\( \newcommand{\abs}[1]{\left| \, {#1} \, \right| } \) \( \newcommand{\cm}{\mathrm{cm} } \) \( \newcommand{\sec}{ \, \mathrm{sec} \, } \) \( \newcommand{\vhat}[1]{\,\hat{#1}} \) \( \newcommand{\vhati}{\,\hat{i}} \) \( \newcommand{\vhatj}{\,\hat{j}} \) \( \newcommand{\vhatk}{\,\hat{k}} \) \( \newcommand{\vect}[1]{\boldsymbol{\vec{#1}}} \) \( \newcommand{\norm}[1]{\|{#1}\|} \) \( \newcommand{\arccot}{ \, \mathrm{arccot} \, } \) \( \newcommand{\arcsec}{ \, \mathrm{arcsec} \, } \) \( \newcommand{\arccsc}{ \, \mathrm{arccsc} \, } \) \( \newcommand{\sech}{ \, \mathrm{sech} \, } \) \( \newcommand{\csch}{ \, \mathrm{csch} \, } \) \( \newcommand{\arcsinh}{ \, \mathrm{arcsinh} \, } \) \( \newcommand{\arccosh}{ \, \mathrm{arccosh} \, } \) \( \newcommand{\arctanh}{ \, \mathrm{arctanh} \, } \) \( \newcommand{\arccoth}{ \, \mathrm{arccoth} \, } \) \( \newcommand{\arcsech}{ \, \mathrm{arcsech} \, } \) \( \newcommand{\arccsch}{ \, \mathrm{arccsch} \, } \)

17Calculus - Calculus 2 - 100 Practice Problems

Limits

Using Limits

Limits FAQs

Derivatives

Graphing

Related Rates

Optimization

Other Applications

Integrals

Improper Integrals

Trig Integrals

Length-Area-Volume

Applications - Tools

Infinite Series

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Parametrics

Conics

Polar Coordinates

Practice

Calculus 1 Practice

Calculus 2 Practice

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This page consists of 100 calculus 2 practice problems based on a video from one of our favorite instructors. We have laid out each practice problem and included the video clip containing each solution.

Here is the list of practice problems. We recommend that you download this pdf before starting.

Make sure you support the guy that did this video. He put a LOT of work into, not just doing the video, but also preparing the problems and making sure his solutions were correct. He did a GREAT job. So go to YouTube and like this video and follow him. He is one of our favorite instructors. (By the way, we are not receiving any compensation from him. We just think his videos will help you.)

We have another page with 100 infinite series practice problems by this same guy.

Notes

1. In this video, some of his problems are multiple choice. See his handout for the choices but we recommmend that you work each problem without looking at the choices. This will prepare you better for your exams, especially your final exam.
2. When evaluating integrals using integration by parts, he uses a tabular method. We strongly recommend that you do not work them with this shortcut. It essentially allows you to use integration by parts without having to learn the technique. When we teach, we do not give credit for problems worked this way. However, check with your instructor to see what they require.
3. The problems here are not in the same order as he has them in the video. We have grouped them by type in order for you to focus on the techniques that you need more practice on for your exam. You can find his question number in the hint. Here are the sections.

Sequences and Series

Limits

Integrals

Parametrics

Polar Coordinates

Hyperbolics

Applications

Basic Differential Equations

Approximations

Practice

Sequences and Series

Which one of these series diverge by the Divergence Test?

A. \(\displaystyle{ \sum_{n=1}^{\infty}{ \cos(1/n) } }\)

B. \(\displaystyle{ \sum_{n=1}^{\infty}{ \frac{n^2}{n^3+2n-1} } }\)

C. \(\displaystyle{ \sum_{n=3}^{\infty}{ \frac{1}{\sqrt{n}\ln n} } }\)

D. \(\displaystyle{ \sum_{n=1}^{\infty}{ \ln \left[ \frac{n}{n+1} \right] } }\)

Problem Statement

Which one of these series diverge by the Divergence Test?

A. \(\displaystyle{ \sum_{n=1}^{\infty}{ \cos(1/n) } }\)

B. \(\displaystyle{ \sum_{n=1}^{\infty}{ \frac{n^2}{n^3+2n-1} } }\)

C. \(\displaystyle{ \sum_{n=3}^{\infty}{ \frac{1}{\sqrt{n}\ln n} } }\)

D. \(\displaystyle{ \sum_{n=1}^{\infty}{ \ln \left[ \frac{n}{n+1} \right] } }\)

Hint

[Question 1]

Problem Statement

Which one of these series diverge by the Divergence Test?

A. \(\displaystyle{ \sum_{n=1}^{\infty}{ \cos(1/n) } }\)

B. \(\displaystyle{ \sum_{n=1}^{\infty}{ \frac{n^2}{n^3+2n-1} } }\)

C. \(\displaystyle{ \sum_{n=3}^{\infty}{ \frac{1}{\sqrt{n}\ln n} } }\)

D. \(\displaystyle{ \sum_{n=1}^{\infty}{ \ln \left[ \frac{n}{n+1} \right] } }\)

Hint

[Question 1]

Solution

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Consider a sequence defined recursively by \(a_1=5\), \(a_n=8-a_{n-1}\) for \(n \geq 2\). Which of the following statements about \(a_n\) is true?

A. \(a_n\) diverges

B. \(a_n\) converges to \(3\)

C. \(a_n\) converges to \(5\)

D. \(a_n\) is increasing

E. \(a_n\) is decreasing

Problem Statement

Consider a sequence defined recursively by \(a_1=5\), \(a_n=8-a_{n-1}\) for \(n \geq 2\). Which of the following statements about \(a_n\) is true?

A. \(a_n\) diverges

B. \(a_n\) converges to \(3\)

C. \(a_n\) converges to \(5\)

D. \(a_n\) is increasing

E. \(a_n\) is decreasing

Hint

[Question 4]

Problem Statement

Consider a sequence defined recursively by \(a_1=5\), \(a_n=8-a_{n-1}\) for \(n \geq 2\). Which of the following statements about \(a_n\) is true?

A. \(a_n\) diverges

B. \(a_n\) converges to \(3\)

C. \(a_n\) converges to \(5\)

D. \(a_n\) is increasing

E. \(a_n\) is decreasing

Hint

[Question 4]

Solution

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Evaluate \(\displaystyle{ 16 - 4 + 1 - 1/4 + 1/16 - \cdots }\)

Problem Statement

Evaluate \(\displaystyle{ 16 - 4 + 1 - 1/4 + 1/16 - \cdots }\)

Hint

[Question 11]

Problem Statement

Evaluate \(\displaystyle{ 16 - 4 + 1 - 1/4 + 1/16 - \cdots }\)

Hint

[Question 11]

Solution

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Determine \(\displaystyle{ \int{\frac{\tan^{-1}(x^2)}{x^2} ~dx} }\) as a power series.

Problem Statement

Determine \(\displaystyle{ \int{\frac{\tan^{-1}(x^2)}{x^2} ~dx} }\) as a power series.

Hint

[Question 13]

Problem Statement

Determine \(\displaystyle{ \int{\frac{\tan^{-1}(x^2)}{x^2} ~dx} }\) as a power series.

Hint

[Question 13]

Solution

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Which of the following is an example of \(a_n\) where
\(\displaystyle{ \lim_{n \to \infty}{a_n} = 0 }\) but \(\displaystyle{ \sum_{n=1}^{\infty}{ a_n } }\) diverges?

A. \(\displaystyle{ a_n = \frac{1}{\sqrt{n}} }\)

B. \(\displaystyle{ a_n = \frac{1}{n!} }\)

C. \(\displaystyle{ a_n = \frac{1}{n^2} }\)

D. \(\displaystyle{ a_n = e^{-n} }\)

E. \(\displaystyle{ a_n = \frac{1}{\tan^{-1}n} }\)

Problem Statement

Which of the following is an example of \(a_n\) where
\(\displaystyle{ \lim_{n \to \infty}{a_n} = 0 }\) but \(\displaystyle{ \sum_{n=1}^{\infty}{ a_n } }\) diverges?

A. \(\displaystyle{ a_n = \frac{1}{\sqrt{n}} }\)

B. \(\displaystyle{ a_n = \frac{1}{n!} }\)

C. \(\displaystyle{ a_n = \frac{1}{n^2} }\)

D. \(\displaystyle{ a_n = e^{-n} }\)

E. \(\displaystyle{ a_n = \frac{1}{\tan^{-1}n} }\)

Hint

[Question 14]

Problem Statement

Which of the following is an example of \(a_n\) where
\(\displaystyle{ \lim_{n \to \infty}{a_n} = 0 }\) but \(\displaystyle{ \sum_{n=1}^{\infty}{ a_n } }\) diverges?

A. \(\displaystyle{ a_n = \frac{1}{\sqrt{n}} }\)

B. \(\displaystyle{ a_n = \frac{1}{n!} }\)

C. \(\displaystyle{ a_n = \frac{1}{n^2} }\)

D. \(\displaystyle{ a_n = e^{-n} }\)

E. \(\displaystyle{ a_n = \frac{1}{\tan^{-1}n} }\)

Hint

[Question 14]

Solution

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Evaluate \(\displaystyle{ \sum_{n=1}^{\infty}{ \frac{1}{n^2+5n+4} } }\)

Problem Statement

Evaluate \(\displaystyle{ \sum_{n=1}^{\infty}{ \frac{1}{n^2+5n+4} } }\)

Hint

[Question 16]

Problem Statement

Evaluate \(\displaystyle{ \sum_{n=1}^{\infty}{ \frac{1}{n^2+5n+4} } }\)

Hint

[Question 16]

Solution

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Determine the interval of convergence of the power series \(\displaystyle{ \sum_{n=1}^{\infty}{ \frac{(-1)^n}{n3^n}(x-2)^n } }\)

Problem Statement

Determine the interval of convergence of the power series \(\displaystyle{ \sum_{n=1}^{\infty}{ \frac{(-1)^n}{n3^n}(x-2)^n } }\)

Hint

[Question 24]
For more practice, we recommend that you take the time to determine that the radius of convergence is \(R=3\) using the ratio test but the video gives it as a hint.

Problem Statement

Determine the interval of convergence of the power series \(\displaystyle{ \sum_{n=1}^{\infty}{ \frac{(-1)^n}{n3^n}(x-2)^n } }\)

Hint

[Question 24]
For more practice, we recommend that you take the time to determine that the radius of convergence is \(R=3\) using the ratio test but the video gives it as a hint.

Solution

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Which of the following converges absolutely?

A. \(\displaystyle{ \sum_{n=1}^{\infty}{ \frac{(-1)^n}{n!} } }\)

B. \(\displaystyle{ \sum_{n=1}^{\infty}{ \frac{(-1)^n}{\sqrt{n+2}} } }\)

C. \(\displaystyle{ \sum_{n=1}^{\infty}{ \frac{(-1)^n n^3}{n^3+1} } }\)

D. \(\displaystyle{ \sum_{n=1}^{\infty}{ \frac{(-1)^n n^2}{n^3+1} } }\)

Problem Statement

Which of the following converges absolutely?

A. \(\displaystyle{ \sum_{n=1}^{\infty}{ \frac{(-1)^n}{n!} } }\)

B. \(\displaystyle{ \sum_{n=1}^{\infty}{ \frac{(-1)^n}{\sqrt{n+2}} } }\)

C. \(\displaystyle{ \sum_{n=1}^{\infty}{ \frac{(-1)^n n^3}{n^3+1} } }\)

D. \(\displaystyle{ \sum_{n=1}^{\infty}{ \frac{(-1)^n n^2}{n^3+1} } }\)

Hint

[Question 27]
The last page of the pdf associated with this video has a correction to some of the work in the video on this problem.

Problem Statement

Which of the following converges absolutely?

A. \(\displaystyle{ \sum_{n=1}^{\infty}{ \frac{(-1)^n}{n!} } }\)

B. \(\displaystyle{ \sum_{n=1}^{\infty}{ \frac{(-1)^n}{\sqrt{n+2}} } }\)

C. \(\displaystyle{ \sum_{n=1}^{\infty}{ \frac{(-1)^n n^3}{n^3+1} } }\)

D. \(\displaystyle{ \sum_{n=1}^{\infty}{ \frac{(-1)^n n^2}{n^3+1} } }\)

Hint

[Question 27]
The last page of the pdf associated with this video has a correction to some of the work in the video on this problem.

Solution

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Evaluate \(\displaystyle{ \sum_{n=0}^{\infty}{ \frac{1}{2^n n!} } }\)

Problem Statement

Evaluate \(\displaystyle{ \sum_{n=0}^{\infty}{ \frac{1}{2^n n!} } }\)

Hint

[Question 30]

Problem Statement

Evaluate \(\displaystyle{ \sum_{n=0}^{\infty}{ \frac{1}{2^n n!} } }\)

Hint

[Question 30]

Solution

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Evaluate \(\displaystyle{ \int{ \ln(1+x^3) ~dx } }\) as a power series.

Problem Statement

Evaluate \(\displaystyle{ \int{ \ln(1+x^3) ~dx } }\) as a power series.

Hint

[Question 34]

Problem Statement

Evaluate \(\displaystyle{ \int{ \ln(1+x^3) ~dx } }\) as a power series.

Hint

[Question 34]

Solution

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Determine the power series expansion for \(\displaystyle{ f(x) = \frac{x^5}{4 + x^2} }\) using sigma notation, at \(a=0\) and find the radius and interval of convergence.

Problem Statement

Determine the power series expansion for \(\displaystyle{ f(x) = \frac{x^5}{4 + x^2} }\) using sigma notation, at \(a=0\) and find the radius and interval of convergence.

Hint

[Question 37]

Problem Statement

Determine the power series expansion for \(\displaystyle{ f(x) = \frac{x^5}{4 + x^2} }\) using sigma notation, at \(a=0\) and find the radius and interval of convergence.

Hint

[Question 37]

Solution

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Does \(\displaystyle{ \sum_{n=1}^{\infty}{ \sin^2(1/n) } }\) converge or diverge? Justify your answer.

Problem Statement

Does \(\displaystyle{ \sum_{n=1}^{\infty}{ \sin^2(1/n) } }\) converge or diverge? Justify your answer.

Hint

[Question 39]

Problem Statement

Does \(\displaystyle{ \sum_{n=1}^{\infty}{ \sin^2(1/n) } }\) converge or diverge? Justify your answer.

Hint

[Question 39]

Solution

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Determine \(\displaystyle{ \int{ e^{x^3} ~dx } }\) using a power series.

Problem Statement

Determine \(\displaystyle{ \int{ e^{x^3} ~dx } }\) using a power series.

Hint

[Question 44]

Problem Statement

Determine \(\displaystyle{ \int{ e^{x^3} ~dx } }\) using a power series.

Hint

[Question 44]

Solution

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How large should \(n\) be to guarantee that the approximation \(T_n\) to the integral \(\displaystyle{ \int_1^4{ e^{-x^2} ~dx } }\) is accurate to within \(0.0005\)?

Problem Statement

How large should \(n\) be to guarantee that the approximation \(T_n\) to the integral \(\displaystyle{ \int_1^4{ e^{-x^2} ~dx } }\) is accurate to within \(0.0005\)?

Hint

[Question 45]

Problem Statement

How large should \(n\) be to guarantee that the approximation \(T_n\) to the integral \(\displaystyle{ \int_1^4{ e^{-x^2} ~dx } }\) is accurate to within \(0.0005\)?

Hint

[Question 45]

Solution

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Determine the Maclaurin series of \(\displaystyle{ \cosh(x) = \frac{e^x+e^{-x}}{2} }\)

Problem Statement

Determine the Maclaurin series of \(\displaystyle{ \cosh(x) = \frac{e^x+e^{-x}}{2} }\)

Hint

[Question 50]

Problem Statement

Determine the Maclaurin series of \(\displaystyle{ \cosh(x) = \frac{e^x+e^{-x}}{2} }\)

Hint

[Question 50]

Solution

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Given the sequence \(\displaystyle{ 1/2, -1/3, 2/9, -4/27, \cdots, a_n, \cdots }\)
(a) does \(a_n\) converge? If so, find the value.
(b) does \(\displaystyle{ \sum_{n=1}^{\infty}{ a_n } }\) converge? If so, find the value.

Problem Statement

Given the sequence \(\displaystyle{ 1/2, -1/3, 2/9, -4/27, \cdots, a_n, \cdots }\)
(a) does \(a_n\) converge? If so, find the value.
(b) does \(\displaystyle{ \sum_{n=1}^{\infty}{ a_n } }\) converge? If so, find the value.

Hint

[Question 53]

Problem Statement

Given the sequence \(\displaystyle{ 1/2, -1/3, 2/9, -4/27, \cdots, a_n, \cdots }\)
(a) does \(a_n\) converge? If so, find the value.
(b) does \(\displaystyle{ \sum_{n=1}^{\infty}{ a_n } }\) converge? If so, find the value.

Hint

[Question 53]

Solution

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Does \(\displaystyle{ \sum_{n=3}^{\infty}{ \frac{1}{n\sqrt{\ln(n)}} } }\) converge or diverge? Justify your answer.

Problem Statement

Does \(\displaystyle{ \sum_{n=3}^{\infty}{ \frac{1}{n\sqrt{\ln(n)}} } }\) converge or diverge? Justify your answer.

Hint

[Question 59]

Problem Statement

Does \(\displaystyle{ \sum_{n=3}^{\infty}{ \frac{1}{n\sqrt{\ln(n)}} } }\) converge or diverge? Justify your answer.

Hint

[Question 59]

Solution

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Does \(\displaystyle{ \sum_{n=1}^{\infty}{ \frac{4^n n!}{n^n} } }\) converge or diverge? Justify your answer.

Problem Statement

Does \(\displaystyle{ \sum_{n=1}^{\infty}{ \frac{4^n n!}{n^n} } }\) converge or diverge? Justify your answer.

Hint

[Question 66]

Problem Statement

Does \(\displaystyle{ \sum_{n=1}^{\infty}{ \frac{4^n n!}{n^n} } }\) converge or diverge? Justify your answer.

Hint

[Question 66]

Solution

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The sum of infinitely many rational numbers has to be rational. True or False?

Problem Statement

The sum of infinitely many rational numbers has to be rational. True or False?

Hint

[Question 71]

Problem Statement

The sum of infinitely many rational numbers has to be rational. True or False?

Hint

[Question 71]

Solution

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Does \(\displaystyle{ \sum_{n=1}^{\infty}{ \frac{\cos n}{n^2} } }\) converge or diverge? If it converges, does it converge conditionally or absolutely?

Problem Statement

Does \(\displaystyle{ \sum_{n=1}^{\infty}{ \frac{\cos n}{n^2} } }\) converge or diverge? If it converges, does it converge conditionally or absolutely?

Hint

[Question 75]

Problem Statement

Does \(\displaystyle{ \sum_{n=1}^{\infty}{ \frac{\cos n}{n^2} } }\) converge or diverge? If it converges, does it converge conditionally or absolutely?

Hint

[Question 75]

Solution

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Can the Ratio Test be used to determine convergence or divergence of \(\displaystyle{ \sum_{n=1}^{\infty}{ \frac{1}{n^2} } }\)? Why or why not?

Problem Statement

Can the Ratio Test be used to determine convergence or divergence of \(\displaystyle{ \sum_{n=1}^{\infty}{ \frac{1}{n^2} } }\)? Why or why not?

Hint

[Question 76]

Problem Statement

Can the Ratio Test be used to determine convergence or divergence of \(\displaystyle{ \sum_{n=1}^{\infty}{ \frac{1}{n^2} } }\)? Why or why not?

Hint

[Question 76]

Solution

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If \(\displaystyle{ \sum_{n=1}^{\infty}{ a_n } }\) converges, then \(\displaystyle{ \sum_{n=1}^{\infty}{ (a_n)^2 } }\) must also converge. True or False?

Problem Statement

If \(\displaystyle{ \sum_{n=1}^{\infty}{ a_n } }\) converges, then \(\displaystyle{ \sum_{n=1}^{\infty}{ (a_n)^2 } }\) must also converge. True or False?

Hint

[Question 79]

Problem Statement

If \(\displaystyle{ \sum_{n=1}^{\infty}{ a_n } }\) converges, then \(\displaystyle{ \sum_{n=1}^{\infty}{ (a_n)^2 } }\) must also converge. True or False?

Hint

[Question 79]

Solution

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Consider the sequence \( \{ a_n \} \) where \(\displaystyle{ a_n = \frac{1}{(n+2)n!} }\).
(a) Find a formula for the nth partial sum.
(b) Evaluate \(\displaystyle{ \sum_{n=1}^{\infty}{ a_n } }\) if it converges.

Problem Statement

Consider the sequence \( \{ a_n \} \) where \(\displaystyle{ a_n = \frac{1}{(n+2)n!} }\).
(a) Find a formula for the nth partial sum.
(b) Evaluate \(\displaystyle{ \sum_{n=1}^{\infty}{ a_n } }\) if it converges.

Hint

[Question 82]
The last page of the pdf associated with this video has a correction for some of the work in the video for this problem.

Problem Statement

Consider the sequence \( \{ a_n \} \) where \(\displaystyle{ a_n = \frac{1}{(n+2)n!} }\).
(a) Find a formula for the nth partial sum.
(b) Evaluate \(\displaystyle{ \sum_{n=1}^{\infty}{ a_n } }\) if it converges.

Hint

[Question 82]
The last page of the pdf associated with this video has a correction for some of the work in the video for this problem.

Solution

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Does \(\displaystyle{ \sum_{n=3}^{\infty}{ \frac{1}{2^n \ln n} } }\) converge or diverge? Justify your answer.

Problem Statement

Does \(\displaystyle{ \sum_{n=3}^{\infty}{ \frac{1}{2^n \ln n} } }\) converge or diverge? Justify your answer.

Hint

[Question 86]

Problem Statement

Does \(\displaystyle{ \sum_{n=3}^{\infty}{ \frac{1}{2^n \ln n} } }\) converge or diverge? Justify your answer.

Hint

[Question 86]

Solution

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Does \( 1/3 - 2/5 + 3/7 - 4/9 + \cdots \) converge or diverge? Justify your answer.

Problem Statement

Does \( 1/3 - 2/5 + 3/7 - 4/9 + \cdots \) converge or diverge? Justify your answer.

Hint

[Question 90]

Problem Statement

Does \( 1/3 - 2/5 + 3/7 - 4/9 + \cdots \) converge or diverge? Justify your answer.

Hint

[Question 90]

Solution

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Does \(\displaystyle{ \sum_{n=1}^{\infty}{ ( 1 - 1/n )^{n^2} } }\) converge or diverge? Justify your answer.

Problem Statement

Does \(\displaystyle{ \sum_{n=1}^{\infty}{ ( 1 - 1/n )^{n^2} } }\) converge or diverge? Justify your answer.

Hint

[Question 95]

Problem Statement

Does \(\displaystyle{ \sum_{n=1}^{\infty}{ ( 1 - 1/n )^{n^2} } }\) converge or diverge? Justify your answer.

Hint

[Question 95]

Solution

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Does \(\displaystyle{ \sum_{n=1}^{\infty}{ \frac{2n^3-n}{\sqrt{ n^9+10n^3-8 }} } }\) converge or diverge? Justify your answer.

Problem Statement

Does \(\displaystyle{ \sum_{n=1}^{\infty}{ \frac{2n^3-n}{\sqrt{ n^9+10n^3-8 }} } }\) converge or diverge? Justify your answer.

Hint

[Question 100]

Problem Statement

Does \(\displaystyle{ \sum_{n=1}^{\infty}{ \frac{2n^3-n}{\sqrt{ n^9+10n^3-8 }} } }\) converge or diverge? Justify your answer.

Hint

[Question 100]

Solution

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Find the Taylor series, in sigma notation, of \(\displaystyle{ \frac{1}{1-x} }\) centered at \(a=3\). Determine the radius and interval of convergence.

Problem Statement

Find the Taylor series, in sigma notation, of \(\displaystyle{ \frac{1}{1-x} }\) centered at \(a=3\). Determine the radius and interval of convergence.

Hint

[Question 101]

Problem Statement

Find the Taylor series, in sigma notation, of \(\displaystyle{ \frac{1}{1-x} }\) centered at \(a=3\). Determine the radius and interval of convergence.

Hint

[Question 101]

Solution

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Limits

Evaluate \(\displaystyle{ \lim_{x\to\infty}{ \frac{\ln(2x)}{\ln(x^3+1)} } }\)

Problem Statement

Evaluate \(\displaystyle{ \lim_{x\to\infty}{ \frac{\ln(2x)}{\ln(x^3+1)} } }\)

Hint

[Question 10]

Problem Statement

Evaluate \(\displaystyle{ \lim_{x\to\infty}{ \frac{\ln(2x)}{\ln(x^3+1)} } }\)

Hint

[Question 10]

Solution

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Evaluate \(\displaystyle{ \lim_{t \to 0^+}{ \left[ \frac{1}{t} - \frac{1}{e^t-1} \right] } }\)

Problem Statement

Evaluate \(\displaystyle{ \lim_{t \to 0^+}{ \left[ \frac{1}{t} - \frac{1}{e^t-1} \right] } }\)

Hint

[Question 46]

Problem Statement

Evaluate \(\displaystyle{ \lim_{t \to 0^+}{ \left[ \frac{1}{t} - \frac{1}{e^t-1} \right] } }\)

Hint

[Question 46]

Solution

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Evaluate \(\displaystyle{ \lim_{x \to \infty}{ ( \sqrt{x^2+5x+2} - \sqrt{x^2-x+5} ) } }\)

Problem Statement

Evaluate \(\displaystyle{ \lim_{x \to \infty}{ ( \sqrt{x^2+5x+2} - \sqrt{x^2-x+5} ) } }\)

Hint

[Question 52]

Problem Statement

Evaluate \(\displaystyle{ \lim_{x \to \infty}{ ( \sqrt{x^2+5x+2} - \sqrt{x^2-x+5} ) } }\)

Hint

[Question 52]

Solution

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Evaluate \(\displaystyle{ \lim_{x \to 0^+}{(\sin x \ln x)} }\)

Problem Statement

Evaluate \(\displaystyle{ \lim_{x \to 0^+}{(\sin x \ln x)} }\)

Hint

[Question 77]

Problem Statement

Evaluate \(\displaystyle{ \lim_{x \to 0^+}{(\sin x \ln x)} }\)

Hint

[Question 77]

Solution

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Evaluate \(\displaystyle{ \lim_{x \to \infty}{ \frac{(\ln x)^2}{x} } }\)

Problem Statement

Evaluate \(\displaystyle{ \lim_{x \to \infty}{ \frac{(\ln x)^2}{x} } }\)

Hint

[Question 85]

Problem Statement

Evaluate \(\displaystyle{ \lim_{x \to \infty}{ \frac{(\ln x)^2}{x} } }\)

Hint

[Question 85]

Solution

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Evaluate \(\displaystyle{ \lim_{x \to 1}{ x^{1/(1-x)} } }\)

Problem Statement

Evaluate \(\displaystyle{ \lim_{x \to 1}{ x^{1/(1-x)} } }\)

Hint

[Question 98]

Problem Statement

Evaluate \(\displaystyle{ \lim_{x \to 1}{ x^{1/(1-x)} } }\)

Hint

[Question 98]

Solution

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Integrals

Given that \(\displaystyle{ \int_2^{\infty}{ \frac{1}{x\sqrt{x^2-1}} ~dx } }\) converges, which of the following integrals also converge by the comparison theorem with the given integral?

A. \(\displaystyle{ \int_2^{\infty}{ \frac{1}{x^2\sqrt{x^2-1}} ~dx} }\)

B. \(\displaystyle{ \int_2^{\infty}{ \frac{x}{\sqrt{x^2-1}} ~dx} }\)

C. \(\displaystyle{ \int_2^{\infty}{ \frac{1}{\sqrt{x^2-1}} ~dx} }\)

D. \(\displaystyle{ \int_2^{\infty}{ \frac{1}{\sqrt{x^2+1}} ~dx} }\)

Problem Statement

Given that \(\displaystyle{ \int_2^{\infty}{ \frac{1}{x\sqrt{x^2-1}} ~dx } }\) converges, which of the following integrals also converge by the comparison theorem with the given integral?

A. \(\displaystyle{ \int_2^{\infty}{ \frac{1}{x^2\sqrt{x^2-1}} ~dx} }\)

B. \(\displaystyle{ \int_2^{\infty}{ \frac{x}{\sqrt{x^2-1}} ~dx} }\)

C. \(\displaystyle{ \int_2^{\infty}{ \frac{1}{\sqrt{x^2-1}} ~dx} }\)

D. \(\displaystyle{ \int_2^{\infty}{ \frac{1}{\sqrt{x^2+1}} ~dx} }\)

Hint

[Question 7]

Problem Statement

Given that \(\displaystyle{ \int_2^{\infty}{ \frac{1}{x\sqrt{x^2-1}} ~dx } }\) converges, which of the following integrals also converge by the comparison theorem with the given integral?

A. \(\displaystyle{ \int_2^{\infty}{ \frac{1}{x^2\sqrt{x^2-1}} ~dx} }\)

B. \(\displaystyle{ \int_2^{\infty}{ \frac{x}{\sqrt{x^2-1}} ~dx} }\)

C. \(\displaystyle{ \int_2^{\infty}{ \frac{1}{\sqrt{x^2-1}} ~dx} }\)

D. \(\displaystyle{ \int_2^{\infty}{ \frac{1}{\sqrt{x^2+1}} ~dx} }\)

Hint

[Question 7]

Solution

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Evaluate \(\displaystyle{ \int{ \frac{3x^2-5x-4}{x^2-2x-3} ~dx } }\)

Problem Statement

Evaluate \(\displaystyle{ \int{ \frac{3x^2-5x-4}{x^2-2x-3} ~dx } }\)

Hint

[Question 9]

Problem Statement

Evaluate \(\displaystyle{ \int{ \frac{3x^2-5x-4}{x^2-2x-3} ~dx } }\)

Hint

[Question 9]

Solution

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Evaluate \(\displaystyle{ \int{ \frac{2\sin x}{\sin(2x)} ~dx } }\)

Problem Statement

Evaluate \(\displaystyle{ \int{ \frac{2\sin x}{\sin(2x)} ~dx } }\)

Hint

[Question 15]

Problem Statement

Evaluate \(\displaystyle{ \int{ \frac{2\sin x}{\sin(2x)} ~dx } }\)

Hint

[Question 15]

Solution

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Determine if \(\displaystyle{ \int_0^{\pi/2}{ \tan x ~ dx } }\) converges or diverges. If it converges, find the value to which it converges.

Problem Statement

Determine if \(\displaystyle{ \int_0^{\pi/2}{ \tan x ~ dx } }\) converges or diverges. If it converges, find the value to which it converges.

Hint

[Question 22(pdf)-21(video)] After he finishes this problem, he notices that he missed question 21 and changes the number on the board to 22.

Problem Statement

Determine if \(\displaystyle{ \int_0^{\pi/2}{ \tan x ~ dx } }\) converges or diverges. If it converges, find the value to which it converges.

Hint

[Question 22(pdf)-21(video)] After he finishes this problem, he notices that he missed question 21 and changes the number on the board to 22.

Solution

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Evaluate \(\displaystyle{ \int_0^1{ x\ln x~ dx } }\).

Problem Statement

Evaluate \(\displaystyle{ \int_0^1{ x\ln x~ dx } }\).

Hint

[Question 29]

Problem Statement

Evaluate \(\displaystyle{ \int_0^1{ x\ln x~ dx } }\).

Hint

[Question 29]

Solution

His notation in this problem is not very good. For starters, he uses the tabular method to do integration by parts. He does this most of the time. However, we do not recommend that method since it does not teach you to actually understand and use integration by parts. However, to check your answer, it is okay.
Also, he has equal signs between equations that are obviously not equal. Make sure you do not do that in any of your work. But, of course, check with your instructor to see what they expect.

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Evaluate \(\displaystyle{ \int{ \sin^5 x \cos^2 x} }\)

Problem Statement

Evaluate \(\displaystyle{ \int{ \sin^5 x \cos^2 x} }\)

Hint

[Question 33]

Problem Statement

Evaluate \(\displaystyle{ \int{ \sin^5 x \cos^2 x} }\)

Hint

[Question 33]

Solution

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Evaluate \(\displaystyle{ \int{ \tan^3 x ~dx } }\)

Problem Statement

Evaluate \(\displaystyle{ \int{ \tan^3 x ~dx } }\)

Hint

[Question 36]

Problem Statement

Evaluate \(\displaystyle{ \int{ \tan^3 x ~dx } }\)

Hint

[Question 36]

Solution

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Evaluate \(\displaystyle{ \int{ \sin^{-1} x ~dx } }\)

Problem Statement

Evaluate \(\displaystyle{ \int{ \sin^{-1} x ~dx } }\)

Hint

[Question 40]

Problem Statement

Evaluate \(\displaystyle{ \int{ \sin^{-1} x ~dx } }\)

Hint

[Question 40]

Solution

He does integration by parts in this problem but he uses the tabular method. We do not recommend that you use that shortcut since it keeps you from actually learning the integration by parts technique. However, using it to check your answer is okay. As usual, check with your instructor to see what they expect.

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Evaluate \(\displaystyle{ \int{ \frac{1}{\sqrt{(1-x^2)^3}} ~dx } }\)

Problem Statement

Evaluate \(\displaystyle{ \int{ \frac{1}{\sqrt{(1-x^2)^3}} ~dx } }\)

Hint

[Question 48]

Problem Statement

Evaluate \(\displaystyle{ \int{ \frac{1}{\sqrt{(1-x^2)^3}} ~dx } }\)

Hint

[Question 48]

Solution

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Evaluate \(\displaystyle{ \int{ \frac{\ln x}{\sqrt{x}} ~dx } }\)

Problem Statement

Evaluate \(\displaystyle{ \int{ \frac{\ln x}{\sqrt{x}} ~dx } }\)

Hint

[Question 51]

Problem Statement

Evaluate \(\displaystyle{ \int{ \frac{\ln x}{\sqrt{x}} ~dx } }\)

Hint

[Question 51]

Solution

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Evaluate \(\displaystyle{ \int_0^{\infty}{ \frac{1}{1+e^x} ~dx } }\) if it converges.

Problem Statement

Evaluate \(\displaystyle{ \int_0^{\infty}{ \frac{1}{1+e^x} ~dx } }\) if it converges.

Hint

[Question 55]

Problem Statement

Evaluate \(\displaystyle{ \int_0^{\infty}{ \frac{1}{1+e^x} ~dx } }\) if it converges.

Hint

[Question 55]

Solution

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Evaluate \(\displaystyle{ \int{ x^2 \sin(3x) ~dx } }\)

Problem Statement

Evaluate \(\displaystyle{ \int{ x^2 \sin(3x) ~dx } }\)

Hint

[Question 58]

Problem Statement

Evaluate \(\displaystyle{ \int{ x^2 \sin(3x) ~dx } }\)

Hint

[Question 58]

Solution

In the video, he uses the shortcut tabular method for integration by parts. We do not recommend using that shortcut since it doesn't allow you to actually understand integration by parts. However, the tabular method can be used to check your answers. Check with your instructor to see what they require.

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Evaluate \(\displaystyle{ \int{ \frac{2x+7}{(x+1)(x^2+9)} ~dx } }\)

Problem Statement

Evaluate \(\displaystyle{ \int{ \frac{2x+7}{(x+1)(x^2+9)} ~dx } }\)

Hint

[Question 60]

Problem Statement

Evaluate \(\displaystyle{ \int{ \frac{2x+7}{(x+1)(x^2+9)} ~dx } }\)

Hint

[Question 60]

Solution

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Does \(\displaystyle{ \int_1^{\infty}{ \frac{1}{x+e^x} ~dx } }\) converge or diverge? Justify your answer.

Problem Statement

Does \(\displaystyle{ \int_1^{\infty}{ \frac{1}{x+e^x} ~dx } }\) converge or diverge? Justify your answer.

Hint

[Question 61]
In this question, he asks for you to determine convergence or divergence, not necessarily what it converges to. So you can use the Direct Comparison Test similar to series.

Problem Statement

Does \(\displaystyle{ \int_1^{\infty}{ \frac{1}{x+e^x} ~dx } }\) converge or diverge? Justify your answer.

Hint

[Question 61]
In this question, he asks for you to determine convergence or divergence, not necessarily what it converges to. So you can use the Direct Comparison Test similar to series.

Solution

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Evaluate \(\displaystyle{ \int{ \frac{1}{x^2-4x+7} ~dx } }\)

Problem Statement

Evaluate \(\displaystyle{ \int{ \frac{1}{x^2-4x+7} ~dx } }\)

Hint

[Question 65]

Problem Statement

Evaluate \(\displaystyle{ \int{ \frac{1}{x^2-4x+7} ~dx } }\)

Hint

[Question 65]

Solution

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Evaluate \(\displaystyle{ \int{ \sin^4 x ~ dx } }\)

Problem Statement

Evaluate \(\displaystyle{ \int{ \sin^4 x ~ dx } }\)

Hint

[Question 88]

Problem Statement

Evaluate \(\displaystyle{ \int{ \sin^4 x ~ dx } }\)

Hint

[Question 88]

Solution

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Evaluate \(\displaystyle{ \int{ e^{3x}\sin(2x)~dx } }\)

Problem Statement

Evaluate \(\displaystyle{ \int{ e^{3x}\sin(2x)~dx } }\)

Hint

[Question 93]

Problem Statement

Evaluate \(\displaystyle{ \int{ e^{3x}\sin(2x)~dx } }\)

Hint

[Question 93]

Solution

In the video, he uses the shortcut tabular method for integration by parts. We do not recommend using that shortcut since it doesn't allow you to actually understand integration by parts. However, the tabular method can be used to check your answers. Check with your instructor to see what they require.

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Evaluate \(\displaystyle{ \int{ \sqrt{e^x - 1} ~dx } }\)

Problem Statement

Evaluate \(\displaystyle{ \int{ \sqrt{e^x - 1} ~dx } }\)

Hint

[Question 99]

Problem Statement

Evaluate \(\displaystyle{ \int{ \sqrt{e^x - 1} ~dx } }\)

Hint

[Question 99]

Solution

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\(\displaystyle{ \int{ \frac{\sqrt{x^2-9}}{x^3} ~dx } }\)

Problem Statement

\(\displaystyle{ \int{ \frac{\sqrt{x^2-9}}{x^3} ~dx } }\)

Hint

[Question 41]

Problem Statement

\(\displaystyle{ \int{ \frac{\sqrt{x^2-9}}{x^3} ~dx } }\)

Hint

[Question 41]

Solution

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Parametrics

If \(x=e^t\) and \(y=3e^t\), calculate \(\displaystyle{ \frac{d^2y}{dx^2} }\)

Problem Statement

If \(x=e^t\) and \(y=3e^t\), calculate \(\displaystyle{ \frac{d^2y}{dx^2} }\)

Hint

[Question 5]

Problem Statement

If \(x=e^t\) and \(y=3e^t\), calculate \(\displaystyle{ \frac{d^2y}{dx^2} }\)

Hint

[Question 5]

Solution

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Find a parameterization (with time interval) of the full circle with radius \(2\), centered at \((1,2)\), the starting point at \((1,4)\), traveling . . . (a) counterclockwise, (b) clockwise.

Problem Statement

Find a parameterization (with time interval) of the full circle with radius \(2\), centered at \((1,2)\), the starting point at \((1,4)\), traveling . . . (a) counterclockwise, (b) clockwise.

Hint

[Question 32]

Problem Statement

Find a parameterization (with time interval) of the full circle with radius \(2\), centered at \((1,2)\), the starting point at \((1,4)\), traveling . . . (a) counterclockwise, (b) clockwise.

Hint

[Question 32]

Solution

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Write an equation of the line tangent to the curve defined by \(x=t^2-6t\) and \(y=\sqrt{t+7}\) at \(t=2\).

Problem Statement

Write an equation of the line tangent to the curve defined by \(x=t^2-6t\) and \(y=\sqrt{t+7}\) at \(t=2\).

Hint

[Question 42]

Problem Statement

Write an equation of the line tangent to the curve defined by \(x=t^2-6t\) and \(y=\sqrt{t+7}\) at \(t=2\).

Hint

[Question 42]

Solution

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Find a parameterization (with time interval) of the line segment from \((1,2)\) to \((5,4)\).

Problem Statement

Find a parameterization (with time interval) of the line segment from \((1,2)\) to \((5,4)\).

Hint

[Question 54]

Problem Statement

Find a parameterization (with time interval) of the line segment from \((1,2)\) to \((5,4)\).

Hint

[Question 54]

Solution

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\(\displaystyle{ \frac{d^2y}{dx^2} = \frac{d^2y/dt^2}{d^2x/dt^2} }\) True or False?

Problem Statement

\(\displaystyle{ \frac{d^2y}{dx^2} = \frac{d^2y/dt^2}{d^2x/dt^2} }\) True or False?

Hint

[Question 74]

Problem Statement

\(\displaystyle{ \frac{d^2y}{dx^2} = \frac{d^2y/dt^2}{d^2x/dt^2} }\) True or False?

Hint

[Question 74]

Solution

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Convert \( x = t^2 - 6t\), \( y=2t+1 \) to Cartesian coordinates.

Problem Statement

Convert \( x = t^2 - 6t\), \( y=2t+1 \) to Cartesian coordinates.

Hint

[Question 94]

Problem Statement

Convert \( x = t^2 - 6t\), \( y=2t+1 \) to Cartesian coordinates.

Hint

[Question 94]

Solution

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Calculate the area under one arch of the cycloid \( x=r(t-\sin t) \), \( y=r(1-\cos t) \).

Problem Statement

Calculate the area under one arch of the cycloid \( x=r(t-\sin t) \), \( y=r(1-\cos t) \).

Hint

[Question 83]

Problem Statement

Calculate the area under one arch of the cycloid \( x=r(t-\sin t) \), \( y=r(1-\cos t) \).

Final Answer

\( 3\pi r^2 \)

Problem Statement

Calculate the area under one arch of the cycloid \( x=r(t-\sin t) \), \( y=r(1-\cos t) \).

Hint

[Question 83]

Solution

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Final Answer

\( 3\pi r^2 \)

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Set up an integral for the surface area obtained by rotating the arc defined by \(x = t + e^t \), \( y=\cos(t)\) from \(t=0\) to \(t=1\) about the y-axis.

Problem Statement

Set up an integral for the surface area obtained by rotating the arc defined by \(x = t + e^t \), \( y=\cos(t)\) from \(t=0\) to \(t=1\) about the y-axis.

Hint

[Question 3]

Problem Statement

Set up an integral for the surface area obtained by rotating the arc defined by \(x = t + e^t \), \( y=\cos(t)\) from \(t=0\) to \(t=1\) about the y-axis.

Hint

[Question 3]

Solution

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Find the exact value of the surface area obtained by rotating the arc \(\displaystyle{\left\{\begin{array}{rcl} x & = & e^t - t \\ y & = & 4e^{t/2} \end{array} \right. }\)
\( 0 \leq t \leq 1 \) about the x-axis.

Problem Statement

Find the exact value of the surface area obtained by rotating the arc \(\displaystyle{\left\{\begin{array}{rcl} x & = & e^t - t \\ y & = & 4e^{t/2} \end{array} \right. }\)
\( 0 \leq t \leq 1 \) about the x-axis.

Hint

[Question 57]

Problem Statement

Find the exact value of the surface area obtained by rotating the arc \(\displaystyle{\left\{\begin{array}{rcl} x & = & e^t - t \\ y & = & 4e^{t/2} \end{array} \right. }\)
\( 0 \leq t \leq 1 \) about the x-axis.

Hint

[Question 57]

Solution

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At what points on the curve \( x=3t^2+1 \), \( y=t^3 - 1 \) does the tangent line have slope \(1/4\)?

Problem Statement

At what points on the curve \( x=3t^2+1 \), \( y=t^3 - 1 \) does the tangent line have slope \(1/4\)?

Hint

[Question 89]

Problem Statement

At what points on the curve \( x=3t^2+1 \), \( y=t^3 - 1 \) does the tangent line have slope \(1/4\)?

Hint

[Question 89]

Solution

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Polar Coordinates

Convert the Cartesian equation of a line \(y=mx+b\) to a polar equation.

Problem Statement

Convert the Cartesian equation of a line \(y=mx+b\) to a polar equation.

Hint

[Question 19]

Problem Statement

Convert the Cartesian equation of a line \(y=mx+b\) to a polar equation.

Hint

[Question 19]

Solution

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Given a polar equation \( r= \sec \theta \tan \theta \), determine \(dy/dx\) in terms of \(\theta\).

Problem Statement

Given a polar equation \( r= \sec \theta \tan \theta \), determine \(dy/dx\) in terms of \(\theta\).

Hint

[Question 23]

Problem Statement

Given a polar equation \( r= \sec \theta \tan \theta \), determine \(dy/dx\) in terms of \(\theta\).

Hint

[Question 23]

Solution

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Find the slope of the line tangent to the polar curve \(r=\cos\theta\) at \(\theta = \pi/6\).

Problem Statement

Find the slope of the line tangent to the polar curve \(r=\cos\theta\) at \(\theta = \pi/6\).

Hint

[Question 63]

Problem Statement

Find the slope of the line tangent to the polar curve \(r=\cos\theta\) at \(\theta = \pi/6\).

Hint

[Question 63]

Solution

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Does the polar equation \( r=\theta^2 \) represent a parabola?

Problem Statement

Does the polar equation \( r=\theta^2 \) represent a parabola?

Hint

[Question 73]

Problem Statement

Does the polar equation \( r=\theta^2 \) represent a parabola?

Hint

[Question 73]

Solution

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Convert the polar equation \( r = 6\sin \theta - 4\cos \theta \) to Cartesian coordinates and describe the shape of the graph.

Problem Statement

Convert the polar equation \( r = 6\sin \theta - 4\cos \theta \) to Cartesian coordinates and describe the shape of the graph.

Hint

[Question 87]

Problem Statement

Convert the polar equation \( r = 6\sin \theta - 4\cos \theta \) to Cartesian coordinates and describe the shape of the graph.

Hint

[Question 87]

Solution

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Derive the arc length formula for a polar curve.

Problem Statement

Derive the arc length formula for a polar curve.

Hint

[Question 91]
You need to derive the equation \(\displaystyle{ L = \int_{\theta_1}^{\theta_2}{ \sqrt{ r^2 + (dr/d\theta)^2 } ~d\theta } }\)

Problem Statement

Derive the arc length formula for a polar curve.

Hint

[Question 91]
You need to derive the equation \(\displaystyle{ L = \int_{\theta_1}^{\theta_2}{ \sqrt{ r^2 + (dr/d\theta)^2 } ~d\theta } }\)

Solution

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What is the area of the shaded region?

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Problem Statement

What is the area of the shaded region?

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Hint

[Question 97]

Problem Statement

What is the area of the shaded region?

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Final Answer

\(A=\pi^3/2 \approx 15.503\)

Problem Statement

What is the area of the shaded region?

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Hint

[Question 97]

Solution

The equation he gets to calculate area is \(\displaystyle{ A = \int_{\pi}^{2\pi}{ \frac{1}{2}(\theta \sin \theta)^2 d\theta } - \int_{0}^{\pi}{ \frac{1}{2}(\theta \sin \theta)^2 d\theta } }\). The first integral is the area of the large figure and the second integral is the smaller figure. Basically, he calculates the full area and then subtracts the hole in the middle.
In the video, he just sets up the integrals and then plugs them into a calculator to get an approximate answer. Hopefully, you didn't do that. So we present here the details of the integration and how to get the exact answer.
The nice thing is that both integrands are the same, so we really only need to evaluate one integral.

\(\displaystyle{ \int{ (\theta \sin \theta)^2 ~d\theta } }\)

\(\displaystyle{ \int{ \theta^2 \sin^2 \theta ~d\theta } }\)

So it looks like we are going to have to do integration by parts twice.

\( u = \theta^2 \to du = 2\theta ~d\theta \)

\( dv = \sin^2 \theta ~d\theta \to v = \int{ \sin^2 \theta ~d\theta } \)

\(\displaystyle{ v = \int{ \frac{1}{2} [1 - \cos(2\theta) ] ~d\theta } }\)

\(\displaystyle{ v = \frac{1}{2} \int{ 1 - \cos (2\theta) ~d\theta } }\)

\(\displaystyle{ v = \frac{1}{2} \left[ \theta - \frac{\sin(2\theta)}{2} \right] }\)

\(\displaystyle{ v = \frac{\theta}{2} - \frac{\sin(2\theta)}{4} }\)

Okay, we found \(du\) and \(v\), so we set up our integration by parts integrals.

\(\displaystyle{ \int{ \theta^2 \sin^2 \theta ~d\theta } = \theta^2 \left[ \frac{\theta}{2} - \frac{\sin(2\theta)}{4} \right] - \int{ \frac{\theta}{2} - \frac{\sin(2\theta)}{4} (2\theta) ~d\theta } }\)

\(\displaystyle{ \frac{\theta^2}{4}[ 2\theta - \sin(2\theta) ] - \int{ \theta^2 - \frac{\theta \sin(2\theta)}{2} ~d\theta } }\)

\(\displaystyle{ \frac{\theta^2}{4}[ 2\theta - \sin(2\theta) ] - \frac{\theta^3}{3} + \int{ \frac{\theta \sin(2\theta)}{2} ~d\theta } }\)

Do integration by parts a second time.

\( u = \theta/2 \to du = d\theta/2 \)

\( dv = \sin(2\theta)~d\theta \to v = -\cos(2\theta)/2 \)

\(\displaystyle{ \frac{\theta^2}{4}[ 2\theta - \sin(2\theta) ] - \frac{\theta^3}{3} + \left[ \frac{-\theta}{4}\cos(2\theta) \right] - \int{ \frac{-\cos(2\theta)}{4} ~d\theta } }\)

\(\displaystyle{ \frac{\theta^2}{4}[ 2\theta - \sin(2\theta) ] - \frac{\theta^3}{3} - \frac{\theta}{4}\cos(2\theta) + \frac{\sin(2\theta)}{8} }\)

So this last expression is the integral \(\displaystyle{ \int{ \theta^2 \sin^2 \theta ~d\theta } }\)

Okay, here is the original equation to calculate area.

\(\displaystyle{ A = \int_{\pi}^{2\pi}{ \frac{1}{2}(\theta \sin \theta)^2 d\theta } - \int_{0}^{\pi}{ \frac{1}{2}(\theta \sin \theta)^2 d\theta } }\)

We need to multiply our result by \(1/2\) and evaluate it at the endpoints. Before we jump into the calculations, let's look at what is going on. Notice that for the first integral, we evaluate our result at \(2\pi\). Then we evaluate our result at \(\pi\) and subtract it. For the second integral, we evaluate our result at \(\pi\) and subtract an evaluation at \(0\). The evaluation at zero gives us zero for every term. Also, every sine term is zero. So evaluating this expression is much simpler than it looks.

At \(\theta = 2\pi\), we have \(\displaystyle{ \frac{8\pi^3}{4} - \frac{8\pi^3}{6} - \frac{2\pi}{8} }\)

At \(\theta = \pi\), we have \(\displaystyle{ \frac{\pi^3}{4} - \frac{\pi^3}{6} - \frac{\pi}{8} }\)

Let's put all the expressions together before we simplify.

\(\displaystyle{ \left[ \frac{8\pi^3}{4} - \frac{8\pi^3}{6} - \frac{2\pi}{8} \right] - }\) \(\displaystyle{ \left[ \frac{\pi^3}{4} - \frac{\pi^3}{6} - \frac{\pi}{8} \right] - }\) \(\displaystyle{ \left[ \frac{\pi^3}{4} - \frac{\pi^3}{6} - \frac{\pi}{8} \right] + 0 }\)

The last zero in the above expression represents evaluating our result at \(\theta = 0\).

Before we move on, let's look at the last expression. Notice that all three terms with \(8\) in the denominator cancel. Combining the terms with the same denominators gives us this.

\(\displaystyle{ \frac{6\pi^3}{4} - \frac{6\pi^3}{6} }\)

\(\displaystyle{ \frac{3\pi^3}{2} - \frac{2\pi^3}{2} }\)

\(\displaystyle{ \frac{\pi^3}{2} }\)

And that's it. The shaded area in the figure is \(A=\pi^3/2\). This matches his approximate answer of \(15.503\). Whew! I can understand why he just plugged the integral into his calculator.

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Final Answer

\(A=\pi^3/2 \approx 15.503\)

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What is the area of the shaded region?

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Problem Statement

What is the area of the shaded region?

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Hint

[Question 2] In his solution, he set up the integral for the top half of the petal on the right and then multiplied by two. This is the recommended way to solve this problem since it involves a simple integral.

Problem Statement

What is the area of the shaded region?

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Final Answer

Exact Answer: \(\pi/12\)

Problem Statement

What is the area of the shaded region?

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Hint

[Question 2] In his solution, he set up the integral for the top half of the petal on the right and then multiplied by two. This is the recommended way to solve this problem since it involves a simple integral.

Solution

In his video solution, he just set up the integral and then used a calculator to calculate the approximate value. If you are not allowed to use a calculator in your class or on the exam, here is how to solve this integral.

His integral is \(\displaystyle{ 2 \int_{0}^{\pi/6}{ (1/2)( \cos (3\theta) )^2 ~ d\theta } }\)

Use the identity \( \cos^2 t = (1 + \cos(2t))/2 \) to get

\(\displaystyle{ \int_{0}^{\pi/6}{ (1 + \cos(6\theta))/2 ~ d\theta } }\)

\(\displaystyle{ (1/2) \int_{0}^{\pi/6}{ 1 + \cos(6\theta) ~ d\theta } }\)

\(\displaystyle{ (1/2) \left[ \theta \right]_{0}^{\pi/6} + (1/2) \int_{0}^{\pi/6}{ \cos(6\theta) ~ d\theta } }\)

\(\displaystyle{ \pi/12 + (1/2) \left[ \frac{-\sin(6\theta)}{6} \right]_0^{\pi/6} }\)

\(\displaystyle{ \pi/12 + (1/12) (-\sin \pi + \sin 0) }\)

\(\displaystyle{ \pi/12 + (1/12) (0 + 0) = \pi/12 }\)

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Final Answer

Exact Answer: \(\pi/12\)

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Find the area of the shaded region in this plot.

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Problem Statement

Find the area of the shaded region in this plot.

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Hint

[Question 38]

Problem Statement

Find the area of the shaded region in this plot.

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Hint

[Question 38]

Solution

He sets up the integral but doesn't finish the problem by showing how to do the integration. Here are the details.

\(\displaystyle{\int_{\pi/2}^{\pi}{ (1/2)( \cos\theta - \sin\theta)^2 ~d\theta } - \frac{\pi}{4} }\)

\(\displaystyle{ (1/2) \int_{\pi/2}^{\pi}{ \cos^2\theta - 2\cos\theta \sin\theta + \sin^2 \theta ~d\theta } - \frac{\pi}{4} }\)

\(\displaystyle{ (1/2) \int_{\pi/2}^{\pi}{ (1/2)(1+\cos(2\theta)) - 2\cos\theta \sin\theta + (1/2)(1-\cos(2\theta)) ~d\theta } - \frac{\pi}{4} }\)

\(\displaystyle{ (1/2) \int_{\pi/2}^{\pi}{ 1-2\cos\theta\sin\theta ~ d\theta } - \frac{\pi}{4} }\)

Use integration by substitution with \( u = \sin\theta \to du = \cos\theta d\theta\)

for \( \theta = \pi/2 \), \( u = \sin\theta \to u = \sin (\pi/2) = -1\)

for \( \theta = \pi \), \( u = \sin\theta \to u = \sin (\pi) = 0\)

\(\displaystyle{ [\theta/2]_{\pi/2}^{\pi} - \int_{-1}^0{ u~du } - \frac{\pi}{4} }\)

\(\displaystyle{ \pi/2 - \pi/4 - (1/2)[u^2]_{-1}^0 - \frac{\pi}{4} = 1/2 }\)

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Hyperbolics

\(\displaystyle{ \frac{d}{dx}(\cosh x) = -\sinh x }\) True or False?

Problem Statement

\(\displaystyle{ \frac{d}{dx}(\cosh x) = -\sinh x }\) True or False?

Hint

[Question 70]

Problem Statement

\(\displaystyle{ \frac{d}{dx}(\cosh x) = -\sinh x }\) True or False?

Hint

[Question 70]

Solution

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Prove that \(\displaystyle{ \sinh^{-1} x = \ln \left( x + \sqrt{1+x^2} \right) }\) for any real number \(x\).

Problem Statement

Prove that \(\displaystyle{ \sinh^{-1} x = \ln \left( x + \sqrt{1+x^2} \right) }\) for any real number \(x\).

Hint

[Question 80]

Problem Statement

Prove that \(\displaystyle{ \sinh^{-1} x = \ln \left( x + \sqrt{1+x^2} \right) }\) for any real number \(x\).

Hint

[Question 80]

Solution

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Prove \(\displaystyle{ \frac{d}{dx}(\sinh^{-1} x) = \frac{1}{\sqrt{1+x^2}} }\)

Problem Statement

Prove \(\displaystyle{ \frac{d}{dx}(\sinh^{-1} x) = \frac{1}{\sqrt{1+x^2}} }\)

Hint

[Question 81]

Problem Statement

Prove \(\displaystyle{ \frac{d}{dx}(\sinh^{-1} x) = \frac{1}{\sqrt{1+x^2}} }\)

Hint

[Question 81]

Solution

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Evaluate \(\displaystyle{ \int{ \sinh^3 x ~ dx } }\)

Problem Statement

Evaluate \(\displaystyle{ \int{ \sinh^3 x ~ dx } }\)

Hint

[Question 78]

Problem Statement

Evaluate \(\displaystyle{ \int{ \sinh^3 x ~ dx } }\)

Hint

[Question 78]

Solution

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Evaluate \(\displaystyle{ \frac{d}{dx}\left[ \frac{\sinh x}{1+\cosh x} \right] }\)

Problem Statement

Evaluate \(\displaystyle{ \frac{d}{dx}\left[ \frac{\sinh x}{1+\cosh x} \right] }\)

Hint

[Question 21 (at the end of the pdf) He does not work this problem in the video.]

Problem Statement

Evaluate \(\displaystyle{ \frac{d}{dx}\left[ \frac{\sinh x}{1+\cosh x} \right] }\)

Final Answer

\(\displaystyle{ \frac{1}{1+\cosh x} }\)

Problem Statement

Evaluate \(\displaystyle{ \frac{d}{dx}\left[ \frac{\sinh x}{1+\cosh x} \right] }\)

Hint

[Question 21 (at the end of the pdf) He does not work this problem in the video.]

Solution

\(\displaystyle{ \frac{d}{dx}\left[ \frac{\sinh x}{1+\cosh x} \right] }\)

Use the quotient rule.

\(\displaystyle{ \frac{(1+\cosh x)(\cosh x) - \sinh x (\sinh x)}{(1+\cosh x)^2} }\)

\(\displaystyle{ \frac{\cosh x + \cosh^2 x - \sinh^2 x}{(1+\cosh x)^2} }\)

\(\displaystyle{ \frac{1+\cosh x}{(1+\cosh x)^2} }\)

\(\displaystyle{ \frac{1}{1+\cosh x} }\)

Final Answer

\(\displaystyle{ \frac{1}{1+\cosh x} }\)

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Find the arc length of the curve \( y = \sinh(x)\) from \(x=1\) to \(x=4\).

Problem Statement

Find the arc length of the curve \( y = \sinh(x)\) from \(x=1\) to \(x=4\).

Hint

[Question 17]
The integral he gets for the arc length is not easily evaluated. So he just uses his calculator to get the final numeric approximate answer.

Problem Statement

Find the arc length of the curve \( y = \sinh(x)\) from \(x=1\) to \(x=4\).

Hint

[Question 17]
The integral he gets for the arc length is not easily evaluated. So he just uses his calculator to get the final numeric approximate answer.

Solution

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If \(f(x) = \tanh^{-1}(\sin x)\), calculate \(f'(x)\).

Problem Statement

If \(f(x) = \tanh^{-1}(\sin x)\), calculate \(f'(x)\).

Hint

[Question 6]
\(\tanh^{-1}\) is the inverse hyperbolic tangent.

Problem Statement

If \(f(x) = \tanh^{-1}(\sin x)\), calculate \(f'(x)\).

Hint

[Question 6]
\(\tanh^{-1}\) is the inverse hyperbolic tangent.

Solution

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Applications

The vertical plate is partially submerged in water and has the indicated shape. Find the hydrostatic forces against one side of the plate.

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Problem Statement

The vertical plate is partially submerged in water and has the indicated shape. Find the hydrostatic forces against one side of the plate.

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Hint

[Question 18]
Not stated in the problem but assumed from the picture is that the plate is half of a circle with diameter 6m.

Problem Statement

The vertical plate is partially submerged in water and has the indicated shape. Find the hydrostatic forces against one side of the plate.

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Final Answer

Exact Answer: \(\displaystyle{ 19600 \left[ \frac{-9\pi}{4} + \frac{9}{2}\arcsin(1/3) + \frac{19\sqrt{2}}{3} \right] }\)

Problem Statement

The vertical plate is partially submerged in water and has the indicated shape. Find the hydrostatic forces against one side of the plate.

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Hint

[Question 18]
Not stated in the problem but assumed from the picture is that the plate is half of a circle with diameter 6m.

Solution

He sets up the integral and then plugs it into his calculator to get the answer. The integral is \(\displaystyle{ \int_0^2{ (2-y)\cdot 9.8 \cdot 1000 \cdot 2 \sqrt{ 9 - (y-3)^2 } ~ dy} }\). We solve the integral \(\displaystyle{ \int_0^2{ (2-y) \sqrt{ 9 - (y-3)^2 } ~ dy} }\) on the trig substitution page as problem 3486. In order to get an exact answer to this problem, take the answer from practice 3486 and multiply by \( 9.8 \cdot 1000 \cdot 2 = 19600 \).

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Final Answer

Exact Answer: \(\displaystyle{ 19600 \left[ \frac{-9\pi}{4} + \frac{9}{2}\arcsin(1/3) + \frac{19\sqrt{2}}{3} \right] }\)

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A rectangular swimming pool with length 20m, width 12m and depth 2m, is filled with water to the 1.7m mark. How much work is required to pump all the water out over the side?

Problem Statement

A rectangular swimming pool with length 20m, width 12m and depth 2m, is filled with water to the 1.7m mark. How much work is required to pump all the water out over the side?

Hint

[Question 25]

Problem Statement

A rectangular swimming pool with length 20m, width 12m and depth 2m, is filled with water to the 1.7m mark. How much work is required to pump all the water out over the side?

Final Answer

\( 4598160 J \)

Problem Statement

A rectangular swimming pool with length 20m, width 12m and depth 2m, is filled with water to the 1.7m mark. How much work is required to pump all the water out over the side?

Hint

[Question 25]

Solution

He does not completely solve this problem. We finish it here so that you can check your work.

He ends with the \(\displaystyle{ \int_0^{1.7}{ (2-y) 9.8 (1000) 12 (20) ~dy } }\)

\(\displaystyle{ 2352000 \int_0^{1.7}{ (2-y) ~dy } }\)

\(\displaystyle{ 2352000 \left[ 2y - \frac{y^2}{2} \right]_0^{1.7} }\)

\(\displaystyle{ 2352000 \left[ 3.4 - \frac{(1.7)^2}{2} \right] - 0 }\)

\( 2352000 \cdot 1.955 = 4598160 \)

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Final Answer

\( 4598160 J \)

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Biologists stocked a lake with 600 fish and estimated the carrying capacity to be 15000. The number of fish tripled after two years. If the size of the fish population satisfies the logistic equation, find the number of fish after another two years. Round your answer to the nearest whole number.

Problem Statement

Biologists stocked a lake with 600 fish and estimated the carrying capacity to be 15000. The number of fish tripled after two years. If the size of the fish population satisfies the logistic equation, find the number of fish after another two years. Round your answer to the nearest whole number.

Hint

[Question 28]

Problem Statement

Biologists stocked a lake with 600 fish and estimated the carrying capacity to be 15000. The number of fish tripled after two years. If the size of the fish population satisfies the logistic equation, find the number of fish after another two years. Round your answer to the nearest whole number.

Hint

[Question 28]

Solution

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A tank with a capacity of 400L is full of a mixture of water and chlorine with a concentration of 0.22g of chlorine per liter. In order to reduce the concentration of chlorine, fresh water is pumped into the tank at a rate of 12L/min. The mixture is kept stirred and is pumped out at the same rate. Let \(A(t)\) be the amount of chlorine in the tank after \(t\) minutes. What is the value of \(A(30)\)?

Problem Statement

A tank with a capacity of 400L is full of a mixture of water and chlorine with a concentration of 0.22g of chlorine per liter. In order to reduce the concentration of chlorine, fresh water is pumped into the tank at a rate of 12L/min. The mixture is kept stirred and is pumped out at the same rate. Let \(A(t)\) be the amount of chlorine in the tank after \(t\) minutes. What is the value of \(A(30)\)?

Hint

[Question 31]

Problem Statement

A tank with a capacity of 400L is full of a mixture of water and chlorine with a concentration of 0.22g of chlorine per liter. In order to reduce the concentration of chlorine, fresh water is pumped into the tank at a rate of 12L/min. The mixture is kept stirred and is pumped out at the same rate. Let \(A(t)\) be the amount of chlorine in the tank after \(t\) minutes. What is the value of \(A(30)\)?

Hint

[Question 31]

Solution

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Find the centroid of the region bounded by \(y=1/x\), \(y=0\), \(x=1\) and \(x=4\).

Problem Statement

Find the centroid of the region bounded by \(y=1/x\), \(y=0\), \(x=1\) and \(x=4\).

Hint

[Question 35]

Problem Statement

Find the centroid of the region bounded by \(y=1/x\), \(y=0\), \(x=1\) and \(x=4\).

Hint

[Question 35]

Solution

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Prove the volume of the cone with height \(h\) and the base radius \(r\) is \( V = \pi r^2 h /3 \).

Problem Statement

Prove the volume of the cone with height \(h\) and the base radius \(r\) is \( V = \pi r^2 h /3 \).

Hint

[Question 43]

Problem Statement

Prove the volume of the cone with height \(h\) and the base radius \(r\) is \( V = \pi r^2 h /3 \).

Hint

[Question 43]

Solution

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Find the centroid of the region bounded by \(y=e^{2x}\), \(y=0\), \(x=0\) and \(x=1\).

Problem Statement

Find the centroid of the region bounded by \(y=e^{2x}\), \(y=0\), \(x=0\) and \(x=1\).

Hint

[Question 62]

Problem Statement

Find the centroid of the region bounded by \(y=e^{2x}\), \(y=0\), \(x=0\) and \(x=1\).

Hint

[Question 62]

Solution

In the video, he uses the shortcut tabular method for integration by parts. We do not recommend using that shortcut since it doesn't allow you to actually understand integration by parts. However, the tabular method can be used to check your answers. Check with your instructor to see what they require.

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A tank has the shape of a frustrum with height 5m, bottom radius 6m, top radius 2m. It is filled with water to a height of 4m. Find the work required to empty the tank by pumping all the water to the top of the tank.

Problem Statement

A tank has the shape of a frustrum with height 5m, bottom radius 6m, top radius 2m. It is filled with water to a height of 4m. Find the work required to empty the tank by pumping all the water to the top of the tank.

Hint

[Question 64]

Problem Statement

A tank has the shape of a frustrum with height 5m, bottom radius 6m, top radius 2m. It is filled with water to a height of 4m. Find the work required to empty the tank by pumping all the water to the top of the tank.

Hint

[Question 64]

Solution

In the video, he gets an integral but does not evaluate it. Here are the missing details.

\(\displaystyle{ 9800\pi \int_0^4{ (5-y) (-4y/5+6)^2 ~ dy } }\)

To make the algebra simpler, we remove the fraction in the second term and pull it outside the integral.

\(\displaystyle{ 9800\pi/25 \int_0^4{ (5-y) (-4y+30)^2 ~ dy } }\)

\(\displaystyle{ 392\pi \int_0^4{ (5-y) (-4y+30)^2 ~ dy } }\)

\(\displaystyle{ 392\pi \int_0^4{ (5-y) (16y^2-240y+900) ~ dy } }\)

\(\displaystyle{ 392\pi \int_0^4{ 80y^2-1200y+4500-16y^3+240y^2-900y ~ dy } }\)

\(\displaystyle{ 392\pi \int_0^4{ -16y^3 + 320y^2 - 2100y +4500 ~ dy } }\)

\(\displaystyle{ 392\pi \left[ -4y^4 + \frac{320y^3}{3} -1050y^2 + 4500y \right]_0^4 }\)

\(\displaystyle{ 392\pi \frac{21008}{3} = \frac{82351360\pi}{3} }\)

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The vertical plate is submerged in water and has the indicated shape. Find the hydrostatic forces against one side of the plate.

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Problem Statement

The vertical plate is submerged in water and has the indicated shape. Find the hydrostatic forces against one side of the plate.

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Hint

[Question 69]

Problem Statement

The vertical plate is submerged in water and has the indicated shape. Find the hydrostatic forces against one side of the plate.

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Hint

[Question 69]

Solution

Although he doesn't work out the integral, we recommend that you do.

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A bottle of beer at room temperature of 75oF is placed in a refrigerator where the temperature is 36oF. After 40 minutes the beer has cooled to 60oF. Use Newton's Law of Cooling to find the temperature of the beer after another 40 minutes.

Problem Statement

A bottle of beer at room temperature of 75oF is placed in a refrigerator where the temperature is 36oF. After 40 minutes the beer has cooled to 60oF. Use Newton's Law of Cooling to find the temperature of the beer after another 40 minutes.

Hint

[Question 92]

Problem Statement

A bottle of beer at room temperature of 75oF is placed in a refrigerator where the temperature is 36oF. After 40 minutes the beer has cooled to 60oF. Use Newton's Law of Cooling to find the temperature of the beer after another 40 minutes.

Hint

[Question 92]

Solution

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Consider a population modeled by the differential equation \(\displaystyle{ \frac{dP}{dt} = 0.02P( 1 - P/4500 ) }\).
(a) For what values of \(P\) is the population increasing?
(b) For what values of \(P\) is the population decreasing?
(c) What are the equilibrium solutions?

Problem Statement

Consider a population modeled by the differential equation \(\displaystyle{ \frac{dP}{dt} = 0.02P( 1 - P/4500 ) }\).
(a) For what values of \(P\) is the population increasing?
(b) For what values of \(P\) is the population decreasing?
(c) What are the equilibrium solutions?

Hint

[Question 96]

Problem Statement

Consider a population modeled by the differential equation \(\displaystyle{ \frac{dP}{dt} = 0.02P( 1 - P/4500 ) }\).
(a) For what values of \(P\) is the population increasing?
(b) For what values of \(P\) is the population decreasing?
(c) What are the equilibrium solutions?

Hint

[Question 96]

Solution

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Basic Differential Equations

Which of the following is the slope field for \(\displaystyle{ dy/dx = 4-y^2 }\)

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Problem Statement

Which of the following is the slope field for \(\displaystyle{ dy/dx = 4-y^2 }\)

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Hint

[Question 12]

Problem Statement

Which of the following is the slope field for \(\displaystyle{ dy/dx = 4-y^2 }\)

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Hint

[Question 12]

Solution

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Is \(y=x^3\) a solution to \(x^2y'' + 6xy' + 6y = 0\)?

Problem Statement

Is \(y=x^3\) a solution to \(x^2y'' + 6xy' + 6y = 0\)?

Hint

[Question 72]

Problem Statement

Is \(y=x^3\) a solution to \(x^2y'' + 6xy' + 6y = 0\)?

Hint

[Question 72]

Solution

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For what values of \(r\) will the function \(y=e^{rx}\) satisfy \( 3y'' + 2y' - 8y = 0 \)?

Problem Statement

For what values of \(r\) will the function \(y=e^{rx}\) satisfy \( 3y'' + 2y' - 8y = 0 \)?

Hint

[Question 49]

Problem Statement

For what values of \(r\) will the function \(y=e^{rx}\) satisfy \( 3y'' + 2y' - 8y = 0 \)?

Hint

[Question 49]

Solution

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Solve \(dy/dx = xy^2\) when \(y(0)=4\).

Problem Statement

Solve \(dy/dx = xy^2\) when \(y(0)=4\).

Hint

[Question 8]

Problem Statement

Solve \(dy/dx = xy^2\) when \(y(0)=4\).

Hint

[Question 8]

Solution

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Solve \(\displaystyle{ dy/dx = 8x^3 e^{2y} }\) and \(y(1)=0\).

Problem Statement

Solve \(\displaystyle{ dy/dx = 8x^3 e^{2y} }\) and \(y(1)=0\).

Hint

[Question 56]

Problem Statement

Solve \(\displaystyle{ dy/dx = 8x^3 e^{2y} }\) and \(y(1)=0\).

Hint

[Question 56]

Solution

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Solve \(\displaystyle{ \frac{dy}{dx} = x\sqrt{1-y^2} }\), \(y(0) = 1/2\)

Problem Statement

Solve \(\displaystyle{ \frac{dy}{dx} = x\sqrt{1-y^2} }\), \(y(0) = 1/2\)

Hint

[Question 84]

Problem Statement

Solve \(\displaystyle{ \frac{dy}{dx} = x\sqrt{1-y^2} }\), \(y(0) = 1/2\)

Hint

[Question 84]

Solution

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Approximations

We do not cover approximations on this site. See his YouTube channel for discussion and other examples on how to solve these problems.

Compute \(M_3\) for \(\displaystyle{ \int_1^4{ \frac{2^x}{1+x} ~dx } }\)

Problem Statement

Compute \(M_3\) for \(\displaystyle{ \int_1^4{ \frac{2^x}{1+x} ~dx } }\)

Hint

[Question 26]

Problem Statement

Compute \(M_3\) for \(\displaystyle{ \int_1^4{ \frac{2^x}{1+x} ~dx } }\)

Hint

[Question 26]

Solution

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Compute \( S_4 \) for \(\displaystyle{ \int_0^8{ \sqrt{1+x^3} ~dx } }\)

Problem Statement

Compute \( S_4 \) for \(\displaystyle{ \int_0^8{ \sqrt{1+x^3} ~dx } }\)

Hint

[Question 67]

Problem Statement

Compute \( S_4 \) for \(\displaystyle{ \int_0^8{ \sqrt{1+x^3} ~dx } }\)

Hint

[Question 67]

Solution

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Let \(y(x)\) be the solution to the differential equation \( dy/dx = xy-2\) where \( y(1) = 3 \). Use Euler's Method with step size \(0.05\) to estimate \(y(1.2)\).

Problem Statement

Let \(y(x)\) be the solution to the differential equation \( dy/dx = xy-2\) where \( y(1) = 3 \). Use Euler's Method with step size \(0.05\) to estimate \(y(1.2)\).

Hint

[Question 68]

Problem Statement

Let \(y(x)\) be the solution to the differential equation \( dy/dx = xy-2\) where \( y(1) = 3 \). Use Euler's Method with step size \(0.05\) to estimate \(y(1.2)\).

Hint

[Question 68]

Solution

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Topics You Need To Understand For This Page

all single variable calculus topics

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100 calculus 2 problems (in ONE take)

Trig Formulas

The Unit Circle

The Unit Circle [wikipedia]

Basic Trig Identities

Set 1 - basic identities

\(\displaystyle{ \tan(t) = \frac{\sin(t)}{\cos(t)} }\)

\(\displaystyle{ \cot(t) = \frac{\cos(t)}{\sin(t)} }\)

\(\displaystyle{ \sec(t) = \frac{1}{\cos(t)} }\)

\(\displaystyle{ \csc(t) = \frac{1}{\sin(t)} }\)

Set 2 - squared identities

\( \sin^2t + \cos^2t = 1\)

\( 1 + \tan^2t = \sec^2t\)

\( 1 + \cot^2t = \csc^2t\)

Set 3 - double-angle formulas

\( \sin(2t) = 2\sin(t)\cos(t)\)

\(\displaystyle{ \cos(2t) = \cos^2(t) - \sin^2(t) }\)

Set 4 - half-angle formulas

\(\displaystyle{ \sin^2(t) = \frac{1-\cos(2t)}{2} }\)

\(\displaystyle{ \cos^2(t) = \frac{1+\cos(2t)}{2} }\)

Trig Derivatives

\(\displaystyle{ \frac{d[\sin(t)]}{dt} = \cos(t) }\)

 

\(\displaystyle{ \frac{d[\cos(t)]}{dt} = -\sin(t) }\)

\(\displaystyle{ \frac{d[\tan(t)]}{dt} = \sec^2(t) }\)

 

\(\displaystyle{ \frac{d[\cot(t)]}{dt} = -\csc^2(t) }\)

\(\displaystyle{ \frac{d[\sec(t)]}{dt} = \sec(t)\tan(t) }\)

 

\(\displaystyle{ \frac{d[\csc(t)]}{dt} = -\csc(t)\cot(t) }\)

Inverse Trig Derivatives

\(\displaystyle{ \frac{d[\arcsin(t)]}{dt} = \frac{1}{\sqrt{1-t^2}} }\)

 

\(\displaystyle{ \frac{d[\arccos(t)]}{dt} = -\frac{1}{\sqrt{1-t^2}} }\)

\(\displaystyle{ \frac{d[\arctan(t)]}{dt} = \frac{1}{1+t^2} }\)

 

\(\displaystyle{ \frac{d[\arccot(t)]}{dt} = -\frac{1}{1+t^2} }\)

\(\displaystyle{ \frac{d[\arcsec(t)]}{dt} = \frac{1}{\abs{t}\sqrt{t^2 -1}} }\)

 

\(\displaystyle{ \frac{d[\arccsc(t)]}{dt} = -\frac{1}{\abs{t}\sqrt{t^2 -1}} }\)

Trig Integrals

\(\int{\sin(x)~dx} = -\cos(x)+C\)

 

\(\int{\cos(x)~dx} = \sin(x)+C\)

\(\int{\tan(x)~dx} = -\ln\abs{\cos(x)}+C\)

 

\(\int{\cot(x)~dx} = \ln\abs{\sin(x)}+C\)

\(\int{\sec(x)~dx} = \) \( \ln\abs{\sec(x)+\tan(x)}+C\)

 

\(\int{\csc(x)~dx} = \) \( -\ln\abs{\csc(x)+\cot(x)}+C\)

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Topics Listed Alphabetically

Single Variable Calculus

Multi-Variable Calculus

Differential Equations

Precalculus

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