Definition
md"## Finding Area by the Limit Definition"
Find the area of a plane region bounded above by the graph of a nonnegative, continuous function
The region is bounded below by the x-axis and the left and right boundaries of the region are the vertical lines
begin
findingAreaP = plot(0.2:0.1:4, x -> 0.6x^3 - (10 / 3) * x^2 + (13 / 3) * x + 1.4, fillrange=zero, fillalpha=0.35, c=:red, framestyle=:origin, label=nothing, ticks=nothing)
plot!(findingAreaP, -0.1:0.1:4.1, x -> 0.6x^3 - (10 / 3) * x^2 + (13 / 3) * x + 1.4, c=:green, label=nothing)
(0.1, 4, text(L"y", 14)),
(4.1, 0.1, text(L"x", 14)),
(0.2, -0.1, text(L"a", 14)),
(4, -0.1, text(L"b", 14)),
(3.9, 4, text(L"f", 14))
])
cm"""
Find the area of a plane region bounded above by the graph of a nonnegative, __continuous__ function
```math
y=f(x)
```
The region is bounded below by the `x-`axis and the left and right boundaries of the region are the vertical lines ``x=a`` and ``x=b``
<div class="img-container">
$(Resource("https://www.dropbox.com/s/hnspiptmyybneqn/area_with_lower_and_upper.jpg?raw=1",:width=>400))
</div>
- To approximate the area of the region, begin by subdividing the interval into subintervals, each of width
- The endpoints of the intervals are
- Let
- Define an inscribed rectangle lying inside the
- Define an circumscribed rectangle lying outside the
- The sum of the areas of the inscribed rectangles is called a lower sum, and the sum of the areas of the circumscribed rectangles is called an upper sum.
- The actual area of the region lies between these two sums.
cm"""
- To approximate the area of the region, begin by subdividing the interval into subintervals, each of width
```math
\Delta x = \frac{b-a}{n}
```
- The endpoints of the intervals are
```math
{\color{purple}{\overset{a=x_0}{\overbrace{\color{black}{a+0(\Delta x)}}}}} <
{\color{purple}{\overset{a=x_1}{\overbrace{\color{black}{a+1(\Delta x)}}}}} <
{\color{purple}{\overset{a=x_2}{\overbrace{\color{black}{a+2(\Delta x)}}}}} <
\cdots <
{\color{purple}{\overset{a=x_n}{\overbrace{\color{black}{a+n(\Delta x)}}}}}.
```
- Let
```math
\begin{array}{lcl}
\displaystyle f(m_i) & = & \text{Minimum value of } f(x) \text{ on the }i^{\text{th}}\text{ subinterval} \\
\\
\displaystyle f(M_i) & = & \text{Maximum value of } f(x) \text{ on the }i^{\text{th}}\text{ subinterval} \\
\\
Example 4:
Finding Upper and Lower Sums for a Region
cm"""
$(example("Example 4","Finding Upper and Lower Sums for a Region"))
Find the upper and lower sums for the region bounded by the graph of ``f(x)=x^2`` and the ``x-``axis between ``x=0`` and ``x=2``.
"""
n =
f4 (generic function with 1 method)begin
end
Theorem
Let
where
and
cm"""
$(beginTheorem("Limits of the Lower and Upper Sums"))
Let ``f`` be continuous and nonnegative on the interval ``[a,b]``. The limits as ``n\to\infty`` of both the lower and upper sums exist and are equal to each other. That is,
```math
\displaystyle \lim_{n\to\infty}s(n)=
\displaystyle \lim_{n\to\infty}\sum_{i=1}^nf(m_i)\Delta x
=\displaystyle \lim_{n\to\infty}\sum_{i=1}^nf(M_i)\Delta x
=\displaystyle \lim_{n\to\infty}S(n)
```
where
```math
\Delta x = \frac{b-a}{n}
```
and ``f(m_i)`` and ``f(M_i)`` are the minimum and maximum values of ``f`` on the ``i``th subinterval.
$(endTheorem())
"""
Definition
Let
where
See the grpah
cm"""
$(beginBlock("Definition","Area of a Region in the Plane"))
Let ``f`` be continuous and nonnegative on the interval ``[a,b]``. The area of the region bounded by the graph of ``f`` , the ``x``-axis, and the vertical lines ``x=a`` and ``y=b`` is
```math
\textrm{Area} = \displaystyle \lim_{n\to\infty}\sum_{i=1}^nf(c_i)\Delta x
```
where
```math
x_{i-1}\leq c_i\leq x_i\quad \textrm{and}\quad \Delta x =\frac{b-a}{n}.
```
See the grpah
<div class="img-container">
$(Resource("https://www.dropbox.com/s/a3sjz8m9vspp5ec/area_def.jpg?raw=1"))
</div>
$(endBlock())
"""
Example 5:
Finding Area by the Limit Definition
Find the area of the region bounded by the graph of
cm"""
$(example("Example 5","Finding Area by the Limit Definition"))
Find the area of the region bounded by the graph of ``f(x)=x^3`` , the ``x``-axis, and the vertical lines ``x=0`` and ``x=1``.
"""
Example 7:
A Region Bounded by the y-axis
cm"""
$(example("Example 7","A Region Bounded by the <i>y</i>-axis"))
Find the area of the region bounded by the graph of ``f(y)=y^2`` and the ``y``-axis for ``0\leq y\leq 1``.))
"""
cm"""
### Midpoint Rule
```math
\textrm{Area} \approx \sum_{i=1}^n f\left(\frac{x_{i-1}+x_i}{2}\right)\Delta x.
```
$(example("Example 8","Approximating Area with the Midpoint Rule"))
Use the Midpoint Rule with ``n=4`` to approximate the area of the region bounded by the graph of ``f(x)=\sin x`` and the ``x``-axis for ``0\leq x\leq \pi``.
"""
2.05234430595406235.3: Riemann Sums and Definite Integrals
Objectives
Understand the definition of a Riemann sum.
Evaluate a definite integral using limits and geometric formulas.
Evaluate a definite integral using properties of definite integrals.
md"""# 5.3: Riemann Sums and Definite Integrals
> __Objectives__
> 1. Understand the definition of a Riemann sum.
> 2. Evaluate a definite integral using limits and geometric formulas.
> 3. Evaluate a definite integral using properties of definite integrals.
"""
Riemann Sums
md"## Riemann Sums"
g (generic function with 1 method)n =
Definition of Riemann Sum
Let
where
If
is called a Riemann sum of
cm"""
$(beginBlock("Definition of Riemann Sum",""))
Let ``f`` be defined on the closed interval ``[a,b]``, and let ``\Delta`` be a partition of ``[a,b]`` given by
```math
a=x_0 < x_1 < x_2 < \cdots < x_{n-1} < x_n = b
```
where ``\Delta x_i`` is the width of the th subinterval
```math
[x_{i-1},x_i]\quad \color{red}i\textrm{{th subinterval}}
```
If ``c_i`` is any point in the th subinterval, then the sum
```math
\sum_{i=1}^n f(c_i)\Delta x_i, \quad x_{i-1}\leq c_i\leq x_i
```
is called a __Riemann sum__ of ``f`` for the partition ``\Delta``.
$(endBlock())
"""
Remark
The width of the largest subinterval of a partition
- If every subinterval is of equal width, then the partition is regular and the norm is denoted by
- For a general partition, the norm is related to the number of subintervals of
- Note that
cm"""
__Remark__
The width of the largest subinterval of a partition ``\Delta`` is the __norm__ of the partition and is denoted by ``\|\Delta\|``.
- If every subinterval is of equal width, then the partition is __regular__ and the norm is denoted by
```math
\|\Delta\| = \Delta x =\frac{b-a}{n} \quad \color{red}{\textrm{Regular partition}}
```
- For a general partition, the norm is related to the number of subintervals of ``[a,b]`` in the following way.
```math
\frac{b-a}{\|\Delta\|}\leq n \quad \color{red}{\textrm{General partition}}
```
- Note that
```math
\|\Delta\|\to 0 \quad \textrm{implies that}\quad n\to \infty.
```
"""
Definite Integrals
md"## Definite Integrals"
Definition of Definite Integral
If
exists, then
The limit is called the definite integral of
cm"""
$(beginBlock("Definition of Definite Integral",""))
If ``f`` is defined on the closed interval ``[a,b]`` and the limit of Riemann sums over partitions ``\Delta``
```math
\lim_{\|\Delta\|\to 0}\sum_{i=1}^nf(c_i)\Delta x_i
```
exists, then ``f`` is said to be __integrable__ on ``[a,b]`` and the limit is denoted by
```math
\lim_{\|\Delta\|\to 0}\sum_{i=1}^nf(c_i)\Delta x_i = \int_a^b f(x) dx.
```
The limit is called the __definite integral__ of ``f`` from ``a`` to ``b``. The number ``a`` is the __lower limit__ of integration, and the number ``b`` is the __upper limit__ of integration.
$(endBlock())
"""
Theorem
If a function
cm"""
$(beginTheorem("Continuity Implies Integrability"))
If a function ``f`` is continuous on the closed interval ``[a,b]``, then ``f`` is integrable on ``[a,b]``. That is,
```math
\int_a^b f(x) dx \quad \textrm{exists}.
```
$(endTheorem())
"""
Theorem
If
cm"""
$(beginTheorem("The Definite Integral as the Area of a Region"))
If ``f`` is continuous and nonnegative on the closed interval ``[a,b]``, then the area of the region bounded by the graph of ``f``, the ``x``-axis, and the vertical lines ``x=a`` and ``x=b`` is
```math
\textrm{Area} = \int_a^b f(x) dx
```
$(endTheorem())
"""
Example 3:
Areas of Common Geometric Figures
Evaluate each integral using a geometric formula.
cm"""
$(example("Example 3", "Areas of Common Geometric Figures"))
Evaluate each integral using a geometric formula.
- ``\displaystyle\int_1^3 4 dx``
- ``\displaystyle\int_0^3 (x+2) dx``
- ``\displaystyle\int_{-2}^2 \sqrt{4-x^2} dx``
"""
Remark
- It does not depend on
-
If
-
begin
theme(:wong)
x521 = 1:0.1:5
plot!(p3, x521, y / 2, ribbon=y / 2, linestyle=:dot, linealpha=0.1, framestyle=:origin, xticks=(1:5, [:a, "", "", "", :b]), label=nothing, ylims=(-1, 4))
cm"""
$(beginBlock("Remark","The definite integral is a **number**"))
- It does not depend on ``x``. In fact, we could use any letter in place of ``x`` without changing the value of the integral:
```math
\int_a^b f(x) dx = \int_a^b f(y) dy =\int_a^b f(w) dw =\int_a^b f(😀) d😀
```
- If ``f(x)\ge 0``, the integral ``\int_a^b f(x) dx`` is the area under the curve ``y=f(x)`` from ``a`` to ``b``.
$p3
- ``\int_a^b f(x) dx`` is the net area
<div class="img-container" >
Properties of Definite Integrals
md"## Properties of Definite Integrals"
Definitions
- If
- If
cm"""
$(beginBlock("Definitions","Two Special Definite Integrals"))
- If ``f`` is defined at ``x=a``, then ``\displaystyle \int_a^a f(x) dx =0``.
- If ``f`` is integrable on ``[a,b]``, then ``\displaystyle \int_b^a f(x) dx =- \int_a^b f(x) dx``.
$(endBlock())
"""
Theorem
If
cm"""
$(beginTheorem("Additive Interval Property"))
If ``f`` is integrable on the three closed intervals determined by ``a, b`` and ``c``, then
```math
\int_a^b f(x) dx = \int_a^c f(x) dx + \int_c^b f(x) dx.
```
$(endTheorem())
"""
Theorem
- If
cm"""
$(beginTheorem("Properties of Definite Integrals"))
- If ``f`` and ``g`` are integrable on ``[a,b]`` and ``k`` is a constant, then the functions ``kf`` and ``f\pm g`` are integrable on ``[a,b]``, and
1. ``\displaystyle \int_a^b kf(x) dx = k \int_a^b f(x) dx``.
2. ``\displaystyle \int_a^b \left[f(x)\pm g(x)\right] dx = \int_a^b f(x) dx\pm \int_a^b g(x) dx``.
$(endTheorem())
"""
Theorem
- If
- If
cm"""
$(beginTheorem("Preservation of Inequality"))
- If ``f`` is integrable and nonnegative on the closed interval ``[a,b]``, then
```math
0\leq \int_a^b f(x) dx.
```
- If ``f`` and ``g`` are integrable on the closed interval ``[a,b]`` and ``f(x)\leq g(x)`` for every ``x`` in ``[a,b]`` , then
```math
\int_a^b f(x) dx \leq \int_a^b g(x) dx.
```
$(endTheorem())
"""
Examples:
cm"""
$(example("Examples",""))
<div class="img-container">
$(Resource("https://www.dropbox.com/s/cat9ots4ausfzyc/qrcode_itempool.com_kfupm.png?raw=1",:width=>300))
</div>
"""
5.4: The Fundamental Theorem of Calculus
Objectives
Evaluate a definite integral using the Fundamental Theorem of Calculus.
Understand and use the Mean Value Theorem for Integrals.
Find the average value of a function over a closed interval.
Understand and use the Second Fundamental Theorem of Calculus.
Understand and use the Net Change Theorem.
md"""
# 5.4: The Fundamental Theorem of Calculus
> __Objectives__
> 1. Evaluate a definite integral using the Fundamental Theorem of Calculus.
> 2. Understand and use the Mean Value Theorem for Integrals.
> 3. Find the average value of a function over a closed interval.
> 4. Understand and use the Second Fundamental Theorem of Calculus.
> 5. Understand and use the Net Change Theorem.
"""
The Fundamental Theorem of Calculus
md"## The Fundamental Theorem of Calculus"
Antidifferentiation and Definite Integration
- ✒
- definite integral
- number
- ✒
- indefinite integral
- function
cm"""
__Antidifferentiation and Definite Integration__
<div class="img-container">
$(Resource("https://www.dropbox.com/s/8f52dty2aywwr92/diff_vs_antidiff.jpg?raw=1",:width=>600))
</div>
* ✒ ``\displaystyle \int_a^b f(x) dx``
* definite integral
* number
* ✒ ``\displaystyle \int f(x) dx``
* indefinite integral
* function
"""
Theorem
If a function
cm"""
$(beginTheorem("The Fundamental Theorem of Calculus"))
If a function ``f`` is continuous on the closed interval ``[a,b]`` and ``F`` is an antiderivative of ``f`` on the interval ``[a,b]``, then
```math
\int_a^b f(x) dx = F(b) - F(a).
```
$(endTheorem())
"""
Remark
We use the notation
cm"""
$(beginBlock("Remark",""))
We use the notation
```math
\int_a^b f(x) dx = \bigl. F(x)\Biggr|_a^b= F(b)-F(a) \quad \textrm{or}\quad
\int_a^b f(x) dx =\Bigl[F(x)\Bigr]_a^b = F(b)-F(a)
```
$(endBlock())
"""
Example 1:
Evaluating a Definite Integral
Evaluate each definite integral.
cm"""
$(example("Example 1", "Evaluating a Definite Integral"))
Evaluate each definite integral.
- ``\displaystyle \int_1^2 (x^2-3) dx``
$(" ")
- ``\displaystyle \int_1^4 3\sqrt{x} dx``
$(" ")
- ``\displaystyle \int_{0}^{\pi/4} \sec^2 x dx``
$(" ")
- ``\displaystyle \int_{0}^{2} \Big|2x-1\Big| dx``
"""
cm"""
<div class="img-container">
$(Resource("https://www.dropbox.com/s/cat9ots4ausfzyc/qrcode_itempool.com_kfupm.png?raw=1",:width=>300))
</div>
"""
Example 3:
Using the Fundamental Theorem to Find Area
Find the area of the region bounded by the graph of
the
begin
theme(:wong)
s54e3_x = 1:0.1:exp(1)
plot!(s54e3_p, s54e3_x, s54e3_f.(s54e3_x) / 2, ribbon=s54e3_f.(s54e3_x) / 2, linestyle=:dot, linealpha=0.1, framestyle=:origin, xticks=(1:4, [:1, :2, :3]), label=nothing, ylims=(-0.1, 1.5), xlims=(-0.1, 3))
cm"""
$(example("Example 3","Using the Fundamental Theorem to Find Area"))
Find the area of the region bounded by the graph of
```math
y=\frac{1}{x}
```
the ``x``-axis, and the vertical lines ``x=1`` and ``x=e``.
$s54e3_p
"""
end
The Mean Value Theorem for Integrals
md"## The Mean Value Theorem for Integrals"
Theorem
If
cm"""
$(beginTheorem("The Mean Value Theorem for Integrals"))
If ``f`` is continuous on the closed interval ``[a,b]``, then there exists a number ``c`` in the closed interval ``[a,b]`` such that
```math
\int_a^b f(x) dx =f(c)(b-a).
```
<div class="img-container">
$(Resource("https://www.dropbox.com/s/7fnr2kfq082kq0y/mvt.jpg?raw=1",
:style=>"display:flex;align-items:center;flex-direction: column;"))
</div>
$(endTheorem())
"""
Average Value of a Function
Definition
If
Example 4:
Finding the Average Value of a Function
Find the average value of
begin
ttt = md"## Average Value of a Function"
cm"""
$ttt
$(beginBlock("Definition","the Average Value of a Function on an Interval"))
If ``f`` is integrable on the closed interval ``[a,b]``, then the __average value__ of ``f`` on the interval is
```math
\textbf{Avergae value} = \frac{1}{b-a}\int_a^b f(x) dx
```
$(endBlock())
$(example("Example 4","Finding the Average Value of a Function"))
Find the average value of ``f(x)=3x^2-2x`` on the interval ``[1,4]``.
"""
end
The Second Fundamental Theorem of Calculus
md"## The Second Fundamental Theorem of Calculus"
cm"""
<div class="img-container">
$(Resource("https://www.dropbox.com/s/knjbngrqs2r2h1z/ftc2.jpg?raw=1",
:style=>"margin-top:20px;"
))
</div>
"""
Consider the following function
where
md"""
Consider the following function
```math
F(x) = \int_a^x f(t) dt
```
where ``f`` is a continuous function on the interval ``[a,b]`` and ``x \in [a,b]``.
"""
x =
begin
theme(:wong)
x4 = 1:0.1:5
plot!(p4, xVar, yVar, ribbon=yVar, linestyle=:dot, linealpha=0.1, framestyle=:origin, xticks=(1:5, [:a, "", "", "", :b]), label=nothing, ylims=(-1, 4))
plot!(p4, xticks=(x4, [:a, ["" for i in 2:length(xVar)-1]..., :x, ["" for i in length(xVar):length(x4)-2]..., :b]))
md"""
$p4
"""
end
Example If
Find
begin
md"""
**Example**
If ``g(x) = \int_0^x f(t) dt``
$img
Find ``g(2)``
"""
end
Theorem
If
cm"""
$(beginTheorem("The Second Fundamental Theorem of Calculus"))
If ``f`` is continuous on an open interval ``I`` containing ``a``, then, for every ``x`` in the interval,
```math
\frac{d}{dx}\left[\int_a^x f(t) \right] = f(x).
```
$(endTheorem())
"""
Remarks
Examples
Find the derivative of
(1)
(2)
(3)
(4)
md"""
__Remarks__
* ``{\large \frac{d}{dx}\left( \int_a^x f(u) du\right) = f(x)}``
* ``g(x)`` is an **antiderivative** of ``f``
__Examples__
Find the derivative of
(1) ``g_1(x) = \int_0^x \sqrt{1+t} dt``.
(2) ``g_2(x) = \int_x^0 \sqrt{1+t} dt``.
(3) ``g_3(x) = \int_0^{x^2} \sqrt{1+t} dt``.
(4) ``g_4(x) = \int_{\sin(x)}^{\cos(x)} \sqrt{1+t} dt``.
"""
💣 BE CAREFUL:
Evaluate
md"""
💣 BE CAREFUL:
Evaluate ``\large \int_{-3}^6 \frac{1}{x}dx``
"""
Net Change Theorem
md"## Net Change Theorem"
Question: If
Theorem
If
cm"""
**Question:** If ``y=F(x)``, then what does ``F'(x)`` represents?
$(beginTheorem("The Net Change Theorem"))
If ``F'(x)`` is the rate of change of a quantity ``F(x)`` , then the definite integral of ``F'(x)`` from ``a`` to ``b`` gives the total change, or __net change__, of ``F(x)`` on the interval ``[a,b]``.
```math
\int_a^b F'(x) dx = F(b) - F(a) \qquad \color{red}{\textrm{Net change of } F(x)}
```
$(endTheorem())
"""
- There are many applications, we will focus on one
If an object moves along a straight line with position function
- Remarks
- The acceleration of the object is
cm"""
- There are many applications, we will focus on one
If an object moves along a straight line with position function ``s(t)``, then its velocity is ``v(t)=s'(t)``, so
```math
\int_{t_1}^{t_2}v(t) dt = s(t_2)-s(t_1)
```
- **Remarks**
```math
\begin{array}{rcl}
\text{displacement} &=& \displaystyle \int_{t_1}^{t_2}v(t) dt\\
\\
\text{total distance traveled} &=& \displaystyle \int_{t_1}^{t_2}|v(t)| dt \\ \\
\end{array}
```
- The acceleration of the object is ``a(t)=v'(t)``, so
```math
\int_{t_1}^{t_2}a(t) dt = v(t_2)-v(t_1) \quad \text { is the change in velocity from time to time .}
```
"""
Example 10:
Solving a Particle Motion Problem
- What is the displacement of the particle on the time interval 1≤ t≤ 5?
- What is the total distance traveled by the particle on the time interval 1≤ t≤ 5?
cm"""
$(example("Example 10","Solving a Particle Motion Problem"))
A particle is moving along aline. Its velocity function (in ``m/s^2``) is given by
```math
v(t)=t^3-10t^2+29t-20,
```
<ul style="list-style-type: lower-alpha;">
<li> What is the <b>displacement</b> of the particle on the time interval 1≤ t≤ 5?</li>
<li>What is the <b>total distance</b> traveled by the particle on the time interval 1≤ t≤ 5?</li>
</ul>
"""
v (generic function with 1 method)
begin
u = symbols("u", real=true)
theme(:default)
a1, b1 = 1, 5
pp = plot(; layout=(2, 1))
markersize=5,
grids=:none,
framestyle=:origin,
showaxis=:x,
yticks=nothing,
ylims=(-0.4, 0.4),
label=nothing,
xticks=nothing,
Saved animation to E:\Dropbox\KFUPMWork\Teaching\OldSemesters\Sem231\MATH102\lessons\src\example_fps15.gif
5.5: The Substitution Rule
Objectives
Use pattern recognition to find an indefinite integral.
Use a change of variables to find an indefinite integral.
Use the General Power Rule for Integration to find an indefinite integral.
Use a change of variables to evaluate a definite integral.
Evaluate a definite integral involving an even or odd function.
| solve | ||
md"""
# 5.5: The Substitution Rule
> __Objectives__
> 1. Use pattern recognition to find an indefinite integral.
> 2. Use a change of variables to find an indefinite integral.
> 3. Use the General Power Rule for Integration to find an indefinite integral.
> 4. Use a change of variables to evaluate a definite integral.
> 5. Evaluate a definite integral involving an even or odd function.
||||
|------|------|-------|
||solve|||
|$$\int 2x \sqrt{1+x^2} \;\; dx$$| | $$\int \sqrt{u} \;\; du$$|
"""
Pattern Recognition
md"## Pattern Recognition"
Theorem
Let
Letting
Substitution Rule says: It is permissible to operate with
Example Find
# f155(x) = x / sqrt(1 - 4 * x^2)
# ex1_55=plot(-0.49:0.01:0.49,f155.(-0.49:0.01:0.49), framestyle=:origin)
cm"""
$(beginTheorem("Antidifferentiation of a Composite Function"))
Let ``g`` be a function whose range is an interval ``I``, and let ``f`` be a function that is continuous on ``I``. If ``g`` is differentiable on its domain and ``F`` is an antiderivative of ``f`` on ``I``, then
```math
\int f(g(x))g'(x)dx = F(g(x)) + C.
```
Letting ``u=g(x)`` gives ``du=g'(x)dx`` and
```math
\int f(u) du = F(u) + C.
```
$(endTheorem())
<div class="img-container">
$(Resource("https://www.dropbox.com/s/uua8vuahfxnp48c/subs_th.jpg?raw=1"))
</div>
**Substitution Rule says:** It is permissible to operate with ``dx`` and ``du`` after integral signs as if they were differentials.
**Example**
Find
```math
cm"""
<div class="img-container">
$(Resource("https://www.dropbox.com/s/cat9ots4ausfzyc/qrcode_itempool.com_kfupm.png?raw=1",:width=>300))
</div>
"""
Change of Variables for Indefinite Integrals
Example: Find
md"""
## Change of Variables for Indefinite Integrals
__Example__: Find
```math
\begin{array}{ll}
(i) & \int \sqrt{2x-1} dx \\ \\
(ii) & \int x\sqrt{2x-1} dx \\ \\
(iii) & \int \sin^23x\cos3x dx \\ \\
\end{array}
```
"""
The General Power Rule for Integration
md"""
## The General Power Rule for Integration
"""
Theorem
If
Equivalently, if
cm"""
$(beginTheorem("The General Power Rule for Integration"))
If ``g`` is a differentiable function of ``x``, then
```math
\int\bigl[g(x)\bigr]^ng'(x) dx = \frac{\bigl[g(x)\bigr]^{n+1}}{n+1} + C, \quad n\neq -1.
```
Equivalently, if ``u=g(x)``, then
```math
\int u^n du = \frac{u^{n+1}}{n+1} + C, \quad n\neq -1.
```
$(endTheorem())
__Example__: Find
```math
\begin{array}{ll}
(i) & \int 3(3x-1)^4 dx \\ \\
(ii) & \int (e^x+1)(e^x+x) dx \\ \\
(iii) & \int 3x^2\sqrt{x^3-2} \;dx \\ \\
(iv) & \displaystyle \int \frac{-4x}{(1-2x^2)^2}\; dx \\ \\
(v) & \int \cos^2 x\sin x \;dx \\ \\
\end{array}
```
"""
cm"""
<div class="img-container">
$(Resource("https://www.dropbox.com/s/cat9ots4ausfzyc/qrcode_itempool.com_kfupm.png?raw=1",:width=>300))
</div>
"""
x = symbols("x", real=true);
Change of Variables for Definite Integrals
md"""
## Change of Variables for Definite Integrals
"""
Substitution: Definite Integrals
Example: Evaluate
begin
ex2x1 = 1:0.1:exp(1)
ex2x12 = 0:0.1:1
ex2x2 = 0.6:0.1:4
ex2x22 = log(0.6):0.1:log(4)
theme(:wong)
ex2plt1 = plot(ex2x1, ex2y1, framestyle=:origin, xlims=(0, exp(1)), ylims=(-1, 1), fillrange=0, fillalpha=0.5, c=:red, label=nothing)
Example: Evaluate
cm"""
**Example:**
Evaluate
```math
\begin{array}{ll}
(i) & \int_1^2 \frac{dx}{\left(3-5x\right)^2} \\ \\
(iii) & \int_0^1 x(x^2+1)^3 \;dx \\ \\
(iv) & \int_1^5 \frac{x}{\sqrt{2x-1}}\;dx \\ \\
\end{array}
```
"""
Integration of Even and Odd Functions
md"""
## Integration of Even and Odd Functions
"""
Theorem
Let
- If
- If
Example Find
cm"""
$(beginTheorem("Integration of Even and Odd Functions"))
Let ``f`` be integrable on **``[-a,a]``**.
* If ``f`` is **even** ``\left[f(-x)=f(x)\right]``, then
```math
\int_{-a}^a f(x) dx = 2\int_0^a f(x) dx
```
* If ``f`` is **odd** ``\left[f(-x)=-f(x)\right]``, then
```math
\int_{-a}^a f(x) dx = 0
```
$(endTheorem())
**Example**
Find
```math
\int_{-1}^1 \frac{\tan x}{1+x^2+x^4} dx
```
"""
5.7: The Natural Logarithmic Function: Integration
Objectives
Use the Log Rule for Integration to integrate a rational function.
Integrate trigonometric functions.
Log Rule for Integration
md"""# 5.7: The Natural Logarithmic Function: Integration
> __Objectives__
> 1. Use the Log Rule for Integration to integrate a rational function.
> 2. Integrate trigonometric functions.
## Log Rule for Integration
"""
Theorem
Let
Remark
cm"""
$(bth("Log Rule for Integration"))
Let ``u`` be a differentiable function of ``x``.
```math
\begin{array}{llll}
\textrm{(i) }& \displaystyle \int \frac{1}{x} dx &=& \ln|x| + C \\ \\
\textrm{(ii) }& \displaystyle \int \frac{1}{u} du &=& \ln|u| + C \\ \\
\end{array}
```
$(eth())
$(bbl("Remark",""))
```math
\displaystyle \int \frac{u'}{u} dx = \ln|u| + C
```
$(ebl())
"""
Example 1:
Using the Log Rule for Integration
Example 3:
Finding Area with the Log Rule
the
Example 5:
Using Long Division Before Integrating
cm"""
$(ex(1,s="Using the Log Rule for Integration"))
```math
\int \frac{2}{x}dx
```
$(ex(3,s="Finding Area with the Log Rule"))
Find the area of the region bounded by the graph of
```math
y = \frac{x}{x^2+1}
```
the ``x``-axis, and the line ``x=3``.
<hr>
$(ex(5,s="Using Long Division Before Integrating"))
```math
\int \frac{x^2+x+1}{x^2+1}dx
```
$(poolcode())
"""
Examples Find
cm"""
__Examples__ Find
```math
\begin{array}{llll}
\textrm{(i) }& \displaystyle \int \frac{1}{4x-1} dx \\ \\
\textrm{(ii) }& \displaystyle \int \frac{3x^2+1}{x^3+x} dx \\ \\
\textrm{(iii) }& \displaystyle \int \frac{\sec^2x}{\tan x} dx \\ \\
\textrm{(iv) }& \displaystyle \int \frac{x^2+x+1}{x^2+1} dx \\ \\
\textrm{(v) }& \displaystyle \int \frac{2x}{(x+1)^2} dx \\ \\
\end{array}
```
"""
begin
function reimannSum(f, n, a, b; method="l", color=:green, plot_it=false)
# plot(f;xlim=(-2π,2π), xticks=(-2π:(π/2):2π,["$c π" for c in -2:0.5:2]))
else
Example 7:
Solve the differential equation
cm"""
$(ex(7,s="Solve the differential equation"))
Solve
```math
\frac{dy}{dx}=\frac{1}{x\ln x}
```
"""
Integrals of Trigonometric Functions
md"""
## Integrals of Trigonometric Functions
"""
Example 8:
Using a Trigonometric Identity
Example 9:
Derivation of the Secant Formula
cm"""
$(ex(8,s="Using a Trigonometric Identity"))
```math
\int \tan x dx
```
$(ex(9,s="Derivation of the Secant Formula"))
```math
\int \sec x dx
```
"""
5.8:Inverse Trigonometric Functions: Integration
Objectives
Integrate functions whose antiderivatives involve inverse trigonometric functions.
Use the method of completing the square to integrate a function.
Review the basic integration rules involving elementary functions.
Integrals Involving Inverse Trigonometric Functions
md"""# 5.8:Inverse Trigonometric Functions: Integration
> __Objectives__
> 1. Integrate functions whose antiderivatives involve inverse trigonometric functions.
> 2. Use the method of completing the square to integrate a function.
> 3. Review the basic integration rules involving elementary functions.
## Integrals Involving Inverse Trigonometric Functions
"""
Theorem
Let
cm"""
$(bth("Integrals Involving Inverse Trigonometric Functions"))
Let ``u`` be a differential function of ``x``, and let ``a>0``.
```math
\begin{array}{lllll}
\textrm{1.} & \displaystyle \int\frac{du}{\sqrt{a^2-u^2}} &=&\arcsin\frac{u}{a} + C \\ \\
\textrm{2.} & \displaystyle \int\frac{du}{a^2+u^2} &=&\frac{1}{a}\arctan\frac{u}{a} + C \\ \\
\textrm{3.} & \displaystyle \int\frac{du}{u\sqrt{u^2-a^2}} &=&\frac{1}{a}\text{arcsec}\frac{|u|}{a} + C \\ \\
\end{array}
```
$(eth())
"""
Examples Find
cm"""
__Examples__
Find
```math
\begin{array}{lllll}
\textrm{➡} & \displaystyle \int\frac{dx}{\sqrt{4-x^2}}, \\ \\
\textrm{➡} & \displaystyle \int\frac{dx}{2+9x^2}, \\ \\
\textrm{➡} & \displaystyle \int\frac{dx}{x\sqrt{4x^2-9}}, \\ \\
\textrm{➡} & \displaystyle \int\frac{dx}{\sqrt{e^{2x}-1}}, \\ \\
\textrm{➡} & \displaystyle \int\frac{x+2}{\sqrt{4-x^2}}dx. \\ \\
\end{array}
```
"""
Completing the Square
md" ## Completing the Square"
Example 5:
Completing the Square
Example 6:
Completing the Square
the
cm"""
$(ex(5,s="Completing the Square"))
Find
```math
\int\frac{dx}{x^2-4x+7}.
```
$(ex(6, s="Completing the Square"))
Find the area of the region bounded by the graph of
```math
f(x) = \frac{1}{\sqrt{3x-x^2}}
```
the ``x``-axis, and the lines ``x=\frac{3}{2}`` and ``x=\frac{9}{4}``.
"""
5.9: Hyperbolic Functions
Objectives
Develop properties of hyperbolic functions (MATH101).
Differentiate (MATH101) and integrate hyperbolic functions.
Develop properties of inverse hyperbolic functions (Reading only).
Differentiate and integrate functions involving inverse hyperbolic functions. (Reading only).
md"""
# 5.9: Hyperbolic Functions
> __Objectives__
> 1. Develop properties of hyperbolic functions (MATH101).
> 2. Differentiate (MATH101) and integrate hyperbolic functions.
> 3. Develop properties of inverse hyperbolic functions (Reading only).
> 4. Differentiate and integrate functions involving inverse hyperbolic functions. (Reading only).
"""
Circle:
Hyperbola:
cm"""
__Circle__: ``x^2+y^2=1``
<div class="img-container">
$(Resource("https://www.dropbox.com/s/c53yvdcyul4vvlz/circle.jpg?raw=1"))
</div>
__Hyperbola__: ``-x^2+y^2=1``
<div class="img-container">
$(Resource("https://www.dropbox.com/s/iy6fw024c6r50f8/hyperbola.jpg?raw=1"))
</div>
"""
Definitions of the Hyperbolic Functions
cm"""
__Definitions of the Hyperbolic Functions__
```math
\begin{array}{lllllll}
\sinh x &=& \displaystyle \frac{e^x-e^{-x}}{2} &\qquad&
\text{csch}\; x &=& \displaystyle \frac{1}{\sinh x},\; x\neq 0\\ \\
\cosh x &=& \displaystyle \frac{e^x+e^{-x}}{2} &\qquad&
\text{sech}\; x &=& \displaystyle \frac{1}{\cosh x}\\ \\
\tanh x &=& \displaystyle \frac{\sinh x}{\cosh x} &\qquad&
\text{coth}\; x &=& \displaystyle \frac{1}{\tanh x},\; x\neq 0\\ \\
\end{array}
```
<div class="img-container">
$(Resource("https://www.dropbox.com/s/0q1vcqb77u0ft1t/hyper_graphs.jpg?raw=1"))
</div>
"""
Hyperbolic Identities
cm"""
__Hyperbolic Identities__
```math
\begin{array}{rllllll}
\cosh^2 x - \sinh^2 x &=& 1, &\qquad&
\sinh (x+y)\; &=& \sinh x\cosh y +\cosh x\sinh y\\ \\
\tanh^2 x + \text{sech}^2 x &=& 1, &\qquad&
\sinh (x-y)\; &=& \sinh x\cosh y -\cosh x\sinh y\\ \\
\coth^2 x - \text{csch}^2 x &=& 1, &\qquad&
\cosh (x+y)\; &=& \cosh x\cosh y +\sinh x\sinh y\\ \\
&& &\qquad&
\cosh (x-y)\; &=& \cosh x\cosh y -\sinh x\sinh y\\ \\
\sinh^2 x &=& \displaystyle\frac{\cosh 2x -1}{2}, &\qquad&
\cosh^2 x\; &=& \displaystyle\frac{\cosh 2x +1}{2}\\ \\
\sin 2x &=& 2\sinh x\cosh x, &\qquad&
\cosh 2x\; &=& \cosh^2 x +\sinh^2 x\\ \\
\end{array}
```
"""
Theorem
Theorem Let
cm"""
$(bth("Differentiation and Integration of Hyperbolic Functions"))
__Theorem__ Let ``u`` be a differentiable function of ``x``.
```math
\begin{array}{rllllll}
\displaystyle \frac{d}{dx}\left(\sinh u\right) &=& \left(\cosh u\right)u', &\qquad&
\displaystyle \int \cosh u du &=& \sinh u \; +\; C\\ \\
\displaystyle \frac{d}{dx}\left(\cosh u\right) &=& \left(\sinh u\right)u', &\qquad&
\displaystyle \int \sinh u du &=& \cosh u \; +\; C\\ \\
\displaystyle \frac{d}{dx}\left(\tanh u\right) &=& \left(\text{sech}^2 u\right)u', &\qquad&
\displaystyle \int \text{sech}^2 u du &=& \tanh u \; +\; C\\ \\
\displaystyle \frac{d}{dx}\left(\coth u\right) &=& -\left(\text{csch}^2 u\right)u', &\qquad&
\displaystyle \int \text{csch}^2 u du &=& -\coth u \; +\; C\\ \\
\displaystyle \frac{d}{dx}\left(\text{sech} u\right) &=& -\left(\text{sech }u \tanh u\right)u', &\qquad&
\displaystyle \int \text{sech } u\tanh u du &=& -\text{sech } u \; +\; C\\ \\
Example 4:
Integrating a Hyperbolic Function
cm"""
$(ex(4,s="Integrating a Hyperbolic Function"))
Find
```math
\int \cosh 2x \sinh^2 2x dx
```
"""
7.1: Area of a Region Between Two Curves
Objectives
Find the area of a region between two curves using integration.
Find the area of a region between intersecting curves using integration.
Describe integration as an accumulation process.
md"""# 7.1: Area of a Region Between Two Curves
__Objectives__
> 1. Find the area of a region between two curves using integration.
> 2. Find the area of a region between intersecting curves using integration.
> 3. Describe integration as an accumulation process.
"""
......
begin
struct PlotData
x::StepRangeLen
fun::Function
lb::Union{Integer,Vector{Float64}}
end
xrotation --> 45
zrotation --> 6, :quiet
aspect_ratio --> 1
framestyle --> :origin
label-->nothing
end
# x, y = symbols("x,y", real=true)
p1Opt = (framestyle=:origin, aspectration=1)
function plot_implicit(F, c=0;
nlevels=6, # number of levels in a direction
slices=Dict(:x => :blue,
:y => :red,
:z => :green), # which directions and color
kwargs... # passed to initial `plot` call
Area of a Region Between Two Curves
md"## Area of a Region Between Two Curves"
| move | |
begin
md"""
| | |
|---|---|
|move $cnstSlider| ``n`` = $n1Slider|
|||
"""
end
How can we find the area between the two curves?
begin
theme(:wong)
a71,b71 = 1, 5
(5.9,0,L"x"),
(0.2,6,L"y"),
]
)
Remark
- Area =
Example 1:
Finding the Area of a Region Between Two Curves
Find the area of the region bounded above by
cm"""
**Remark**
- Area = ``y_{top}-y_{bottom}``.
$(ex(1,s="Finding the Area of a Region Between Two Curves"))
Find the area of the region bounded above by ``y=e^x``, bounded below by ``y=x``, bounded on the sides by ``x=0`` and ``x=1``.
---
"""
Solution
begin
ex1x=0:0.01:1
ex1Rect = Shape([(0.5,0.55),(0.55,0.55),(0.55,exp(0.55)),(0.5,exp(0.55))])
annotate!([ (0.77,0.6,L"y=x")
, (0.7,exp(0.7)+0.2,L"y=e^x")
, (1.1,1.7,L"x=1")
, (-0.1,0.5,L"x=0")
, (0.54,0.44,text(L"\Delta x",10))
])
md"""
**Solution**
$ex1plt
"""
end
Area of a Region Between Intersecting Curves
In geberal,
md"""## Area of a Region Between Intersecting Curves
In geberal,
$p2
```math
Area = \int_a^b \left|f(x) - g(x)\right| dx
```
"""
Example 2:
A Region Lying Between Two Intersecting Graphs
Find the area of the region enclosed by the graphs of
Solution in class
cm"""
$(ex(2,s="A Region Lying Between Two Intersecting Graphs"))
Find the area of the region enclosed by the graphs of ``f(x)=2-x^2`` and ``g(x)=x``.
*Solution in class*
---
"""
Example 3:
A Region Lying Between Two Intersecting Graphs
Find the area of the region bounded by the curves
cm"""
$(ex(3,s="A Region Lying Between Two Intersecting Graphs"))
Find the area of the region bounded by the curves
```math
y=\cos(x), \;\; y=\sin(x), \;\; x=0, \;\; x=\frac{\pi}{2}
```
---
"""
Example 4:
Curves That Intersect at More than Two Points
cm"""
$(ex(4,s="Curves That Intersect at More than Two Points"))
Find the area of the region between the graphs of
```math
f(x) = 3x^3-x^2 -10x, \qquad g(x)=-x^2+2x.
```
"""
Integrating with Respect to
begin
img1 = load(download("https://www.dropbox.com/s/r39ny15umqafmls/wrty.png?raw=1"))
md"""
__Integrating with Respect to ``y``__
$img1
"""
end
Example 5:
Horizontal Representative Rectangles
cm"""
$(ex(5,s="Horizontal Representative Rectangles"))
Find the area of the region bounded by the graphs of ``x=3-y^2`` and ``x=y+1``.
"""
7.2: Volume: The Disk Method
Objectives
Find the volume of a solid of revolution using the disk method.
Find the volume of a solid of revolution using the washer method.
Find the volume of a solid with known cross sections.
The Disk Method
md"""# 7.2: Volume: The Disk Method
> __Objectives__
> - Find the volume of a solid of revolution using the disk method.
> - Find the volume of a solid of revolution using the washer method.
> - Find the volume of a solid with known cross sections.
## The Disk Method
"""
Solids of Revolution
Volume of a disk
Disk Method
Taking the limit
Disk Method
To find the volume of a solid of revolution with the disk method, use one of the formulas below
cm"""
**Solids of Revolution**
<div class="img-container">
$(Resource("https://www.dropbox.com/s/z2k777veuxiaorq/solids_of_revs.png?raw=1"))
</div>
<div class="img-container">
$(Resource("https://www.dropbox.com/s/ik73cokibh1fuj6/disk_volume.png?raw=1"))
__Volume of a disk__
```math
V = \pi R^2 w
```
</div>
<div class="img-container">
__Disk Method__
$(Resource("https://www.dropbox.com/s/odttq795nrpcznw/disk_method.png?raw=1"))
</div>
```math
\begin{array}{lcl}
\textrm{Volume of solid} & \approx &\displaystyle \sum_{i=1}^n\pi\bigl[R(x_i)\bigr]^2 \Delta x \\
Example 1:
Using the Disk Method
and the
cm"""
$(ex(1,s="Using the Disk Method"))
Find the volume of the solid formed by revolving the region bounded by the graph of
```math
f(x) = \sqrt{\sin x}
```
and the ``x``-axis (``0\leq x\leq \pi``) about the ``x``-axis
"""
Example 2:
Using a Line That Is Not a Coordinate Axis
and
cm"""
$(ex(2,s="Using a Line That Is Not a Coordinate Axis"))
Find the volume of the solid formed by revolving the region bounded by the graphs of
```math
f(x)=2-x^2
```
and ``g(x)=1`` about the line ``y=1``.
"""
The Washer Method
md"## The Washer Method"
Washer Method
cm"""
<div class="img-container">
$(Resource("https://www.dropbox.com/s/ajra8g5fr8ssewe/washer_volume.png?raw=1"))
```math
\textrm{Volume of washer} = \pi(R^2-r^2)w
```
</div>
__Washer Method__
<div class="img-container">
$(Resource("https://www.dropbox.com/s/hvwa3707bftjir0/washer_method.png?raw=1"))
```math
V = \pi\int_a^b \bigl[\left(R[x]\right)^2-\left(r[x]\right)^2) dx
```
</div>
"""
Example 3:
Using the Washer Method
about the
cm"""
$(ex(3,s="Using the Washer Method"))
Find the volume of the solid formed by revolving the region bounded by the graphs of
```math
y=\sqrt{x} \qquad \textrm{and}\qquad y = x^2
```
about the ``x``-axis.
"""
Example 4:
Integrating with Respect to `y`: Two-Integral Case
about the
cm"""
$(ex(4,s="Integrating with Respect to `y`: Two-Integral Case"))
Find the volume of the solid formed by revolving the region bounded by the graphs of
```math
y=x^2+1, \quad y=0, \quad x=0, \quad \textrm{and}\quad x=1
```
about the ``y``-axis
"""
Solids with Known Cross Sections
md"## Solids with Known Cross Sections"
Volumes of Solids with Known Cross Sections
- For cross sections of area
- For cross sections of area
Example 6:
Triangular Cross Sections
The cross sections perpendicular to the
cm"""
[Example 1](https://www.geogebra.org/m/XFgMaKTy) | [Example 2](https://www.geogebra.org/m/XArpgR3A)
$(bbl("Volumes of Solids with Known Cross Sections",""))
1. For cross sections of area ``A(x)`` taken perpendicular to the ``x``-axis,
```math
V = \int_a^b A(x) dx
```
2. For cross sections of area ``A(y)`` taken perpendicular to the ``y``-axis,
```math
V = \int_c^d A(y) dy
```
$(ebl())
$(ex(6,s="Triangular Cross Sections"))
The base of a solid is the region bounded by the lines
```math
f(x)=1-\frac{x}{2},\quad g(x)=-1+\frac{x}{2}\quad \textrm{and}\quad x=0.
```
The cross sections perpendicular to the ``x``-axis are equilateral triangles.
"""
Exercise
Find the volume of the solid obtained by rotating the region bounded by
Exercise The region
Exercise Find the volume of the solid obtained by rotating the region in the previous Example about the line
Exercise Find the volume of the solid obtained by rotating the region in the previous Example about the line
cm"""
**Exercise**
Find the volume of the solid obtained by rotating the region bounded by ``y=x^3``, ``y=8`` , and ``x=0`` about the ``y``-axis.
**Exercise** The region ``\mathcal{R}`` enclosed by the curves ``y=x`` and ``y=x^2`` is rotated about the ``x``-axis. Find the volume of the resulting solid.
**Exercise** Find the volume of the solid obtained by rotating the region in the previous Example about the line ``y=2``.
**Exercise** Find the volume of the solid obtained by rotating the region in the previous Example about the line ``x=-1``.
"""
Exercise Figure below shows a solid with a circular base of radius 
md"""
**Exercise** Figure below shows a solid with a circular base of radius ``1``. Parallel cross-sections perpendicular to the base are equilateral triangles. Find the volume of the solid.
$(load(download("https://www.dropbox.com/s/bbxedang718jvvp/img4.png?dl=0")))
"""
7.3: Volume: The Shell Method
Objectives
Find the volume of a solid of revolution using the shell method.
Compare the uses of the disk method and the shell method.
Problem Find the volume of the solid generated by rotating the region bounded by
md"""
# 7.3: Volume: The Shell Method
> __Objectives__
> 1. Find the volume of a solid of revolution using the shell method.
> 2. Compare the uses of the disk method and the shell method.
**Problem**
Find the volume of the solid generated by rotating the region bounded by ``y=2x^2-x^3`` and ``y=0`` about the ``y-``axis.
"""
Step 1:
begin
md"""
Step 1: $show_graph_s
Step 2: $show_rect_s
Step 3: $show_labels_s
"""
end
""begin
else
""
end
end
The Shell Method
md"## The Shell Method"
A shell is a hallow circular cylinder
begin
shellImg=load(download("https://www.dropbox.com/s/8a2njc50e2hptok/shell.png?dl=0"))
md"""
A shell is a hallow circular cylinder
$shellImg
"""
end
cm"""
```math
V = 2 \pi r h \Delta r = \text{[circumference][height][thickness]}
```
"""
Cylindrical Shells Illustration
html"""
<div style="display: flex; justify-content:center; padding:20px; border: 2px solid rgba(125,125,125,0.2);">
<div>
<h5>Cylindrical Shells Illustration</h5>
<iframe width="560" height="315" src="https://www.youtube.com/embed/JrRniVSW9tg" title="YouTube video player" frameborder="0" allow="accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture" allowfullscreen></iframe>
</div>
</div>
"""
Horizontal Axis of Revolution
Vertical Axis of Revolution
cm"""
<div style="display: flex; justify-content: center;">
<div style="margin-right:12px;padding:4px;border: 2px solid white; border-radius: 10px;background: #eee;">
<span style="font-size:1.5M; font-weight: 800;">Horizontal Axis of Revolution</span>
```math
\text{Volume}=V = 2\pi \int_c^d p(y) h(y) dy
```
<div class="img-container">
$(Resource("https://www.dropbox.com/s/6qmijrhprht4kqj/shell_y.png?raw=1"))
</div>
</div>
<div style="margin-right:12px;padding:4px;border: 2px solid white; border-radius: 10px;background: #eee;"><span style="font-size:1.5M; font-weight: 800;">Vertical Axis of Revolution</span>
```math
\text{Volume}=V = 2\pi \int_a^b p(x) h(x) dx
```
<div class="img-container">
$(Resource("https://www.dropbox.com/s/ivbwuge5ti8vrff/shell_x.png?raw=1"))
Example: Find the volume of the solid generated by rotating the region bounded by
Solution:
begin
Shape(
]
)
, label=nothing
)
md"""
**Example:**
Find the volume of the solid generated by rotating the region bounded by ``y=2x^2-x^3`` and ``y=0`` about the ``y-``axis.
Solution:
$s3e0p1
"""
end
Example : Find the volume of the solid obtained by rotating about the
md"""
**Example :**
Find the volume of the solid obtained by rotating about the ``y-``axis the region between ``y=x`` and ``y=x^2``.
"""
Example: Find the volume of the solid obtained by rotating the region bounded by
md"""
**Example:**
Find the volume of the solid obtained by rotating the region bounded by ``y=x-x^2`` and ``y=0`` about the line ``x=2``.
"""
Example 4:
Shell Method Preferable
about the
cm"""
$(ex(4,s="Shell Method Preferable"))
Find the volume of the solid formed by revolving the region bounded by the graphs of
```math
y=x^2+1, \quad y=0, \quad x=0, \quad \text{and}\quad x=1.
```
about the ``y``-axis.
"""
Example 5:
Shell Method Necessary
cm"""
$(ex(5,s="Shell Method Necessary"))
Find the volume of the solid formed by revolving the region bounded by the graphs of ``y=x^3+x+1``, ``y=1``, and ``x=1`` about the line ``x=2``.
"""
7.4: Arc Length and Surfaces of Revolution
Objectives
Find the arc length of a smooth curve.
Find the area of a surface of revolution.
Arc Length
md"""
# 7.4: Arc Length and Surfaces of Revolution
> __Objectives__
> 1. Find the arc length of a smooth curve.
> 2. Find the area of a surface of revolution.
## Arc Length
"""
Definition
Let the function
Similarly, for a smooth curve
cm"""
$(bbl("Definition","Arc Length"))
Let the function ``y=f(x)`` represents a smooth curve on the interval ``[a,b]``. The __arc length__ of ``f`` between ``a`` and ``b`` is
```math
s = \int_a^b\sqrt{1+[f'(x)]^2} dx.
```
Similarly, for a smooth curve ``x=g(y)``, the arc length of ``g`` between ``c`` and ``d`` is
```math
s = \int_c^d\sqrt{1+[g'(y)]^2} dy.
```
$(ebl())
"""
Example 2:
Finding Arc Length
cm"""
$(ex(2, s="Finding Arc Length"))
Find the arc length of the graph of ``y=\displaystyle \frac{x^3}{6}+\frac{1}{2x}`` on the interval ``[\frac{1}{2},2]``.
"""
Example 3:
Finding Arc Length
cm"""
$(ex(3, s="Finding Arc Length"))
Find the arc length of the graph of ``\displaystyle (y-1)^3=x^2`` on the interval ``[0,8]``.
"""
Example 4:
Finding Arc Length
cm"""
$(ex(4, s="Finding Arc Length"))
Find the arc length of the graph of ``y=\ln(\cos x)`` from ``x=0`` to ``x=\pi/4``.
"""
Area of a Surface of Revolution
md"## Area of a Surface of Revolution"
Definition
When the graph of a continuous function is revolved about a line, the resulting surface is a surface of revolution.
cm"""
$(bbl("Definition","Surface of Revolution"))
When the graph of a continuous function is revolved about a line, the resulting surface is a __surface of revolution__.
$(ebl())
"""
Surface Area of frustum
Consider a function
Surface Area Formula
Definition
Let
The area
where
If
where
cm"""
<div class="img-container">
$(Resource("https://www.dropbox.com/s/199tfveph8mi2kz/surface_rev.png?raw=1"))
__Surface Area of *frustum*__
```math
S=2\pi r L, \quad \text{where}\quad r=\frac{r_1+r_2}{2}
```
</div>
Consider a function ``f`` that has a continuous derivative on the interval ``[a,b]``. The graph of ``f`` is revolved about the ``x``-axis
<div class="img-container">
$(Resource("https://www.dropbox.com/s/f454ldbfk1z3o2z/surface_rev2.png?raw=1"))
__Surface Area Formula__
```math
S=2\pi \int_a^b x \sqrt{1+[f'(x)]^2} dx.
```
</div>
$(bbl("Definition","Area of a Surface of Revolution"))
Remark
The formulas can be written as
and
where
cm"""
__Remark__
The formulas can be written as
```math
S=2\pi \int_a^b r(x) ds, \quad {\color{red} y \text{ is a function of x }}.
```
and
```math
S=2\pi \int_c^d r(y) ds, \quad {\color{red} x \text{ is a function of y }}.
```
where
```math
ds = \sqrt{1+\big[f'(x)\big]^2}dx \quad \text{and}\quad ds = \sqrt{1+\big[g'(y)\big]^2}dy \quad \text{respectively}.
"""
Example 6:
The Area of a Surface of Revolution
Example 7:
The Area of a Surface of Revolution
cm"""
$(ex(6,s="The Area of a Surface of Revolution"))
Find the area of the surface formed by revolving the graph of ``f(x)=x^3`` on the interval ``[0,1]`` about the ``x``-axis.
$(ex(7,s="The Area of a Surface of Revolution"))
Find the area of the surface formed by revolving the graph of ``f(x)=x^2`` on the interval ``[0,\sqrt{2}]`` about the ``y``-axis.
"""
8.1: Basic Integration Rules
Objectives
Review procedures for fitting an integrand to one of the basic integration rules.
md"""# 8.1: Basic Integration Rules
> __Objectives__
> 1. Review procedures for fitting an integrand to one of the basic integration rules.
"""
Review of Basic Integration Rules (
cm"""
<div style="background-color:#FF9733;color:white;font-weight:800;padding:2px 10px;width:350px;">
Review of Basic Integration Rules (``a>0``)
</div>
<div class="img-container">
$(Resource("https://www.dropbox.com/s/56svdxjfgowjojk/int_table.png?raw=1"))
</div>
"""
Example 3:
Example 4:
A Disguised Form of the Log Rule
cm"""
$(ex(3,s=""))
Find ``\displaystyle \int \frac{x^2}{\sqrt{16-x^6}} dx``.
$(ex(4,s="A Disguised Form of the Log Rule"))
Find ``\displaystyle \int \frac{dx}{1+e^x} ``.
"""
8.2: Integration by Parts
Objectives
Find an antiderivative using integration by parts.
The integration rule that corresponds to the Product Rule for differentiation is called integration by parts
md"""# 8.2: Integration by Parts
> __Objectives__
> 1. Find an antiderivative using integration by parts.
*The integration rule that corresponds to the Product Rule for differentiation is called __integration by parts__*
"""
Indefinite Integrals
cm"""
__Indefinite Integrals__
```math
\int f(x)g'\,(x) dx = f(x) g(x) - \int g(x) f'\,(x) dx
```
"""
Theorem
If
cm"""
$(bth("Integration by Parts"))
If ``u`` and ``v`` are functions of ``x`` and have continuous derivatives, then
```math
\int u dv = uv -\int v du.
```
$(eth())
"""
Example 1:
Integration by Parts
Example 2:
Integration by Parts
Example 3:
An Integrand with a Single Term
Example 4:
Repeated Use of Integration by Parts
Example 5:
Integration by Parts
Example 7:
Using the Tabular Method
cm"""
$(ex(1,s="Integration by Parts"))
Find ``\displaystyle \int x e^x dx``.
$(ex(2,s="Integration by Parts"))
Find ``\displaystyle \int x^2 \ln x dx``.
$(ex(3,s="An Integrand with a Single Term"))
Evaluate ``\displaystyle \int_0^1 \arcsin x dx``.
$(ex(4,s="Repeated Use of Integration by Parts"))
Find ``\displaystyle \int x^2 \sin x dx``.
$(ex(5,s="Integration by Parts"))
Find ``\displaystyle \int\sec^3 x dx``.s
$(ex(7,s="Using the Tabular Method"))
Find ``\displaystyle \int x^2 \sin 4x dx``.
"""
8.3: Trigonometric Integrals
Objectives
Solve trigonometric integrals involving powers of sine and cosine.
Solve trigonometric integrals involving powers of secant and tangent.
Solve trigonometric integrals involving sine-cosine products.
RECALL
md"""# 8.3: Trigonometric Integrals
> __Objectives__
> 1. Solve trigonometric integrals involving powers of sine and cosine.
> 2. Solve trigonometric integrals involving powers of secant and tangent.
> 3. Solve trigonometric integrals involving sine-cosine products.
**RECALL**
```math
\displaystyle
\sin^2 x + \cos^2 x =1, \quad \tan^2 x + 1 =\sec^2 x, \quad 1+\cot^2 x =\csc^2 x,
```
```math
\displaystyle
\cos^2 x = \frac{1 +\cos 2x }{2}, \quad \sin^2 x = \frac{1 -\cos 2x }{2}
```
```math
\displaystyle
\begin{array}{ccc}
\sin mx\sin nx & = & \frac{1}{2}\left[\cos(m-n)x-\cos(m+n)x\right], \\[0.2cm]
\sin mx\cos nx & = & \frac{1}{2}\left[\sin(m-n)x+\sin(m+n)\right], \\[0.2cm]
Integrals of Powers of Sine and Cosine
md"""
## Integrals of Powers of Sine and Cosine
```math
\int \sin^mx \cos^n x dx
```
* ``m`` is ``{\color{red}\text{odd}}``, write as ``\int \sin^{m-1}x \cos^n x \sin x dx``. _Example_: $\int \sin^5x \cos^2 x dx$
* ``n`` is ``{\color{red}\text{odd}}``, write as ``\int \sin^mx \cos^{n-1}\cos x dx``. _Example_ ``\int \sin^5x \cos^3 x dx``
* ``m`` and ``n`` are ``{\color{red}\text{even}}``, use formulae (_Example_ ``\int \cos^2 x dx`` and ``\int \sin^4 x dx``)
```math
\sin^2(x)=\frac{1-\cos(2x)}{2}, \quad \cos^2(x)=\frac{1+\cos(2x)}{2}.
```
"""
Example 1:
Power of Sine Is Odd and Positive
cm"""
$(ex(1,s="Power of Sine Is Odd and Positive"))
Find ``\displaystyle \int \sin^3 x \cos^4 x dx``.
"""
Example 2:
Power of Cosine Is Odd and Positive
cm"""
$(ex(2,s="Power of Cosine Is Odd and Positive"))
Evaluate ``\displaystyle \int_{\pi/6}^{\pi/3}\frac{\cos^3 x}{\sqrt{\sin x}} dx``.
"""
Example 3:
Power of Cosine Is Even and Nonnegative
cm"""
$(ex(3,s="Power of Cosine Is Even and Nonnegative"))
Find ``\displaystyle \int \cos^4 x dx``.
"""
Integrals of Powers of Secant and Tangent
md"""
## Integrals of Powers of Secant and Tangent
```math
\int \tan^mx \sec^n x dx
```
* ``n`` is even, write as ``\int \tan^mx \sec^{n-2}\sec^2 x dx``. _Example_ ``\int \tan^6x \sec^4 x dx``
* ``m`` is odd, write as ``\int \tan^{m-1}x \sec^{n-1}\tan x\sec x dx``. _Example_ ``\int \tan^5x \sec^7 x dx``.
"""
Example 4:
Power of Tangent Is Odd and Positive
cm"""
$(ex(4,s="Power of Tangent Is Odd and Positive"))
Find ``\displaystyle \int \frac{\tan^3x}{\sqrt{\sec x}} dx``.
"""
Example 5:
Power of Secant Is Even and Positive
cm"""
$(ex(5,s="Power of Secant Is Even and Positive"))
Find ``\displaystyle \int \sec^4 3x \tan^3 3x dx``.
"""
Example 6:
Power of Tangent Is Even
cm"""
$(ex(6,s="Power of Tangent Is Even"))
Evaluate ``\displaystyle \int_0^{\pi/4}\tan^4 x dx``.
"""
Example 7:
Converting to Sines and Cosines
cm"""
$(ex(7,s="Converting to Sines and Cosines"))
Find ``\displaystyle \int \frac{\sec x}{\sqrt{\tan^2 x}} dx``.
"""
Integrals Involving Sine-Cosine Products
md"## Integrals Involving Sine-Cosine Products"
Example 8:
Using a Product-to-Sum Formula
cm"""
$(ex(8,s="Using a Product-to-Sum Formula"))
Find ``\displaystyle \int \sin 5x \cos 4x dx``.
"""
8.4: Trigonometric Substitution
Objectives
Use trigonometric substitution to find an integral.
Use integrals to model and solve real-life applications.
md"""
# 8.4: Trigonometric Substitution
> __Objectives__
> 1. Use trigonometric substitution to find an integral.
> 2. Use integrals to model and solve real-life applications.
"""
Trigonometric Substitution
md"""
## Trigonometric Substitution
"""
We use trigonometric substitution to find integrals involving the radicals
cm"""
We use __trigonometric substitution__ to find integrals involving the radicals
```math
\sqrt{a^2-u^2},\quad \sqrt{a^2+u^2},\quad \sqrt{u^2-a^2}.
```
"""
Example 1:
Trigonometric Substitution
Example 2:
Trigonometric Substitution
Example 3:
Trigonometric Substitution: Rational Powers
Example 4:
Converting the Limits of Integration
cm"""
$(ex(1,s="Trigonometric Substitution"))
Find ``\displaystyle \int \frac{dx}{x^2\sqrt{9-x^2}}``.
$(ex(2,s="Trigonometric Substitution"))
Find ``\displaystyle \int \frac{dx}{\sqrt{4x^2+1}}``.
$(ex(3,s="Trigonometric Substitution: Rational Powers"))
Find ``\displaystyle \int \frac{dx}{(x^2+1)^{3/2}}``.
$(ex(4,s="Converting the Limits of Integration"))
Find ``\displaystyle \int_{\sqrt{3}}^{2} \frac{\sqrt{x^2-3}}{x}dx``.
"""
Applications
md"## Applications"
Example 5:
Finding Arc Length
cm"""
$(ex(5,s="Finding Arc Length"))
Find the arc length of the graph of ``f(x)=\frac{1}{2}x^2`` from ``x=0`` to ``x=1``.
"""
8.5: Partial Fractions
Objectives
Understand the concept of partial fraction decomposition.
Use partial fraction decomposition with linear factors to integrate rational functions.
Use partial fraction decomposition with quadratic factors to integrate rational functions.
Partial Fractions
We learn how to integrate rational function: quotient of polunomial.
How?
◾ STEP 0 : if degree of
◾ STEP 1 : Peform long division of
and apply STEP 2 on
◾ STEP 2 : Write the partial fractions decomposition
◾ STEP 3 : Integrate
md"""
# 8.5: Partial Fractions
> __Objectives__
> 1. Understand the concept of partial fraction decomposition.
> 2. Use partial fraction decomposition with linear factors to integrate rational functions.
> 3. Use partial fraction decomposition with quadratic factors to integrate rational functions.
## Partial Fractions
We learn how to integrate rational function: quotient of polunomial.
```math
f(x) =\frac{P(x)}{Q(x)}, \qquad P, Q \text{ are polynomials}
```
**How?**
◾ __STEP 0__ : if degree of ``P`` is greater than or equal to degree of ``Q`` goto
__STEP 1__, else GOTO __STEP 2__.
◾ __STEP 1__ : Peform long division of ``P`` by ``Q`` to get
```math
\frac{P(x)}{Q(x)} = S(x) + \frac{R(x)}{Q(x)}
```
and apply __STEP 2__ on ``\frac{R(x)}{Q(x)}``.
Partial Fractions Decomposition
We need to write
case 1:
then there exist constants
case 2:
then there exist constants
case 3:
then there exist constants
case 4:
then there exist constants
md"""
__Partial Fractions Decomposition__
We need to write ``\frac{R(x)}{Q(x)}`` as sum of __partial fractions__ by __factor__ ``Q(x)``. Based on the factors, we write the decomposition accoding to the following cases
__case 1__: ``Q(x)`` is a product of distinct linear factors.
we write
```math
Q(x)=(a_1x+b_1)(a_2x+b_2)\cdots (a_kx+b_k)
```
then there exist constants ``A_1, A_2, \cdots, A_k`` such that
```math
\frac{R(x)}{Q(x)}= \frac{A_1}{a_1x+b_1}+\frac{A_2}{a_2x+b_2}+\cdots +\frac{A_k}{a_kx+b_k}
```
__case 2__: ``Q(x)`` is a product of linear factors, some of which are repeated.
say first one
```math
Q(x)=(a_1x+b_1)^r(a_2x+b_2)\cdots (a_kx+b_k)
```
then there exist constants ``B_1, B_2, \cdots B_r, A_2, \cdots, A_k`` such that
```math
Example:
Partial Fractions
cm"""
$(example("Example","Partial Fractions"))
Write out the form of the partial fractions decomposition of the function
```math
\frac{x^3+x+1}{x(x-1)(x+1)^2(x^2+x+1)(x^2+4)^2}
```
"""
More Examples
Find
md"""
**More Examples**
Find
```math
\begin{array}{lll}
\text{(1)} & \displaystyle \int \frac{1}{x^2-5x+6}dx. \\
\text{(2)} &\displaystyle \int \frac{5x^2+20x+6}{x^3+2x^2+x}dx. \\
\text{(3)} &\displaystyle \int \frac{2x^3-4x-8}{(x^2-x)(x^2+4)}dx. \\
\text{(4)} &\displaystyle \int \frac{8x^3+13x}{(x^2+2)^2}dx. \\
\text{(5)} &\displaystyle \int \frac{x^3+x}{x-1}dx. \\
\text{(6)} &\displaystyle \int \frac{x^2+2x-1}{2x^3+3x^2-2x}dx. \\
\text{(7)} &\displaystyle \int \frac{dx}{x^2-a^2}, \text{ where } a\not = 0 \\
\text{(8)} &\displaystyle \int \frac{x^4-2x^2+4x+1}{x^3-x^2-x+1}dx \\
\text{(9)} &\displaystyle \int \frac{2x^2-x+4}{x^3+4x}dx \\
\text{(10)} &\displaystyle \int \frac{4x^2-3x+2}{4x^2-4x+3}dx \\
\text{(11)} &\displaystyle \int \frac{1-x+2x^2-x^3}{x(x^2+1)^2}dx \\
\end{array}
```
Remarks
md"""
**Remarks**
```math
\int \frac{dx}{x^2-a^2} = \frac{1}{2a}\ln\left|\frac{x-a}{x+a}\right|
```
```math
\int \frac{dx}{x^2+a^2} = \frac{1}{a}\tan^{-1}\left(\frac{x}{a}\right)
```
"""
Rationalizing Substitutions Find
md"""
__Rationalizing Substitutions__
Find
```math
\begin{array}{lll}
\text{(1)} & \int \frac{\sqrt{x+4}}{x}dx. \\
\text{(2)} & \int \frac{dx}{2\sqrt{x+3}+\;x}. \\
\end{array}
```
"""
8.7: Rational Functions of Sine & Cosine
Special Substitution (
md"""
# 8.7: Rational Functions of Sine & Cosine
Special Substitution (``u = \tan \left(\frac{x}{2}\right), \quad -\pi < x < \pi``)
**(for rational functions of ``\sin x`` and ``\cos x``)**
```math
\begin{array}{lll}
dx=\frac{2}{1+u^2}du, & \cos{x}=\frac{1-u^2}{1+u^2}, & \sin{x}=\frac{2u}{1+u^2} \\
\end{array}
```
```math
\begin{array}{lll}
\text{(1)} & \displaystyle\int \frac{dx}{3\sin x - 4 \cos x}. \\
\text{(2)} & \displaystyle\int_0^{\pi\over 2} \frac{\sin 2x \;dx}{2+\cos x}. \\
\end{array}
```
"""
8.8: Improper Integrals
Objectives
Evaluate an improper integral that has an infinite limit of integration.
Evaluate an improper integral that has an infinite discontinuity.
Do you know how to evaluate the following?
md"""
# 8.8: Improper Integrals
> __Objectives__
> - Evaluate an improper integral that has an infinite limit of integration.
> - Evaluate an improper integral that has an infinite discontinuity.
*__Do you know how to evaluate the following?__*
```math
\begin{array}{llr}
\text{(1)} & \int_1^{\infty} \frac{1}{x^2} dx & (\text{Type 1}) \\ \\
\text{(2)} & \int_0^{2} \frac{1}{x-1} dx & (\text{Type 2}) \\ \\
\end{array}
```
"""
Improper Integrals with Infinite Limits of Integration
md"## Improper Integrals with Infinite Limits of Integration"
Definition of an Improper Integral of Type 1
(a) If
provided this limit exists (as a finite number).
(b) If
provided this limit exists (as a finite number).
The improper integrals
(c) If both
In part (c) any real number can be used
cm"""
$(bbl("Definition of an Improper Integral of Type 1",""))
**(a)** If ``\int_a^t f(x) dx`` exists for every number ``t\ge a``, then
```math
\int_a^{\infty} f(x) dx = \lim_{t\to \infty} \int_a^t f(x) dx
```
provided this limit exists (as a finite number).
**(b)** If ``\int_t^b f(x) dx`` exists for every number ``t\le b``, then
```math
\int_{-\infty}^b f(x) dx = \lim_{t\to -\infty} \int_t^b f(x) dx
```
provided this limit exists (as a finite number).
The improper integrals ``\int_a^{\infty} f(x) dx`` and ``\int_{-\infty}^b f(x) dx`` are called *__convergent__* if the corresponding limit exists and *__divergent__* if the limit does not exist.
**(c)** If both ``\int_a^{\infty} f(x) dx`` and ``\int_{-\infty}^b f(x) dx`` are convergent, then we define
```math
\int_{-\infty}^{\infty} f(x) dx = \int_{-\infty}^a f(x) dx +\int_a^{\infty} f(x) dx
```
Example: Determine whether the following integrals are convergent or divergent.
md"""
**Example:** Determine whether the following integrals are convergent or divergent.
```math
\begin{array}{ll}
\text{(1)} & \displaystyle \int_1^{\infty} \frac{1}{x^2} dx \\ \\
\text{(2)} & \displaystyle\int_1^{\infty} \frac{1}{x} dx \\ \\
\text{(3)} & \displaystyle\int_{0}^{\infty} e^{-x} dx\\ \\
\text{(4)} & \displaystyle\int_{-\infty}^{\infty} \frac{1}{1+x^2} dx\\ \\
\end{array}
```
"""
t =
begin
md"""
-------------
t = $tSlider
"""
end
begin
pt1=L"\int_1^{%$tslider} \frac{1}{x}dx = %$int1"
pt2=L"\int_1^{%$tslider} \frac{1}{x^2}dx = %$int2"
end
| p = | |
begin
pt3(t)= t<1000 ? L"\int_1^{\infty} \frac{1}{%$ptxt}dx = %$(int3(pslider,ttslider))" : L"\int_1^{\infty} \frac{1}{%$ptxt}dx = %$(int3(pslider,ttslider))"
c=:white,ylims=(3,5),size=(600,100))
md"""
-------------
$p883
"""
end
Remark
cm"""
**Remark**
```math
\int_1^{\infty} \frac{1}{x^p}dx \quad \text{ is convergent if } p > 1 \text{ and divergent if } p\leq 1.
```
"""
Improper Integrals with Infinite Discontinuities
md"## Improper Integrals with Infinite Discontinuities"
Definition of an Improper Integral of Type 2
(a) If
provided this limit exists (as a finite number).
(b) If
provided this limit exists (as a finite number).
The improper integral
(c) If
cm"""
$(bbl("Definition of an Improper Integral of Type 2",""))
**(a)** If ``f`` is continuous on ``[a,b)`` and is discontinuous at ``b``, then
```math
\int_a^b f(x) dx = \lim_{t\to b^-} \int_a^t f(x) dx
```
provided this limit exists (as a finite number).
**(b)** If ``f`` is continuous on ``(a,b]`` and is discontinuous at ``a``, then
```math
\int_a^b f(x) dx = \lim_{t\to a^{+}} \int_t^b f(x) dx
```
provided this limit exists (as a finite number).
The improper integral ``\int_a^b f(x) dx`` is called **convergent** if the corresponding limit exists and **divergent** if the limit does not exist.
**(c)** If ``f`` has a discontinuity at ``c``, where ``a < c < b``, and both ``\int_a^c f(x) dx`` and ``\int_c^b f(x) dx`` are convergent, then we define
```math
\int_{a}^{b} f(x) dx = \int_{a}^c f(x) dx +\int_c^b f(x) dx
Example:
cm"""
**Example:**
```math
\begin{array}{ll}
\text{(1)} & \int_2^5 \frac{1}{\sqrt{x-2}} dx \\ \\
\text{(2)} & \int_0^{3} \frac{1}{1-x} dx \\ \\
\text{(3)} & \int_{0}^{1} \ln x dx\\ \\
\end{array}
```
"""
Example 9:
Doubly Improper Integral
cm"""
$(ex(9,s="Doubly Improper Integral"))
Evaluate ``\displaystyle \int_0^{\infty}\frac{dx}{\sqrt{x}(x+1)}``
"""
9.1: Sequences
Objectives
Write the terms of a sequence.
Determine whether a sequence converges or diverges.
Write a formula for the th term of a sequence.
Use properties of monotonic sequences and bounded sequences.
Sequence: A sequence can be thought of as a list of numbers written in a definite order:
For example:
Notation:
md"""
# 9.1: Sequences
> Objectives
> * Write the terms of a sequence.
> * Determine whether a sequence converges or diverges.
> * Write a formula for the th term of a sequence.
> * Use properties of monotonic sequences and bounded sequences.
__Sequence__: A sequence can be thought of as a list of numbers written in a definite order:
```math
a_1, a_2, a_3, \cdots, a_n, \cdots
```
- ``a_1``: first term,
- ``a_2``: second term,
- ``a_3``: third term,
- ``\vdots``
- ``a_n``: ``\text{n}^\text{th}`` term,
---
__For example__:
* ``1, 2, 3, \cdots``
* ``1, 1/2, 1/3, \cdots``
* ``-1, 1, -1, \cdots``
---
More examples
md"""
__More examples__
- ``\left\{\frac{n}{n+1}\right\}``
- ``\left\{\frac{(-1)^n(n+1)}{5^n}\right\}``
- ``\left\{\sqrt{n-4}\right\}_{n=4}^{\infty}``
- ``a_1=1$, $a_2=1$ , $a_n=a_{n-1}+a_{n-2}`` (__Fibonacci sequence__)
"""
n =
begin
frame_style=:origin,
ylimits=(-0.1,1),
xlimits=(0.2,1.2),
yaxis=nothing,
label=L"a_n=\frac{n}{n+1}",
showaxis=:x,
legend=:topleft,
title_location=:left,
grid=:none,
title="Example 1"
)
annotate!(plt91,[(0.4,0.5,L"a_{%$n91slider}=\frac{%$n91slider}{%$(1+n91slider)}=%$(round(a91(n91slider),digits=6))")])
yaxis=nothing,
frame_style=:origin,
showaxis=:x,
grid=:none,
annotations=[(0.998,0.3,"Zoom",7)]
)
end
md"""
$plt91
"""
Visualization
On a number line (as above)
By plotting graph
md"""
__Visualization__
1. On a number line (as above)
2. By plotting graph
"""
n =
begin
frame_style=:origin,
ylimits=(-0.1,1.6),
xlimits=(-0.1,200),
label=L"a_n=\frac{n}{n+1}",
legend=:topleft,
title_location=:left,
title="Example 1 (Graph)"
)
annotate!(plt92,[(100,0.5,L"a_{%$n92slider}=\frac{%$n92slider}{%$(1+n92slider)}=%$(round(a91(n92slider),digits=6))")])
# if (n92slider>=10)
# lens!(plt92,[n92slider-20.1, n92slider+20.1], [0.9,1.01],
# inset = (1, bbox(0.6, -0.1, 0.4, 0.4)),
# grid=:none,
# )
# end
md"""
$plt92
"""
end
n =
begin
frame_style=:origin,
ylimits=(-1,1),
xlimits=(-0.1,200),
label=L"a_n=\frac{(-1)^n\cdot(n+1)}{5^n}",
legend=:topleft,
title_location=:left,
title="Example 2 (Graph)"
)
annotate!(plt93,[(100,0.5,L"a_{%$n93slider}=\frac{%$((-1)^n93slider) \cdot %$(1+n93slider)}{5^{%$(n93slider)}}=%$(round(a93(n93slider),digits=6))")])
# if (n2slider>=10)
# lens!(plt2,[n2slider-20.1, n2slider+20.1], [0.9,1.01],
# inset = (1, bbox(0.6, -0.1, 0.4, 0.4)),
# grid=:none,
# )
# end
md"""
$plt93
"""
end
n =
begin
frame_style=:origin,
ylimits=(-1,100),
label=L"a_n=\frac{n-4}",
legend=:topleft,
title_location=:left,
title="Example 3 (Graph)"
)
[(50,50,
L"a_{%$n94slider}=\sqrt{%$(n94slider-4)}=%$(round(a94(n94slider),digits=6))")
]
)
[(250,50,
L"a_{%$n94slider}=\sqrt{%$(n94slider-4)}=%$(round(a94(n94slider),digits=6))")
]
)
else
[(500,50,
What are trying to study?
convergence (what happended when
For Example 1:
md"""
#### What are trying to study?
* __convergence__ (what happended when $n$ gets larger and larger $n\to \infty$)
For **_Example 1_**: ``a_n=\frac{n}{n+1}``, it is fair to say and write
```math
\lim_{n\to \infty}\frac{n}{n+1} =1
```
"""
| n = |
begin
frame_style=:origin,
ylimits=(-0.1,3),
xlimits=(-0.1,100),
label=L"a_n=\frac{n}{n+1}",
legend=:topleft,
title_location=:left,
title="Example 1 (Graph)"
)
[
(50,0.5,
L"a_{%$n95slider}=\frac{%$n95slider}{%$(1+n95slider)}=%$(round(a91(n95slider),digits=6))"
)
]
)
labels=:none
)
md"""
$plt95
"""
end
Example
md"""
**Example**
```math
\{(-1)^n\} = \{-1, 1, -1, 1, -1, 1, \cdots\}
```
"""
| n = | |
begin
md"""
----
|||
|---|---|
|``\epsilon`` =$eps2Slider | n = $n96Slider |
|``L =`` $limitSlider | |
----
"""
end
begin
frame_style=:origin,
ylimits=(-2,2),
xlimits=(-0.1,100),
label=L"a_n=(-1)^n",
legend=:topleft,
title_location=:left,
title="Example "
)
[
(50,
1.5,
L"a_{%$n96slider}=%$(a96(n96slider))"
)
]
)
labels=:none
)
md"""
$plt96
"""
end
Remark:
md"""
**Remark**:
```math
\lim_{n\to \infty}(-1)^n \quad \text{DNE}
```
"""
Limit of a Sequence
md"## Limit of a Sequence"
Definition of the Limit of a Sequence
Let
if for each
cm"""
$(bbl("Definition of the Limit of a Sequence",""))
Let ``L`` be a real number. The __limit__ of a sequence ``\{a_n\} is ``L``, written as
```math
\lim_{n\to\infty} a_n = L
```
if for each ``\epsilon >0``, there exists ``M>0`` such that ``|a_n-L|<\epsilon`` whenever ``n>M``. If the limit ``L`` of a sequence exists, then the sequence __converges__ to ``L``. If the limit of a sequence does not exist, then the sequence __diverges__.
$(ebl())
"""
Theorem
If
then
- Remark
cm"""
$(bth("Limit of a Sequence"))
If
```math
\lim_{x\to\infty}f(x)=L \quad \text{and}\quad f(n)=a_n \quad \text{when} \; n \text{ is an integer},
```
then
```math
\lim_{n\to\infty}a_n=L.
```
$(eth())
* **Remark**
```math
\lim_{n\to\infty} \frac{1}{n^r} = 0 \quad \text{if} \quad r>0
"""
Limit Laws for Sequences Suppose that
Sum Law
Difference Law
Constant Multiple Law
Product Law
Quotient Law
md"""
**_Limit Laws for Sequences_**
Suppose that ``\{a_n\}`` and ``\{b_n\}`` are convergent sequences and ``c`` is a constant. Then
1. **Sum Law**
```math
\lim_{n\to\infty} \left(a_n+b_n\right)=\lim_{n\to\infty} a_n+\lim_{n\to\infty}b_n
```
2. **Difference Law**
```math
\lim_{n\to\infty} \left(a_n-b_n\right)=\lim_{n\to\infty} a_n-\lim_{n\to\infty}b_n
```
3. **Constant Multiple Law**
```math
\lim_{n\to\infty} c a_n=c\lim_{n\to\infty} a_n
```
4. **Product Law**
```math
\lim_{n\to\infty} \left(a_nb_n\right)=\lim_{n\to\infty} a_n\cdot\lim_{n\to\infty}b_n
```
5. **Quotient Law**
```math
\lim_{n\to\infty} \frac{a_n}{b_n}=\frac{\lim_{n\to\infty} a_n}{\lim_{n\to\infty}b_n}, \quad \text{if } \lim_{n\to\infty}b_n\not=0
Power Law
Squeeze Theorem for Sequences
If
Theorem
If
then
md"""
**Power Law**
```math
\lim_{n\to\infty} a^p_n=\left[\lim_{n\to\infty} a_n\right]^p
```
**Squeeze Theorem for Sequences**
If ``a_n\leq b_n\leq c_n`` for ``n\geq n_0`` and ``\lim_{n\to\infty}a_n=\lim_{n\to\infty}c_n=L``, then
```math
\lim_{n\to\infty}b_n=L.
```
**_Theorem_**
If
```math
\lim_{n\to\infty}|a_n|=0,
```
then
```math
\lim_{n\to\infty}a_n=0.
```
"""
Theorem
If
md"""
**_Theorem_**
If ``\lim_{n\to\infty}a_n=L`` and the function ``f`` is continuous at ``L``, then
```math
\lim_{n\to\infty}f(a_n)=f(L).
```
"""
Remark
The sequence
md"""
**Remark**
The sequence ``\{r^n\}`` is convergent if ``-1<r\leq 1`` and divergent for all other values of ``r``.
```math
\lim_{n\to\infty}r^n=\left\{\begin{array}{lll}
0 & \text{if} & -1<r<1,\\ \\
1 & \text{if} & r=1
\end{array}
\right.
```
"""
Examples
Find
Exercise
md"""
**Examples**
Find
1. ``\lim_{n\to \infty}\left(1+\frac{1}{n}\right)^n.``
1. ``\lim_{n\to \infty}\frac{n^2}{2^n-1}.``
1. ``\lim_{n\to \infty}\frac{n}{n+1}.``
2. ``\lim_{n\to \infty}\frac{n}{\sqrt{n+1}}.``
3. ``\lim_{n\to \infty}\frac{\ln n}{n}.``
4. ``\lim_{n\to \infty}\frac{(-1)^n}{n}.``
5. ``\lim_{n\to \infty}\sin\left(\pi/n\right).``
6. ``\lim_{n\to \infty}\frac{n!}{n^n}.``
6. ``\lim_{n\to \infty}\frac{(-1)^n}{n!}.``
**Exercise**
```math
\lim_{n\to \infty}\frac{n^n}{n!}.
```
"""
Pattern Recognition for Sequences
Example
Find a sequence
and then determine whether the sequence you have chosen converges or diverges.
Example
Find a sequence
and then determine whether the sequence you have chosen converges or diverges.
md"""
## Pattern Recognition for Sequences
__Example__
Find a sequence ``\{a_n\}`` whose first five terms are
```math
\frac{2}{1},\frac{4}{3},\frac{8}{5},\frac{16}{7},\frac{32}{9},\cdots
```
and then determine whether the sequence you have chosen converges or diverges.
__Example__
Find a sequence ``\{a_n\}`` whose first five terms are
```math
-\frac{2}{1},\frac{8}{2},-\frac{26}{6},\frac{80}{24},-\frac{242}{120},\cdots
```
and then determine whether the sequence you have chosen converges or diverges.
"""
Monotonic and Bounded Sequences
Definition
A sequence
It is called decreasing if
A sequence is called monotonic if it is either increasing or decreasing.
md"""
## Monotonic and Bounded Sequences
**_Definition_**
* A sequence ``\{a_n\}`` is called **increasing** if ``a_n<a_{n+1}`` for all ``n\geq 1``, that is,``a_1<a_2<a_3<\cdots`` .
* It is called **decreasing** if ``a_n>a_{n+1}`` for all ``n\geq 1``.
* A sequence is called **monotonic** if it is either increasing or decreasing.
"""
Examples
Is the following increasing or decreasing?
md"""
**Examples**
Is the following increasing or decreasing?
1. ``\left\{\frac{3}{n+5}\right\}``.
2. ``\left\{\frac{n}{n^2+1}\right\}``.
"""
Definition
A sequence
A sequence is bounded below if there is a number
If a sequence is bounded above and below, then it is called a bounded sequence.
Monotonic Sequence Theorem
Every bounded, monotonic sequence is convergent.
In particular, a sequence that is increasing and bounded above converges, and a sequence that is decreasing and bounded below converges.
md"""
**Definition**
A sequence ``\{a_n\}`` is **bounded above** if there is a number ``M`` such that
```math
a_n\leq M\quad \text{for all } n\geq 1
```
A sequence is **bounded below** if there is a number ``m`` such that
```math
m\leq a_n\quad \text{for all } n\geq 1
```
If a sequence is bounded above and below, then it is called a **bounded sequence**.
---
**_Monotonic Sequence Theorem_**
Every bounded, monotonic sequence is convergent.
In particular, a sequence that is increasing and bounded above converges, and a sequence that is decreasing and bounded below converges.
"""
Example
md"""
**Example**
```math
a_1 =2 , \quad a_{n+1}={1\over 2}\left(a_n+6\right), \quad \text{for }n=1,2,3, \cdots
```
"""
9.2: Series and Convergence
Objectives
Understand the definition of a convergent infinite series.
Use properties of infinite geometric series.
Use the th-Term Test for Divergence of an infinite series.
md"""
# 9.2: Series and Convergence
> __Objectives__
> - Understand the definition of a convergent infinite series.
> - Use properties of infinite geometric series.
> - Use the th-Term Test for Divergence of an infinite series.
"""
# begin
# n98Slider = @bind n98slider Slider(1:1000,show_value=true)
# md"""
# ----
# ||
# |---|
# |n = $n98Slider |
# ----
# """
# end
Infinite Series
md"## Infinite Series"
Consider the sequence
is called an infinite series (or simply series) and we use the notation
To make sense of this sum, we define a related sequence called the sequence of partial sums
and give the following definition
md"""
Consider the sequence ``\left\{a_n\right\}_{n=1}^{\infty}``. The expression
```math
a_1 + a_2 + a_3 +\cdots
```
is called an __infinite series__ (or simply __series__) and we use the notation
```math
\sum_{n=1}^{\infty}a_n \qquad \text{or} \qquad \sum a_n
```
To make sense of this sum, we define a related __sequence__ called the sequence of __partial sums__ ``\left\{s_n\right\}_{n=1}^{\infty}`` as
```math
\begin{array}{lll}
s_1 & = & a_1 \\
s_2 & = & a_1 + a_2 \\
s_3 & = & a_1 + a_2 + a_3\\
\vdots \\
s_n & = & a_1 + a_2 + \cdots + a_n =\sum_{i=1}^n a_i \\
\vdots
\end{array}
```
and give the following definition
"""
Definition
Given a series
If the sequence
The number
If the sequence
Remark
md"""
**Definition**
Given a series ``\sum_{n=1}^{\infty}a_n=a_1+a_2+a_3+\cdots`` , let ``s_n``denote its ``n``th partial sum:
```math
s_n=\sum_{i=1}^{n}a_i = a_1+a_2+\cdots + a_n
```
If the sequence ``\{s_n\}`` is convergent and ``\lim_{n\to \infty}s_n=s`` exists as a real number, then the ``\sum a_n``series is called __convergent__ and we write
```math
\sum_{n=1}^{\infty}a_n=a_1+a_2+a_3+\cdots =s
```
The number ``s`` is called the __sum__ of the series.
If the sequence ``\{s_n\}`` is divergent, then the series is called __divergent__.
**Remark**
```math
\sum_{n=1}^{\infty}a_n=\lim_{n\to\infty}s_n=\lim_{n\to\infty}\sum_{i=1}^{n}a_i
```
"""
Exercise Assume that
Find
Can you find
md"""
**Exercise**
Assume that ``\left\{a_n\right\}_{n=1}^{\infty}`` is a sequence.
1. Find
```math
\sum_{n=1}^{\infty} a_n \quad \text{ if }\quad s_n = \sum_{i=1}^{n} a_i = \frac{n+2}{3n-5}
```
2. Can you find ``a_n``?
"""
Solution
We find first
since the sequence
Note that
so,
md"""
__Solution__
1. We find first
```math
\lim_{n\to \infty}s_n =\lim_{n\to \infty}\frac{n+2}{3n - 5} = \frac{1}{3}
```
since the sequence ``\{s_n\}`` converges to ``1 \over 3``, then the series converges and its sum is
```math
\sum_{n=1}^{\infty} a_n = \frac{1}{3}
```
2. Note that
```math
\begin{array}{lll}
a_n &=& s_n - s_{n-1} = \frac{n+2}{3n - 5} - \frac{(n-1)+2}{3(n-1) - 5}\\
&=&\frac{n+2}{3n - 5}- \frac{n+1}{3n - 8} \\
&=& \frac{(n+2)(3n-8)- (n+1)(3n - 5)}{(3n - 5)(3n-8)}
\end{array}
```
so,
```math
a_n = \frac{-11}{(3n - 5)(3n-8)}
```
"""
Telescoping sum
Find the sum of the following series
Solution in class
md"""
### Telescoping sum
Find the sum of the following series
```math
\sum_{n=1}^{\infty} \frac{1}{n(n+1)}
```
__Solution in class__
---
"""
Recall
So
md"""
**Recall**
```math
\lim_{n\to \infty} r^n =\left\{\begin{array}{lll}
0 & \text{if} & |r|<1 (-1<r<1),\\
1 & \text{if} & r=1 ,\\
\end{array}\right.
```
So ``\{r^n\}`` converges if ``r\in (-1,1]`` and diverges otherwise
"""
Geometric Series
The series
is called the geometric series with common ration
It is convergent if
and divergent if
Remark In words: the sum of a convergent geometric series is
md"""
## Geometric Series
The series
```math
a + a r + a r^2 + \cdots =\sum_{n=1}^{\infty}ar^{n-1}, \qquad a\not = 0
```
is called the __geometric series__ with __common ration__ ``r``
It is convergent if ``|r|< 1`` and its sum is
```math
\sum_{n=1}^{\infty}ar^{n-1} =\frac{a}{1-r}, \qquad |r|<1
```
and divergent if ``|r|\geq 1``.
**Remark**
In words: the sum of a convergent geometric series is
```math
\frac{\text{first term}}{1-\text{common ratio}}
```
"""
Examples
Find the sum of the geomtric series
Is the series
Write
Find the sum of the series
md"""
**_Examples_**
1. Find the sum of the geomtric series
```math
4 - 3 + {9\over 4} - {27 \over 16} + \cdots
```
2. Is the series
```math
\sum_{n=1}^{\infty} 2^{2n}\; 3^{1-n} \quad \text{convergent or divergent?}
```
3. Write ``2.\bar{7}`` as rational number (ratio of integers).
4. Find the sum of the series
```math
\sum_{n=0}^{\infty} x^n \quad \text{where}\quad |x|<1.
```
"""
Test for Divergence
Example Show that the harmonic series
is divergent.
md"""
## Test for Divergence
**Example**
Show that the harmonic series
```math
\sum_{n=1}^{\infty}\frac{1}{n} = 1 + \frac{1}{2}+ \frac{1}{3}+ \frac{1}{4}+\cdots
```
is divergent.
"""
Theorem If the series
converges, then
Proof
Divergence Test
Example
md"""
**_Theorem_**
If the series
```math
\sum_{n=1}^{\infty}a_n
```
converges, then
```math
\lim_{n\to \infty}a_n = 0.
```
__Proof__
```math
a_n = s_n - s_{n-1}
```
---
**_Divergence Test_**
```math
\text{If }\lim_{n\to \infty}a_n \not= 0 \quad \text{or } \lim_{n\to \infty}a_n \text{ DNE}\quad \text{ then the series }\quad
\sum_{n=1}^{\infty}a_n \text{ is divergenet}
```
---
**_Example_**
```math
\text{The series }\quad \sum_{n=1}^{\infty}\frac{n^2+1}{2n^2+5} \quad \text{is divergent.}
```
---
"""
Properties of Convergent Series
Theorem If
md"""
### Properties of Convergent Series
**_Theorem_**
If ``\sum a_n`` and ``\sum b_n`` are convergent series, then so are the series ``\sum ca_n`` (where ``c`` is a constant), ``\sum (a_n+b_n)``, and ``\sum (a_n-b_n)``, and
```math
\begin{array}{llcl}
\text{(i)} & \sum_{n=1}^{\infty} c a_n &=& c\sum_{n=1}^{\infty} a_n \\ \\
\text{(ii)} & \sum_{n=1}^{\infty} \left(a_n+b_n\right) &=& \sum_{n=1}^{\infty} a_n+\sum_{n=1}^{\infty} b_n \\ \\
\text{(iii)} & \sum_{n=1}^{\infty} \left(a_n-b_n\right) &=& \sum_{n=1}^{\infty} a_n-\sum_{n=1}^{\infty} b_n \\ \\ \end{array}
```
"""
Remark
If it can be shown that
is convergent. Then
is convergent.
md"""
---
**_Remark_**
If it can be shown that
```math
\sum_{n=100}^{\infty}a_n
```
is convergent. Then
```math
\sum_{n=1}^{\infty}a_n
```
is convergent.
"""
9.3: The Integral Test and
Objectives
Use the Integral Test to determine whether an infinite series converges or diverges.
Use properties of -series and harmonic series.
The Integral Test and Estimates of Sums
Suppose
continuous on
positive on
decreasing on
and let
is convergent if and only if the improper integral
is convergent. In other words:
If
If
Examples
Test for convergence
Solution in class
md"""
# 9.3: The Integral Test and $p$-Series
> __Objectives__
> - Use the Integral Test to determine whether an infinite series converges or diverges.
> - Use properties of -series and harmonic series.
**The Integral Test and Estimates of Sums**
Suppose ``f`` a function that is
1. continuous on ``[1, \infty)``,
2. positive on ``[1, \infty)``,
3. decreasing on ``[1, \infty)``
and let ``a_n=f(n)``. Then the series
```math
\sum_{n=1}^{\infty}a_n
```
is convergent if and only if the improper integral
```math
\int_1^\infty f(x) dx
```
is convergent. In other words:
1. If ``\displaystyle\int_1^\infty f(x) dx`` is convergent, then is ``\displaystyle\sum_{n=1}^{\infty}a_n`` convergent.
2. If ``\displaystyle\int_1^\infty f(x) dx`` is divergent, then is ``\displaystyle\sum_{n=1}^{\infty}a_n`` divergent.
Remark
md"""
**_Remark_**
```math
\text{The series } \sum_{n=1}^{\infty}\frac{1}{n^2} \text{ is convergent but }
\sum_{n=1}^{\infty}\frac{1}{n^2}\not=1.
```
```math
\text{It sum is actually equal to }
\sum_{n=1}^{\infty}\frac{1}{n^2}=\frac{\pi^2}{6}
```
"""
P-series and the Harmonic Series
Example
Show that
is divergent.
md"""
## P-series and the Harmonic Series
```math
\text{The } p-\text{series}\quad \sum_{n=1}^{\infty}\frac{1}{n^p}
\quad \text{is convergent if } p>1 \text{ and is divergent if } p\leq 1.
```
---
1. ``\sum_{n=1}^{\infty}\frac{1}{n^{1\over 3}}`` is divergent; because it is a ``p-``series with ``p={1\over 3}<1``.
2. ``\sum_{n=1}^{\infty}\frac{1}{n^{3}}`` is convergent; because it is a ``p-``series with ``p={3}>1``.
**_Example_**
Show that
```math
\sum_{n=1}^{\infty}\frac{\ln n}{n}
```
is divergent.
"""
Estimating the Sum of a Series
Suppose that the integral test is used to show that
is convergent. So its sequenc of partial sums
So we can write
md"""
## Estimating the Sum of a Series
Suppose that the __integral test__ is used to show that
```math
\sum_{n=1}^{\infty}a_n
```
is __convergent__. So its sequenc of partial sums ``\left\{s_n=\sum_{i=1}^n a_i\right\}`` is convergent; that is
```math
\lim_{n\to\infty}s_n=s.
```
So we can write
```math
\underset{s}{\underbrace{\sum_{n=1}^{\infty}a_n}} =
\underset{s_n}{\underbrace{\sum_{i=1}^{n}a_i}} + \underset{R_n}{\underbrace{\sum_{i=n+1}^{\infty}a_i}}
```
``R_n`` is the __Remainder__ or the error when ``s_n`` is used to approximate ``s``.
"""
begin
estR=0:0.1:10
estPltstr=(
label=:none,
annotations=[(1,4,L"y=f(x)")],
xlims=(-1,10),
aspect_ratio=1,
frame_style=:origin,
xticks=(1:10,[L"n",[L"n+{%$i}" for i in 1:10]...])
)
# palette([:purple, :green], 7)[1]
md"""
$estP1
```math
0.005
0.00413223
1/200, 1/242
Remainder Estimate for the Integral Test Suppose
md"""
**_Remainder Estimate for the Integral Test_**
Suppose ``f(k)=a_k``, where ``f`` is a continuous, positive, decreasing function for ``x\geq n`` and ``\sum a_n`` is convergent. If ``R_n=s-s_n``, then
```math
\int_{n+1}^{\infty} f(x) dx \leq R_n \leq \int_n^{\infty} f(x) dx
```
"""
9.4: Comparisons of Series
Objectives
Use the Direct Comparison Test to determine whether a series converges or diverges.
Use the Limit Comparison Test to determine whether a series converges or diverges.
The Comparison Tests
The Direct Comparison Test
Suppose that
If
If
Remarks
Most of the time we use one of these series:
geometric series.
md"""
# 9.4: Comparisons of Series
> __Objectives__
> - Use the Direct Comparison Test to determine whether a series converges or diverges.
> - Use the Limit Comparison Test to determine whether a series converges or diverges.
**The Comparison Tests**
### The Direct Comparison Test
Suppose that ``\sum a_n`` and ``\sum b_n`` are series with positive terms.
* If ``\sum b_n`` is convergent and ``a_n\leq b_n`` for all ``n``, then ``\sum a_n`` is also convergent.
* If ``\sum b_n`` is divergent and ``a_n\geq b_n`` for all ``n`` , then is ``\sum a_n`` also divergent
---
**_Remarks_**
* Most of the time we use one of these series:
- ``p-``series ``\sum \frac{1}{n^p}``
- geometric series.
"""
Examples Test for convegence
md"""
**_Examples_**
Test for convegence
```math
\begin{array}{lrrr}
\text{(1)} & \sum_{n=1}^{\infty}\frac{5}{2n^2+4n+3}\\ \\
\text{(2)} & \sum_{n=1}^{\infty}\frac{\ln n}{n}\\
\end{array}
```
"""
The Limit Comparison Test
Suppose
where
Remark
Exercises 40 and 41 deal with the cases
md"""
## The Limit Comparison Test
Suppose ``\sum a_n`` that and ``\sum b_n`` are series with positive terms. If
```math
\lim_{n\to\infty}\frac{a_n}{b_n} = c
```
where ``c`` is a finite number and ``c>0``, then either both series converge or both diverge.
**_Remark_**
Exercises `40` and `41` deal with the cases ``c=0`` and ``c=\infty`` .
"""
Examples Test for convegence
md"""
**_Examples_**
Test for convegence
```math
\begin{array}{llrr}
\text{(3)} & \sum_{n=1}^{\infty}\frac{1}{2^n-1}\\ \\
\text{(4)} & \sum_{n=1}^{\infty}\frac{2n^2+3n}{\sqrt{5+n^5}}\\
\end{array}
```
"""
begin
function poolcode()
cm"""
<div class="img-container">
$(Resource("https://www.dropbox.com/s/cat9ots4ausfzyc/qrcode_itempool.com_kfupm.png?raw=1",:width=>300))
</div>"""
end
function bbl(t,s)
end
function bth(s)
end
function beginBlock(title,subtitle)
"""<div style="box-sizing: border-box;">
<div style="display: flex;flex-direction: column;border: 6px solid rgba(200,200,200,0.5);box-sizing: border-box;">
<div style="display: flex;">
<div style="background-color: #FF9733;
border-left: 10px solid #df7300;
padding: 5px 10px;
color: #fff!important;
clear: left;
Exercises
Test for convegence
md"""
**_Exercises_**
Test for convegence
```math
\begin{array}{llrr}
\text{(5)} & \sum_{n=1}^{\infty}\frac{n+3^n}{n+2^n}\\ \\
\text{(6)} & \sum_{n=1}^{\infty}\frac{1}{n^{1+{1\over n}}}\\
\end{array}
```
"""
9.5: Alternating Series
objectives
Use the Alternating Series Test to determine whether an infinite series converges.
Use the Alternating Series Remainder to approximate the sum of an alternating series.
Classify a convergent series as absolutely or conditionally convergent.
Rearrange an infinite series to obtain a different sum.
An alternating series is a series whose terms are alternately positive and negative. For examples:
Alternating Series Test
If the alternating series
satisfies the conditions
then the series is convergent.
md"""
# 9.5: Alternating Series
> **objectives**
> - Use the Alternating Series Test to determine whether an infinite series converges.
> - Use the Alternating Series Remainder to approximate the sum of an alternating series.
> - Classify a convergent series as absolutely or conditionally convergent.
> - Rearrange an infinite series to obtain a different sum.
An __alternating series__ is a series whose terms are alternately __positive__ and __negative__. For examples:
```math
1-\frac{1}{2}+\frac{1}{3}-\frac{1}{4}+\frac{1}{5}-\frac{1}{6}+\cdots = \sum_{n=1}^{\infty}(-1)^{(n-1)}\frac{1}{n}
```
```math
-\frac{1}{2}+\frac{2}{3}-\frac{3}{4}+\frac{4}{5}-\frac{5}{6}+\frac{6}{7}-\cdots = \sum_{n=1}^{\infty}(-1)^{n}\frac{n}{n+1}
```
```math
\text{alternating series}\quad \sum_{n=1}^{\infty}a_n = \sum_{n=1}^{\infty}(-1)^{n-1}b_n
```
**_Alternating Series Test_**
n =
begin
frame_style=:origin,
ylimits=(-0.1,1),
xlimits=(-0.36,0.55),
yaxis=nothing,
label=:none,
showaxis=:x,
ticks=[],
legend=:topleft,
title_location=:left,
grid=:none,
title=L"\textrm{Proof}",
annotations=[(0.2,0.8,L"s_{%$n9slider}=%$bs92")]
)
md"""
$plt9
"""
end
Example Test for convegrnce
md"""
**_Example_**
Test for convegrnce
```math
\begin{array}{cl}
\text{(1)} & \sum_{n=1}^\infty \frac{(-1)^{n-1}}{n} \\ \\
\text{(2)} & \sum_{n=1}^\infty (-1)^{n}\frac{3n}{4n-1}\\ \\
\text{(3)} & \sum_{n=1}^\infty (-1)^{n+1}\frac{n^2}{n^3+1}
\end{array}
```
"""
Estimating Sums of Alternating Series
If
then
md"""
### Estimating Sums of Alternating Series
If ``s=\sum (-1)^{n-1}b_n``, where ``b_n>0``, is the sum of an alternating series that satisfies
```math
\begin{array}{cl}
\text{(i)} & b_{n+1}\leq b_n \quad \text{and} \\ \\
\text{(ii)} & \lim_{n\to\infty}b_n =0
\end{array}
```
then
```math
\left|R_n\right|=\left|s-s_n\right|\leq b_{n+1}
```
"""
Example How many terms of the series
do we need to add in order to find the sum accurate with
md"""
**_Example_**
How many terms of the series
```math
\sum_{n=1}^{\infty}{\frac{(-1)^{n+1}}{{n^6}}}
```
do we need to add in order to find the sum accurate with $|error|< 0.000001$?
"""
# n10Slider = @bind n10slider NumberField(1:20);md"n = $n10Slider"
Absolute Convergence and Conditional Convergence
A series
A series
Theorem
If a series
md"""
### Absolute Convergence and Conditional Convergence
* A series ``\sum a_n`` is called __absolutely convergent__ if the series of absolute values ``\sum |a_n|`` is convergent.
* A series ``\sum a_n`` is called __conditionally convergent__ if it is convergent but not absolutely convergent; that is, ``\sum a_n`` if converges but ``\sum |a_n|`` diverges.
---
**_Theorem_**
If a series ``\sum a_n`` is absolutely convergent, then it is convergent.
"""
Examples Determine whether the series is absolutely convergent, conditionally convergent, or divergent
md"""
**_Examples_**
Determine whether the series is absolutely convergent, conditionally convergent, or divergent
```math
\begin{array}{lcl}
\text{(i)} & &\sum^{\infty}_{n=1}\frac{(-1)^n}{n} \\ \\
\text{(ii)} & &\sum^{\infty}_{n=1}\frac{(-1)^n}{n^2} \\ \\
\text{(iii)} & &\sum^{\infty}_{n=1}\frac{(-1)^n}{\sqrt[3]{n}} \\ \\
\text{(v)} & &\sum^{\infty}_{n=1}(-1)^n\frac{n}{2n+1} \\ \\
\end{array}
```
"""
9.6: The Ratio and Root Tests
Objectives
Use the Ratio Test to determine whether a series converges or diverges.
Use the Root Test to determine whether a series converges or diverges.
Review the tests for convergence and divergence of an infinite series.
The Ratio Test
md"""
# 9.6: The Ratio and Root Tests
> __Objectives__
> - Use the Ratio Test to determine whether a series converges or diverges.
> - Use the Root Test to determine whether a series converges or diverges.
> - Review the tests for convergence and divergence of an infinite series.
## The Ratio Test
```math
\begin{array}{ll}
\text{(i)} &\text{If } \lim_{n\to\infty}\left|\frac{a_{n+1}}{a_n}\right|=L<1,\text{ then the series is absolutely convergent} \\
& \text{(and therefore convergent).} \\ \\
\text{(ii)} &\text{If } \lim_{n\to\infty}\left|\frac{a_{n+1}}{a_n}\right|=L>1 \text{ or }\lim_{n\to\infty}\left|\frac{a_{n+1}}{a_n}\right|=\infty,\\
& \text{ then the series is divergent} \\ \\
\text{(iii)} & If \lim_{n\to\infty}\left|\frac{a_{n+1}}{a_n}\right|=1,
\text{ the Ratio Test is inconclusive;} \\
& \text{that is, no conclusion can be drawn about}\\
& \text{the convergence or divergence of} \sum a_n.
The Root Test
md"""
## The Root Test
```math
\begin{array}{ll}
\text{(i)} &\text{If } \lim_{n\to\infty}\sqrt[n]{\left|a_n\right|}=L<1,\text{ then the series is absolutely convergent} \\
& \text{(and therefore convergent).} \\ \\
\text{(ii)} &\text{If } \lim_{n\to\infty}\sqrt[n]{\left|a_n\right|}=L>1 \text{ or }\lim_{n\to\infty}\sqrt[n]{\left|a_n\right|}=\infty,\\
& \text{ then the series is divergent} \\ \\
\text{(iii)} & If \lim_{n\to\infty}\sqrt[n]{\left|a_n\right|}=1,
\text{ the Ratio Test is inconclusive}.
\end{array}
```
"""
Examples Test for convergence
md"""
**_Examples_**
Test for convergence
```math
\begin{array}{lrl}
\text{(1)} & \text{ } & \sum_{n=1}^\infty \frac{\cos n}{n^3}\\ \\
\text{(2)} & \text{ } & \sum_{n=1}^\infty (-1)^n\frac{n^3}{3^n}\\ \\
\text{(3)} & \text{ } & \sum_{n=1}^\infty \left(\frac{2n+5}{5n+2}\right)^n \\
\end{array}
```
"""
-1.0
1.0
-1.0
1.0
-1.0
1.0
-1.0
1.0
-1.0
1.0
9.7: Taylor Polynomials and Approximations
Objectives
Find polynomial approximations of elementary functions and compare them with the elementary functions.
Find Taylor and Maclaurin polynomial approximations of elementary functions.
Use the remainder of a Taylor polynomial.
md"""
# 9.7: Taylor Polynomials and Approximations
> __Objectives__
> - Find polynomial approximations of elementary functions and compare them with the elementary functions.
> - Find Taylor and Maclaurin polynomial approximations of elementary functions.
> - Use the remainder of a Taylor polynomial.
"""
begin
p962 = md"""
For the function ``f(x)=e^x``, find a first-degree polynomial function ``P_1(x)=a_0+a_1x`` whose value and slope agree with the value and slope of at ``x=0``.
"""
cm"""
## Polynomial Approximations of Elementary Functions
$p962
"""
end
n =
Taylor and Maclaurin Polynomials
md"""
## Taylor and Maclaurin Polynomials
"""
Definitions
If
is called the
is also called
cm"""
$(bbl("Definitions","of th Taylor Polynomial and th Maclaurin Polynomial"))
If ``f`` has ``n`` derivatives at ``c``, then the polynomial
```math
P_n(x)=f(c)+f'(c)(x-c)+\frac{f''(c)}{2!}(x-c)^2
+\cdots +\frac{f^{(n)}(c)}{n!}(x-c)^n
```
is called the ``n``th __Taylor polynomial__ for ``f`` at ``c``. If ``c``, then
```math
P_n(x)=f(0)+f'(0)x+\frac{f''(0)}{2!}x^2
+\cdots +\frac{f^{(n)}(0)}{n!}x^n
```
is also called ``n``th th __Maclaurin polynomial__ for ``f``.
$(ebl())
"""
Example 3:
A Maclaurin Polynomial for e^x
Example 4:
Finding Taylor Polynomials for lnx
centered at
Example 5:
Finding Maclaurin Polynomials for cosx
Example 6:
Finding Taylor Polynomials for sin x
centered at
Example 7:
Approximation Using Maclaurin Polynomials
cm"""
$(ex(3,s="A Maclaurin Polynomial for e^x"))
$(ex(4,s="Finding Taylor Polynomials for lnx"))
Find the Taylor polynomials ``P_0, P_1, P_2, P_3``, and ``P_4`` for
```math
f(x)=\ln x
```
centered at ``c=1``.
$(ex(5,s="Finding Maclaurin Polynomials for cosx"))
Find the Taylor polynomials ``P_0, P_2, P_4``, and ``P_6`` to approximate ``\cos(0.1)``.
$(ex(6,s="Finding Taylor Polynomials for sin x"))
Find the Taylor polynomial ``P_3``for
```math
f(x)=\sin x
```
centered at ``c=\pi/6``.
$(ex(7,s="Approximation Using Maclaurin Polynomials"))
Use a fourth Maclaurin polynomial to approximate the value of ``\ln(1.1)``.
"""
9.8: Power Series
Objectives -Understand the definition of a power series.
Find the radius and interval of convergence of a power series.
Determine the endpoint convergence of a power series.
Differentiate and integrate a power series.
A series of the form
is called a power series in
We are interested in finding the values of
md"""
# 9.8: Power Series
> __Objectives__
> -Understand the definition of a power series.
> - Find the radius and interval of convergence of a power series.
> - Determine the endpoint convergence of a power series.
> - Differentiate and integrate a power series.
A series of the form
```math
\sum_{n=0}^{\infty} a_n(x-c)^n = a_0 + a_1(x-c) +a_2(x-c)^2 +a_3(x-c)^3 + \cdots
```
is called a __power series in ``(x-c)``__ or a __power series centered at ``c``__ or a __power series about ``c``__
We are interested in __finding the values of ``x`` for which this series is convergent.__
"""
Radius and Interval of Convergence
md"## Radius and Interval of Convergence"
Theorem For a power series
(i) The series converges only when
(ii)The series converges for all
(iii) There is a positive number
Remarks
The number
The radius of convergence is
The interval of convergence of a power series is the interval that consists of all values of for which the series converges.
In case (i) the interval consists of just a single point
In case (ii) the interval is
md"""
----
__*Theorem*__
For a power series ``\sum_{n=0}^{\infty}a_n(x-c)^n``, there are only three possibilities:
(i) The series converges only when ``x=c``.
(ii)The series converges for all ``x``.
(iii) There is a positive number ``R`` such that the series converges if
``|x-c|<R`` and diverges if ``|x-c|>R``.
__Remarks__
- The number ``R`` is called the __radius of convergence__ of the power series.
- The radius of convergence is ``R=0`` in case (i)
- ``R=\infty`` in case (ii).
- The __interval of convergence__ of a power series is the interval that consists of all values of for which the series converges.
- In case (i) the interval consists of just a single point ``a``.
- In case (ii) the interval is ``(-\infty,\infty)``.
"""
Examples:
cm"""
$(example("Examples",""))
```math
\begin{array}{l@{\,\,\,\,}l}
(1) & \displaystyle\sum_{n=0}^\infty n! x^n\\ \\
(2) & \displaystyle\sum_{n=1}^\infty \frac{(x-3)^n}{n}\\ \\
(3) & \displaystyle\sum_{n=0}^\infty \frac{(-1)^nx^{2n}}{(2n)!}
\end{array}
```
"""
Endpoint Convergence
md"""
## Endpoint Convergence
"""
Examples
Find the radius of convergence and interval of convergence of the series
md"""
#### Examples
Find the radius of convergence and interval of convergence of the series
```math
\begin{array}{l@{\,\,\,\,}l}
(4) & \sum_{n=0}^\infty \frac{(-1)^nx^n}{\sqrt{n+1}}\\
(5) & \sum_{n=0}^\infty \frac{n(x+2)^n}{3^{n+1}}\\
\end{array}
```
"""
Differentiation and Integration of Power Series
(term-by-term differentiation and integration)
Theorem
If the power series
is differentiable (and therefore continuous) on the interval
The radii of convergence of the power series in Equations (i) and (ii) are both
md"""
## Differentiation and Integration of Power Series
**(term-by-term differentiation and integration)**
__*Theorem*__
If the power series ``\sum a_n(x-c)^n`` has radius of convergence ``R>0``, then the function ``f`` defined by
```math
f(x) = a_0+a_1(x-c)+a_2(x-c)^2+\cdots = \sum_{n=0}^{\infty}a_n(x-c)^n
```
is differentiable (and therefore continuous) on the interval ``(a-R,a+R)`` and
```math
\begin{array}{lllll}
\text{(i)} &\text{}& f'(x) &=&
a_1+2a_2(x-c)+3a_2(x-c)^2+\cdots \\
&&&=& \sum_{n=1}^{\infty}na_n(x-c)^{n-1} \\ \\
\text{(ii)} &\text{}& \int f(x) dx &=&
C + a_0(x-c)+a_1\frac{(x-c)^2}{2}+a_2\frac{(x-c)^3}{3}+\cdots \\
&&&=&C+\sum_{n=0}^{\infty}a_n\frac{(x-c)^{n+1}}{n+1} \\ \\
\end{array}
```
Example 8:
Intervals of Convergence
Find the interval of convergence for each of the following.
cm"""
$(ex(8,s="Intervals of Convergence"))
Consider the function
```math
f(x)= \sum_{n=1}^{\infty}\frac{x^n}{n}
```
Find the interval of convergence for each of the following.
1. ``\displaystyle f(x)``
1. ``\displaystyle f'(x)``
1. ``\displaystyle \int f(x) dx``
"""
9.9: Representation of Functions by Power Series
Objectives
Find a geometric power series that represents a function.
Construct a power series using series operations.
Geometric Power Series
md"""
# 9.9: Representation of Functions by Power Series
> **Objectives**
> - Find a geometric power series that represents a function.
> - Construct a power series using series operations.
## Geometric Power Series
```math
\frac{1}{1-x} = 1 + x + x^2 +x^3 +\cdots =\sum_{n=0}^{\infty}x^n, \quad |x|<1.
```
"""
Examples
Express as the sum of a power series and find the interval of convergence.
Find a power series representation for
Find a power series representation for
Find a power series representation around
SOLUTION IN CLASS
md"""
#### Examples
1. Express as the sum of a power series and find the interval of convergence.
```math
f(x)=\frac{1}{1+x^2}
```
2. Find a power series representation for
```math
f(x)=\frac{1}{x+2}
```
3. Find a power series representation for
```math
f(x)=\frac{x^3}{x+2}
```
4. Find a power series representation around ``1`` for
```math
f(x)=\frac{1}{x}
```
__SOLUTION IN CLASS__
"""
Operations with Power Series
Let
cm"""
$(bbl("Operations with Power Series",""))
Let ``f(x)=\displaystyle \sum_{n=0}^{\infty}a_n x^n`` and ``g(x)=\displaystyle \sum_{n=0}^{\infty}b_n x^n``.
1. ``f(kx) = \displaystyle \sum_{n=0}^{\infty}a_n k^nx^n``.
1. ``f(x^N) = \displaystyle \sum_{n=0}^{\infty}a_n x^{Nn}``.
1. ``f(x)\pm g(x) = \displaystyle \sum_{n=0}^{\infty}(a_n\pm b_n) x^n``.
$(ebl())
"""
Examples
Express as a power series
Express as a power series
Express as a power series
Express as a power series
Evaluate
Approximate
SOLUTION IN CLASS
md"""
#### Examples
4. Express as a power series
```math
f(x) = \frac{3x-1}{x^2-1}
```
4. Express as a power series
```math
f(x) = \frac{1}{(1-x)^2}
```
5. Express as a power series
```math
f(x) = \ln(1+x)
```
6. Express as a power series
```math
f(x) = \tan^{-1}x
```
7. Evaluate
```math
\int \frac{dx}{1+x^7}
```
8. Approximate ``\pi``
```math
4\tan^{-1}\frac{1}{5}-\tan^{-1}\frac{1}{239} = \frac{\pi}{4} \quad \text{(see ex. 44)}
```
__SOLUTION IN CLASS__
"""
9.10: Taylor and Maclaurin Series [^⭐]
[^⭐]: Students have to memorize the power series representations of the functions
Objectives
Find a Taylor or Maclaurin series for a function.
Find a binomial series.
Use a basic list of Taylor series to find other Taylor series.
By the end of this section we will be able to write the following power series representations of certain functions
md"""# 9.10: Taylor and Maclaurin Series [^⭐]
[^⭐]: Students have to memorize the power series representations of the functions
``f(x) =\frac{1}{1+x}, e^x, \sin x , \cos x , \arctan x , (1 + 𝑥)^𝑘`` in page 674.
> **Objectives**
> - Find a Taylor or Maclaurin series for a function.
> - Find a binomial series.
> - Use a basic list of Taylor series to find other Taylor series.
* By the end of this section we will be able to write the following power series representations of certain functions
```math
{\small
\begin{array}{llcllllr}
{\small \text{(1)}} & \frac{1}{1-x} &=&
\sum_{n=0}^{\infty} x^n &=&
1+x+x^2+x^3+\cdots,&R=1 \\
{\small \text{(2)}} & \ln(1+x) &=&
\sum_{n=0}^{\infty} (-1)^{n} \frac{x^{n+1}}{n+1} &=&
x-\frac{x^2}{2}+\frac{x^3}{3}-\frac{x^4}{4}+\cdots,&R=1 \\
{\small \text{(3)}} & \tan^{-1}x &=&
\sum_{n=0}^{\infty} (-1)^n \frac{x^{2n+1}}{2n+1} &=&
Theorem
If
then its coefficients are given by the formula
Remarks
- The series is called the Taylor series of the function
- (Maclaurin Series) If
cm"""
$(bth("Taylor Theorem"))
If ``f`` has a power series representation (expansion) at ``a`` , that is, if
```math
f(x) = \sum_{n=0}^{\infty}c_n (x-a)^n, \quad |x-c| < R
```
then its coefficients are given by the formula
```math
c_n = \frac{f^{(n)}(a)}{n!}
```
**_Remarks_**
- The series is called the __Taylor series of the function ``f`` at ``a`` (or about ``a`` or centered at ``a``)__.
- (__Maclaurin Series__) If ``a=0``, Taylor series becomes
```math
{\small
f(x) = \sum_{n=0}^{\infty}\frac{f^{(n)}(0)}{n!} x^n=f(0)+\frac{f'(0)}{1!}x
+\frac{f''(0)}{2!}x^2+\frac{f'''(0)}{3!}x^3+\cdots
}
```
$(eth())
"""
Examples (important)
Find Maclaurin series for
Find Taylor Series of
md"""
#### Examples (important)
- Find __*Maclaurin*__ series for
```math
\begin{array}{lll}
{\small \text{(1)}} & f(x)=e^x \\
{\small \text{(2)}} & f(x)=\sin x \\
{\small \text{(3)}} & f(x)=\cos x \\
\end{array}
```
- Find Taylor Series of ``f(x)=\sin x\quad `` about ``\quad {\pi \over 3}``.
"""
The Binomial Series
Example: Find the Maclaurin series for
Solution: In Class
md"""
### The Binomial Series
**Example**:
Find the Maclaurin series for ``f(x)=(1+x)^k``, where ``k`` is any real number.
__*Solution: In Class*__
"""
The Binomial Series (Theorem)
If
where
Remarks
This is called binomial coefficients. Note that
If
If
md"""
__*The Binomial Series (Theorem)*__
If ``k`` is any real number and ``|x|<1``, then
```math
(1+x)^k=\sum_{n=0}^{\infty}\begin{pmatrix}
k \\ n
\end{pmatrix}x^n = 1 + k x + \frac{k(k-1)}{2!}x^2 +
\frac{k(k-1)(k-2)}{3!}x^3 +\cdots
```
where
```math
\left(\begin{array}{c} k \\ n\end{array}\right) = \frac{k(k-1)(k-2)\cdots (k-n+1)}{n!}
```
__*Remarks*__
* This is called __binomial coefficients__. Note that
```math
\left(\begin{array}{c} k \\ n\end{array}\right) =0 \qquad \text{if} \quad k \text{ is integer and } k<n
```
```math
\left(\begin{array}{c} k \\ 0\end{array}\right) =1, \quad \left(\begin{array}{c} k \\ 1\end{array}\right) =k
```
* If ``-1<k\leq 0``, it converges at ``x=1``.
Example
Find the Maclaurin series for the function
and its radius of convergence.
md"""
#### Example
Find the Maclaurin series for the function
```math
f(x)=\frac{1}{\sqrt{4-x}}
```
and its radius of convergence.
"""
Deriving Taylor Series from a Basic List
Check the table
Examples
Find the Maclaurin series for
Find the function represented by the power series
Find the sum of the series
md"""
## Deriving Taylor Series from a Basic List
__Check the table__
#### Examples
- Find the Maclaurin series for
```math
\begin{array}{lll}
{\small \text{(a)}} & f(x)=x\cos x \\
{\small \text{(b)}} & f(x)=\ln (1+3x^2)\\
\end{array}
```
- Find the function represented by the power series
```math
\sum_{n=0}^{\infty}(-1)^n\frac{2^nx^n}{n!}
```
- Find the sum of the series
```math
\frac{1}{1\cdot 2}-\frac{1}{2\cdot 2^2}+\frac{1}{3\cdot 2^3}-
\frac{1}{4\cdot 2^4}+ \cdots
```
"""
More Examples
Evaluate
Evaluate
Find the first 3 nonzero terms of Maclaurin series for
Find the sum of
md"""
#### More Examples
- Evaluate
```math
\int e^{-x^2} dx
```
- Evaluate
```math
\lim_{x\to 0}\frac{e^x-1-x}{x^2}
```
- Find the first 3 nonzero terms of Maclaurin series for
```math
(a) \quad e^x\sin x \qquad (b)\quad \tan x
```
- Find the sum of
```math
(a) \quad \sum_{n=0}^\infty \frac{x^{4n}}{n!}\qquad (b)\quad \sum_{n=0}^\infty (-1)^n\frac{\pi^{2n+1}}{4^{2n+1}(2n+1)!}
```
"""
begin
using FileIO
using SymPy
using PlutoUI
using CommonMark
using Plots
using HypertextLiteral
: @htl, @htl_str
using Colors
using Random
end