Exercises

Subsection 1.3.2 Exercises

Exercises — Stage 1

Questions 11 through 14 are meant to help reinforce key ideas in the Fundamental Theorem of Calculus and its proof.

So far, we have been able to guess many antiderivatives. Often, however, antiderivatives are very difficult to guess. In Questions 16 through 19, we will find some antiderivatives that might appear in a table of integrals. Coming up with the antiderivative might be quite difficult (strategies to do just that will form a large part of this semester), but verifying that your antiderivative is correct is as simple as differentiating.

1 (✳)

Suppose that $f(x)$ is a function and $F(x) = e^{(x^2-3)} + 1$ is an antiderivative of $f(x)\text{.}$ Evaluate the definite integral $\displaystyle\int_1^{\sqrt5} f(x)\,\dee{x}\text{.}$

2 (✳)

For the function $f(x) = x^3 -\sin 2x\text{,}$ find its antiderivative $F(x)$ that satisfies $F(0)=1\text{.}$

3 (✳)

Decide whether each of the following statements is true or false. Provide a brief justification.

If $f(x)$ is continuous on $[1, \pi]$ and differentiable on $(1,\pi)\text{,}$ then $\displaystyle\int_1^\pi f'(x)\,\dee{x} = f(\pi)-f(1)\text{.}$
$\displaystyle\int_{-1}^1 \frac{1}{x^2}\,\dee{x} = 0\text{.}$
If $f$ is continuous on $[a, b]$ then $\displaystyle\int_a^b xf(x)\,\dee{x} = x\int_a^b f(x)\,\dee{x} \text{.}$

4

True or false: an antiderivative of $\dfrac{1}{x^2}$ is $\log (x^2)$ (where by $\log x$ we mean logarithm base $e$).

5

True or false: an antiderivative of $\cos(e^x)$ is $\frac{\sin(e^x)}{e^x}\text{.}$

6

Suppose $F(x) = \displaystyle\int_7^x \sin(t^2)\dee{t}\text{.}$ What is the instantaneous rate of change of $F(x)$ with respect to $x\text{?}$

7

Suppose $F(x) = \displaystyle\int_{2}^x e^{1/t}\dee{t}\text{.}$ What is the slope of the tangent line to $y=F(x)$ when $x=3\text{?}$

8

Suppose $F'(x)=f(x)\text{.}$ Give two different antiderivatives of $f(x)\text{.}$

9

In Question 1.1.8.45, Section 1.1, we found that

\begin{equation*} \int_0^a\sqrt{1-x^2}\dee{x}=\frac{\pi}{4} - \frac{1}{2}\arccos(a)+\frac{1}{2}a\sqrt{1-a^2}. \end{equation*}

Verify that $\displaystyle\diff{}{a}\left\{\frac{\pi}{4} - \frac{1}{2}\arccos(a)+\frac{1}{2}a\sqrt{1-a^2}\right\} = \sqrt{1-a^2}\text{.}$
Find a function $F(x)$ that satisfies $F'(x) = \sqrt{1-x^2}$ and $F(0)=\pi\text{.}$

10

Evaluate the following integrals using the Fundamental Theorem of Calculus Part 2, or explain why it does not apply.

$\displaystyle\int_{-\pi}^\pi \cos x \dee{x}\text{.}$
$\displaystyle\int_{-\pi}^\pi \sec^2 x \dee{x}\text{.}$
$\displaystyle\int_{-2}^0 \frac{1}{x+1}\dee{x}\text{.}$

11

As in the proof of the Fundamental Theorem of Calculus, let $F(x) = \int_{a}^x f(t)\dee{t}\text{.}$ In the diagram below, shade the area corresponding to $F(x+h)-F(x)\text{.}$

12

Let $F(x) = \displaystyle\int_0^x f(t)\dee{t}\text{,}$ where $f(t)$ is shown in the graph below, and $0 \leq x \leq 4\text{.}$

Is $F(0)$ positive, negative, or zero?
Where is $F(x)$ increasing and where is it decreasing?

13

Let $G(x) = \displaystyle\int_x^0 f(t)\dee{t}\text{,}$ where $f(t)$ is shown in the graph below, and $0 \leq x \leq 4\text{.}$

Is $G(0)$ positive, negative, or zero?
Where is $G(x)$ increasing and where is it decreasing?

14

Let $F(x) = \displaystyle\int_a^x t\dee{t}\text{.}$ Using the definition of the derivative, find $F'(x)\text{.}$

15

Give a continuous function $f(x)$ so that $F(x) = \displaystyle\int_0^x f(t)\dee{t}$ is a constant.

16

Evaluate and simplify $\diff{}{x}\{x\log(ax)-x\}\text{,}$ where $a$ is some constant and $\log(x)$ is the logarithm base $e\text{.}$ What antiderivative does this tell you?

17

Evaluate and simplify $\diff{}{x}\{e^x\left(x^3-3x^2+6x-6\right)\}\text{.}$ What antiderivative does this tell you?

18

Evaluate and simplify $\diff{}{x}\left\{\log\left|x+\sqrt{x^2+a^2}\right|\right\}\text{,}$ where $a$ is some constant. What antiderivative does this tell you?

19

Evaluate and simplify $\displaystyle\diff{}{x}\left\{\sqrt{x(a+x)}-a\log\left(\sqrt{x}+\sqrt{a+x}\right)\right\}\text{,}$ where $a$ is some constant. What antiderivative does this tell you?

Exercises — Stage 2

20 (✳)

Evaluate $\displaystyle\int_0^2 \big(x^3+\sin x)\,\dee{x}\text{.}$

21 (✳)

Evaluate $\displaystyle\int_1^2 \frac{x^2+2}{x^2}\,\dee{x}\text{.}$

22

Evaluate $\displaystyle\int \dfrac{1}{1+25x^2}\dee{x}\text{.}$

23

Evaluate $\displaystyle\int \dfrac{1}{\sqrt{2-x^2}}\dee{x}\text{.}$

24

Evaluate $\displaystyle\int \tan^2 x \dee{x}\text{.}$

25

Evaluate $\displaystyle\int 3 \sin x \cos x \dee{x}\text{.}$

26

Evaluate $\displaystyle\int \cos^2 x \dee{x}\text{.}$

27 (✳)

\begin{equation*} F(x)=\int_0^x \log(2+\sin t)\,\dee{t}\quad\text{and}\quad G(y)=\int^0_y \log(2+\sin t)\,\dee{t} \end{equation*}

find $F'\big(\frac{\pi}{2}\big)$ and $G'\big(\frac{\pi}{2}\big)\text{.}$

28 (✳)

Let $f(x)=\displaystyle\int_1^x 100(t^2-3t+2)e^{-t^2}\dee{t}\text{.}$ Find the interval(s) on which $f$ is increasing.

29 (✳)

If $F(x)={\displaystyle\int_0^{\cos x} \frac{1}{t^3+6}\,\dee{t}}\text{,}$ find $F'(x)\text{.}$

30 (✳)

Compute $f'(x)$ where $f(x)= \displaystyle\int_0^{1+x^4}e^{t^2}\dee{t}\text{.}$

31 (✳)

Evaluate $\displaystyle\diff{}{x}\left\{\int_0^{\sin x}(t^6+8)\dee{t}\right\}\text{.}$

32 (✳)

Let $F(x)= \displaystyle\int_0^{x^3}e^{-t}\sin\left(\frac{\pi t}{2}\right)\,\dee{t}\text{.}$ Calculate $F'(1)\text{.}$

33 (✳)

Find $\displaystyle \diff{}{u} \left\{ \int_{\cos u}^0 \frac{\dee{t}}{1+t^3} \right\}\text{.}$

34 (✳)

Find $f(x)$ if $x^2=1+\displaystyle\int_1^x f(t)\dee{t}\text{.}$

35 (✳)

If $x \sin(\pi x) = \displaystyle\int_0^x f(t)\, \dee{t}$ where $f$ is a continuous function, find $f(4)\text{.}$

36 (✳)

Consider the function $\displaystyle F(x)=\int_0^{x^2} e^{-t}\,\dee{t} +\int_{-x}^0 e^{-t^2}\,\dee{t}\text{.}$

Find $F'(x)\text{.}$
Find the value of $x$ for which $F(x)$ takes its minimum value.

37 (✳)

If $F(x)$ is defined by $\displaystyle F(x) = \int_{x^4-x^3}^x e^{\sin t}\,\dee{t}\text{,}$ find $F'(x)\text{.}$

38 (✳)

Evaluate $\displaystyle \diff{}{x}\bigg\{\int_{x^5}^{-x^2} \cos\big(e^t\big)\,\dee{t} \bigg\}\text{.}$

39 (✳)

Differentiate $\displaystyle \int_x^{e^x} \sqrt{\sin t}\,\dee{t}$ for $0\lt x\lt \log \pi\text{.}$

40 (✳)

Evaluate $\displaystyle \int_1^5 f(x)\,\dee{x}\text{,}$ where $\displaystyle f(x)= \begin{cases} 3 &\text{ if $x\le 3$} \\ x &\text{ if $x\ge 3$} \end{cases}\text{.}$

Exercises — Stage 3

41 (✳)

If $f'(1)=2$ and $f'(2)=3\text{,}$ find $\displaystyle\int_1^2 f'(x) f''(x)\,\dee{x}\text{.}$

42 (✳)

A car traveling at $30\,\textrm{m}/\textrm{s}$ applies its brakes at time $t=0\text{,}$ its velocity (in $\textrm{m}/\textrm{s}$) decreasing according to the formula $v(t) = 30 - 10t\text{.}$ How far does the car go before it stops?

43 (✳)

Compute $f'(x)$ where $f(x)= \displaystyle\int_0^{2x-x^2}\log\big(1+e^t\big)\,\dee{t}\text{.}$ Does $f(x)$ have an absolute maximum? Explain.

44 (✳)

Find the minimum value of $\displaystyle\int_0^{x^2-2x}\frac{\dee{t}}{1+t^4}\text{.}$ Express your answer as an integral.

45 (✳)

Define the function $F(x)=\displaystyle\int_0^{x^2}\sin(\sqrt{t})\,\dee{t}$ on the interval $0 \lt x \lt 4\text{.}$ On this interval, where does $F(x)$ have a maximum?

46 (✳)

Evaluate $\lim\limits_{n\rightarrow\infty}\dfrac{\pi}{n}\displaystyle\sum\limits_{j=1}^n \sin\left(\frac{j\pi}{n}\right)$ by interpreting it as a limit of Riemann sums.

47 (✳)

Use Riemann sums to evaluate the limit $\displaystyle\lim_{n\rightarrow\infty}\frac{1}{n} \sum_{j=1}^n \frac{1}{1+\frac{j}{n}}\ .$

48

Below is the graph of $y=f(t)\text{,}$ $-5 \leq t \leq 5\text{.}$ Define $F(x) = \displaystyle\int_{0}^x f(t)\dee{t}$ for any $x$ in $[-5,5]\text{.}$ Sketch $F(x)\text{.}$

49 (✳)

Define $f(x)=x^3\displaystyle\int_{0}^{x^3+1} e^{t^3} \dee{t}\text{.}$

Find a formula for the derivative $f'(x)\text{.}$ (Your formula may include an integral sign.)
Find the equation of the tangent line to the graph of $y=f(x)$ at $x=-1\text{.}$

50

Two students calculate $\int f(x)\dee{x}$ for some function $f(x)\text{.}$

Student A calculates $\int f(x)\dee{x} = \tan^2 x + x + C$
Student B calculates $\int f(x)\dee{x} = \sec^2 x + x + C$
It is a fact that $\diff{}{x}\{\tan^2 x\} = f(x)-1$

Who ended up with the correct answer?

51

Let $F(x)=\displaystyle\int_0^x x^3 \sin(t)\dee{t}\text{.}$

Evaluate $F(3)\text{.}$
What is $F'(x)\text{?}$

52

Let $f(x)$ be an even function, defined everywhere, and let $F(x)$ be an antiderivative of $f(x)\text{.}$ Is $F(x)$ even, odd, or not necessarily either one? (You may use your answer from Section 1.2, Question 1.2.3.20. )