Calculus Single Variable

Deborah Hughes-Hallett, Andrew M. Gleason, William G. McCallum

Chapter 3

Short-Cuts to Differentiation - all with Video Answers

Educators

Section 1

Powers and Polynomials

Problem 1

Let $f(x)=7 .$ Using the definition of the derivative, show that $f^{\prime}(x)=0$ for all values of $x$.

Gregory Higby

Gregory Higby

Numerade Educator

Problem 2

Let $f(x)=17 x+11 .$ Use the definition of the derivative to calculate $f^{\prime}(x)$.

Gregory Higby

Numerade Educator

Problem 3

Determine if the derivative rules from this section apply. If they do, find the derivative. If they don't apply, indicate why.
$$y=3^{x}$$

Gregory Higby

Numerade Educator

Problem 4

Determine if the derivative rules from this section apply. If they do, find the derivative. If they don't apply, indicate why.
$$y=x^{3}$$

Gregory Higby

Numerade Educator

Problem 5

Determine if the derivative rules from this section apply. If they do, find the derivative. If they don't apply, indicate why.
$$y=x^{\pi}$$

Gregory Higby

Numerade Educator

Problem 6

Find the derivatives of the given functions. Assume that $a, b, c,$ and $k$ are constants.
$$y=x^{12}$$

Gregory Higby

Numerade Educator

Problem 7

Find the derivatives of the given functions. Assume that $a, b, c,$ and $k$ are constants.
$$y=x^{11}$$

Gregory Higby

Numerade Educator

Problem 8

Find the derivatives of the given functions. Assume that $a, b, c,$ and $k$ are constants.
$$y=-x^{-11}$$

Gregory Higby

Numerade Educator

Problem 9

Find the derivatives of the given functions. Assume that $a, b, c,$ and $k$ are constants.
$$y=x^{-12}$$

Gregory Higby

Numerade Educator

Problem 10

Find the derivatives of the given functions. Assume that $a, b, c,$ and $k$ are constants.
$$y=x^{3.2}$$

Gregory Higby

Numerade Educator

Problem 11

Find the derivatives of the given functions. Assume that $a, b, c,$ and $k$ are constants.
$$y=x^{-3 / 4}$$

Gregory Higby

Numerade Educator

Problem 12

Find the derivatives of the given functions. Assume that $a, b, c,$ and $k$ are constants.
$$y=x^{4 / 3}$$

Gregory Higby

Numerade Educator

Problem 13

Find the derivatives of the given functions. Assume that $a, b, c,$ and $k$ are constants.
$$y=x^{3 / 4}$$

Gregory Higby

Numerade Educator

Problem 14

Find the derivatives of the given functions. Assume that $a, b, c,$ and $k$ are constants.
$$y=x^{2}+5 x+7$$

Gregory Higby

Numerade Educator

Problem 15

Find the derivatives of the given functions. Assume that $a, b, c,$ and $k$ are constants.
$$f(t)=t^{3}-3 t^{2}+8 t-4$$

Gregory Higby

Numerade Educator

Problem 16

Find the derivatives of the given functions. Assume that $a, b, c,$ and $k$ are constants.
$$f(x)=\frac{1}{x^{4}}$$

Gregory Higby

Numerade Educator

Problem 17

Find the derivatives of the given functions. Assume that $a, b, c,$ and $k$ are constants.
$$g(t)=\frac{1}{t^{5}}$$

Gregory Higby

Numerade Educator

Problem 18

Find the derivatives of the given functions. Assume that $a, b, c,$ and $k$ are constants.
$$f(z)=-\frac{1}{z^{6.1}}$$

Gregory Higby

Numerade Educator

Problem 19

Find the derivatives of the given functions. Assume that $a, b, c,$ and $k$ are constants.
$$y=\frac{1}{r^{7 / 2}}$$

Gregory Higby

Numerade Educator

Problem 20

Find the derivatives of the given functions. Assume that $a, b, c,$ and $k$ are constants.
$$y=\sqrt{x}$$

Gregory Higby

Numerade Educator

Problem 21

Find the derivatives of the given functions. Assume that $a, b, c,$ and $k$ are constants.
$$f(x)=\sqrt[4]{x}$$

Gregory Higby

Numerade Educator

Problem 22

Find the derivatives of the given functions. Assume that $a, b, c,$ and $k$ are constants.
$$h(\theta)=\frac{1}{\sqrt[3]{\theta}}$$

Gregory Higby

Numerade Educator

Problem 23

Find the derivatives of the given functions. Assume that $a, b, c,$ and $k$ are constants.
$$f(x)=\sqrt{\frac{1}{x^{3}}}$$

Gregory Higby

Numerade Educator

Problem 24

Find the derivatives of the given functions. Assume that $a, b, c,$ and $k$ are constants.
$$h(x)=\ln e^{a x}$$

Gregory Higby

Numerade Educator

Problem 25

Find the derivatives of the given functions. Assume that $a, b, c,$ and $k$ are constants.
$$y=4 x^{3 / 2}-5 x^{1 / 2}$$

Gregory Higby

Numerade Educator

Problem 26

Find the derivatives of the given functions. Assume that $a, b, c,$ and $k$ are constants.
$$f(t)=3 t^{2}-4 t+1$$

Gregory Higby

Numerade Educator

Problem 27

Find the derivatives of the given functions. Assume that $a, b, c,$ and $k$ are constants.
$$y=17 x+24 x^{1 / 2}$$

Gregory Higby

Numerade Educator

Problem 28

Find the derivatives of the given functions. Assume that $a, b, c,$ and $k$ are constants.
$$y=z^{2}+\frac{1}{2 z}$$

Gregory Higby

Numerade Educator

Problem 29

Find the derivatives of the given functions. Assume that $a, b, c,$ and $k$ are constants.
$$f(x)=5 x^{4}+\frac{1}{x^{2}}$$

Gregory Higby

Numerade Educator

Problem 30

Find the derivatives of the given functions. Assume that $a, b, c,$ and $k$ are constants.
$$h(w)=-2 w^{-3}+3 \sqrt{w}$$

Gregory Higby

Numerade Educator

Problem 31

Find the derivatives of the given functions. Assume that $a, b, c,$ and $k$ are constants.
$$y=-3 x^{4}-4 x^{3}-6 x$$

Gregory Higby

Numerade Educator

Problem 32

Find the derivatives of the given functions. Assume that $a, b, c,$ and $k$ are constants.
$$y=3 t^{5}-5 \sqrt{t}+\frac{7}{t}$$

Gregory Higby

Numerade Educator

Problem 33

Find the derivatives of the given functions. Assume that $a, b, c,$ and $k$ are constants.
$$y=3 t^{2}+\frac{12}{\sqrt{t}}-\frac{1}{t^{2}}$$

Gregory Higby

Numerade Educator

Problem 34

Find the derivatives of the given functions. Assume that $a, b, c,$ and $k$ are constants.
$$y=\sqrt{x}(x+1)$$

Gregory Higby

Numerade Educator

Problem 35

Find the derivatives of the given functions. Assume that $a, b, c,$ and $k$ are constants.
$$y=t^{3 / 2}(2+\sqrt{t})$$

Gregory Higby

Numerade Educator

Problem 36

Find the derivatives of the given functions. Assume that $a, b, c,$ and $k$ are constants.
$$h(t)=\frac{3}{t}+\frac{4}{t^{2}}$$

Gregory Higby

Numerade Educator

Problem 37

Find the derivatives of the given functions. Assume that $a, b, c,$ and $k$ are constants.
$$h(\theta)=\theta\left(\theta^{-1 / 2}-\theta^{-2}\right)$$

Gregory Higby

Numerade Educator

Problem 38

Find the derivatives of the given functions. Assume that $a, b, c,$ and $k$ are constants.
$$y=\frac{x^{2}+1}{x}$$

Gregory Higby

Numerade Educator

Problem 39

Find the derivatives of the given functions. Assume that $a, b, c,$ and $k$ are constants.
$$f(z)=\frac{z^{2}+1}{3 z}$$

Gregory Higby

Numerade Educator

Problem 40

Find the derivatives of the given functions. Assume that $a, b, c,$ and $k$ are constants.
$$g(x)=\frac{x^{2}+\sqrt{x}+1}{x^{3 / 2}}$$

Gregory Higby

Numerade Educator

Problem 41

Find the derivatives of the given functions. Assume that $a, b, c,$ and $k$ are constants.
$$y=\frac{\theta-1}{\sqrt{\theta}}$$

Gregory Higby

Numerade Educator

Problem 42

Find the derivatives of the given functions. Assume that $a, b, c,$ and $k$ are constants.
$$g(t)=\frac{\sqrt{t}(1+t)}{t^{2}}$$

Gregory Higby

Numerade Educator

Problem 43

Find the derivatives of the given functions. Assume that $a, b, c,$ and $k$ are constants.
$$j(x)=\frac{x^{3}}{a}+\frac{a}{b} x^{2}-c x$$

Gregory Higby

Numerade Educator

Problem 44

Find the derivatives of the given functions. Assume that $a, b, c,$ and $k$ are constants.
$$f(x)=\frac{a x+b}{x}$$

Gregory Higby

Numerade Educator

Problem 45

Find the derivatives of the given functions. Assume that $a, b, c,$ and $k$ are constants.
$$h(x)=\frac{a x+b}{c}$$

Gregory Higby

Numerade Educator

Problem 46

Find the derivatives of the given functions. Assume that $a, b, c,$ and $k$ are constants.
$$V=\frac{4}{3} \pi r^{2} b$$

Gregory Higby

Numerade Educator

Problem 47

Find the derivatives of the given functions. Assume that $a, b, c,$ and $k$ are constants.
$$w=3 a b^{2} q$$

Gregory Higby

Numerade Educator

Problem 48

Find the derivatives of the given functions. Assume that $a, b, c,$ and $k$ are constants.
$$y=a x^{2}+b x+c$$

Gregory Higby

Numerade Educator

Problem 49

Find the derivatives of the given functions. Assume that $a, b, c,$ and $k$ are constants.
$$P=a+b \sqrt{t}$$

Gregory Higby

Numerade Educator

Problem 50

Use the tangent line approximation.
Given $f(4)=5, f^{\prime}(4)=7,$ approximate $f(4.02)$.

Gregory Higby

Numerade Educator

Problem 51

Use the tangent line approximation.
Given $f(4)=5, f^{\prime}(4)=7,$ approximate $f(3.92)$.

Gregory Higby

Numerade Educator

Problem 52

Use the tangent line approximation.
Given $f(5)=3, f^{\prime}(5)=-2,$ approximate $f(5.03)$.

Gregory Higby

Numerade Educator

Problem 53

Use the tangent line approximation.
Given $f(2)=-4, f^{\prime}(2)=-3,$ approximate $f(1.95)$.

Gregory Higby

Numerade Educator

Problem 54

Use the tangent line approximation.
Given $f(-3)=-4, f^{\prime}(-3)=2,$ approximate $f(-2.99) .$

Gregory Higby

Numerade Educator

Problem 55

Use the tangent line approximation.
Given $f(3)=-4, f^{\prime}(3)=-2$ approximate $f(2.99)$.

Gregory Higby

Numerade Educator

Problem 56

Use the tangent line approximation.
Given $f(x)=x^{4}-x^{2}+3$ approximate $f(1.04)$.

Gregory Higby

Numerade Educator

Problem 57

Use the tangent line approximation.
Given $f(x)=x^{3}+x^{2}-6,$ approximate $f(0.97)$.

Gregory Higby

Numerade Educator

Problem 58

If $f(x)=6 x^{4}-2 x+7,$ find $f^{\prime}(x), f^{\prime \prime}(x),$ and $f^{\prime \prime \prime}(x)$.

Gregory Higby

Numerade Educator

Problem 59

(a) Let $f(x)=x^{7} .$ Find $f^{\prime}(x), f^{\prime \prime}(x),$ and $f^{\prime \prime \prime}(x)$.
(b) Find the smallest $n$ such that $f^{(n)}(x)=0$.

Gregory Higby

Numerade Educator

Problem 60

If $p(t)=t^{3}+2 t^{2}-t+4,$ find $d^{2} p / d t^{2}$ and $d^{3} p / d t^{3}$.

Gregory Higby

Numerade Educator

Problem 61

If $w(x)=\sqrt{x}+(1 / \sqrt{x}),$ find $d^{2} w / d x^{2}$ and $d^{3} w / d x^{3}$.

Gregory Higby

Numerade Educator

Problem 62

Determine if the derivative rules from this section apply. If they do, find the derivative. If they don't apply, indicate why.
$$y=(x+3)^{1 / 2}$$

Gregory Higby

Numerade Educator

Problem 63

Determine if the derivative rules from this section apply. If they do, find the derivative. If they don't apply, indicate why.
$$y=\pi^{x}$$

Gregory Higby

Numerade Educator

Problem 64

Determine if the derivative rules from this section apply. If they do, find the derivative. If they don't apply, indicate why.
$$g(x)=x^{\pi}-x^{-\pi}$$

Gregory Higby

Numerade Educator

Problem 65

Determine if the derivative rules from this section apply. If they do, find the derivative. If they don't apply, indicate why.
$$y=3 x^{2}+4$$

Gregory Higby

Numerade Educator

Problem 66

Determine if the derivative rules from this section apply. If they do, find the derivative. If they don't apply, indicate why.
$$y=\frac{1}{3 x^{2}+4}$$

Gregory Higby

Numerade Educator

Problem 67

Determine if the derivative rules from this section apply. If they do, find the derivative. If they don't apply, indicate why.
$$y=\frac{1}{3 z^{2}}+\frac{1}{4}$$

Gregory Higby

Numerade Educator

Problem 68

If $f(x)=(3 x+8)(2 x-5),$ find $f^{\prime}(x)$ and $f^{\prime \prime}(x)$.

Will Erickson

Numerade Educator

Problem 69

Find the equation of the line tangent to the graph of $f$ at $(1,1),$ where $f$ is given by $f(x)=2 x^{3}-2 x^{2}+1$.

Gregory Higby

Numerade Educator

Problem 70

Find the equation of the line tangent to $y=x^{2}+3 x-5$ at $x=2$.

Gregory Higby

Numerade Educator

Problem 71

Find the equation of the line tangent to $f(x)$ at $x=2$, if $$f(x)=\frac{x^{3}}{2}-\frac{4}{3 x}$$.

Gregory Higby

Numerade Educator

Problem 72

(a) Find the equation of the tangent line to $f(x)=x^{3}$ at the point where $x=2$
(b) Graph the tangent line and the function on the same axes. If the tangent line is used to estimate values of the function near $x=2,$ will the estimates be overestimates or underestimates?

Yaw Asomani

Numerade Educator

Problem 73

(a) Use Figure 3.8 to rank the quantities $f^{\prime}(-1), f^{\prime}(0), f^{\prime}(1), f^{\prime}(4)$ from smallest to largest.
(b) Confirm your answer by calculating the quantities using the formula, $f(x)=x^{3}-3 x^{2}+2 x+10$.

Kian Manafi

Numerade Educator

Problem 74

The graph of $y=x^{3}-9 x^{2}-16 x+1$ has a slope of 5 at two points. Find the coordinates of the points.

Gregory Higby

Numerade Educator

Problem 75

On what intervals is the graph of $f(x)=x^{4}-4 x^{3}$ both decreasing and concave up?

Gregory Higby

Numerade Educator

Problem 76

For what values of $x$ is the function $y=x^{5}-5 x$ both increasing and concave up?

Gregory Higby

Numerade Educator

Problem 77

(a) Find the eighth derivative of $f(x)=x^{7}+5 x^{5}-$
$4 x^{3}+6 x-7 .$ Think ahead!
(The $n^{\text {th }}$ derivative, $f^{(n)}(x),$ is the result of differentiating $f(x) n \text { times. })$
(b) Find the seventh derivative of $f(x)$.

Gregory Higby

Numerade Educator

Problem 78

For the functions in Problems:
(a) Find the derivative at $x=-1$
(b) Find the second derivative at $x=-1$
(c) Use your answers to parts (a) and (b) to match the function to one of the graphs in Figure $3.9,$ each of which is shown centered on the point (-1,-1) .
$$k(x)=x^{3}-x-1$$

Kian Manafi

Numerade Educator

Problem 79

For the functions in Problems:
(a) Find the derivative at $x=-1$
(b) Find the second derivative at $x=-1$
(c) Use your answers to parts (a) and (b) to match the function to one of the graphs in Figure $3.9,$ each of which is shown centered on the point (-1,-1) .
$$f(x)=2 x^{3}+3 x^{2}-2$$

Kian Manafi

Numerade Educator

Problem 80

For the functions in Problems:
(a) Find the derivative at $x=-1$
(b) Find the second derivative at $x=-1$
(c) Use your answers to parts (a) and (b) to match the function to one of the graphs in Figure $3.9,$ each of which is shown centered on the point (-1,-1) .
$$g(x)=x^{4}-x^{2}-2 x-3$$

Kian Manafi

Numerade Educator

Problem 81

For the functions in Problems:
(a) Find the derivative at $x=-1$
(b) Find the second derivative at $x=-1$
(c) Use your answers to parts (a) and (b) to match the function to one of the graphs in Figure $3.9,$ each of which is shown centered on the point (-1,-1) .
$$h(x)=2 x^{4}+8 x^{3}+15 x^{2}+14 x+4$$

Kian Manafi

Numerade Educator

Problem 82

With $t$ in years since 2016 , the height of a sand dune (in centimeters is $f(t)=700-3 t^{2} .$ Find $f(5)$ and $f^{\prime}(5)$ Using units, explain what each means in terms of the sand dune.

Gregory Higby

Numerade Educator

Problem 83

A rubber balloon contains neon. As the air pressure, $P$ (in atmospheres), outside the balloon increases, the volume of gas, $V$ (in liters), in the balloon decreases according to $V=f(P)=25 / P$
(a) Evaluate and interpret $f(2)$, including units.
(b) Evaluate and interpret $f^{\prime}(2)$, including units.
(c) Assuming that the pressure increases at a constant rate, does the volume of the balloon decrease faster when the pressure is 1 atmosphere or when the pressure is 2 atmospheres? Justify your answer.

Kian Manafi

Numerade Educator

Problem 84

A ball is dropped from the top of the Empire State building to the ground below. The height, $y,$ of the ball above the ground (in feet) is given as a function of time, $t$ (in seconds), by $$y=1250-16 t^{2}$$.
(a) Find the velocity of the ball at time $t .$ What is the sign of the velocity? Why is this to be expected?
(b) Show that the acceleration of the ball is a constant. What are the value and sign of this constant?
(c) When does the ball hit the ground, and how fast is it going at that time? Give your answer in feet per second and in miles per hour $(1 \mathrm{ft} / \mathrm{sec}=15 / 22$ $\mathrm{mph})$.

Kian Manafi

Numerade Educator

Problem 85

At a time $t$ seconds after it is thrown up in the air, a tomato is at a height of $f(t)=-4.9 t^{2}+25 t+3$ meters.
(a) What is the average velocity of the tomato during the first 2 seconds? Give units.
(b) Find (exactly) the instantaneous velocity of the tomato at $t=2$. Give units.
(c) What is the acceleration at $t=2 ?$
(d) How high does the tomato go?
(e) How long is the tomato in the air?

Kian Manafi

Numerade Educator

Problem 86

Let $f(t)$ and $g(t)$ give, respectively, the amount of water (in acre-feet) in two different reservoirs on day $t$ Suppose that $f(0)=2000, g(0)=1500$ and that $f^{\prime}(0)=11, g^{\prime}(0)=13.5 .$ Let $h(t)=f(t)-g(t)$
(a) Evaluate $h(0)$ and $h^{\prime}(0) .$ What do these quantities tell you about the reservoir?
(b) Assume $h^{\prime}$ is constant for $0 \leq t \leq 250 .$ Does $h$ have any zeros? What does this tell you about the reservoirs?

Kian Manafi

Numerade Educator

Problem 87

A jökulhlaup is the rapid draining of a glacial lake when an ice dam bursts. The maximum outflow rate, $Q\left(\text { in } \mathrm{m}^{3} / \mathrm{sec}\right),$ during a jökulhlaup is given $^{2}$ in terms of its volume, $v$ (in $\mathrm{km}^{3}$ ), before the dam-break by $Q=7700 v^{0.67}$
(a) Find $\frac{d Q}{d v}$
(b) Evaluate $\left.\frac{d Q}{d v}\right|_{v=0,1}$. Include units. What does this derivative mean for glacial lakes?

Kian Manafi

Numerade Educator

Problem 88

(a) For $y=k x^{n},$ show that near any point $x=a,$ we have $\Delta y / y \approx n \Delta x / a$
(b) Interpret this relationship in terms of percent change in $y$ and $x$.

Kaitlin Coad

Numerade Educator

Problem 89

Use the fact that for a power function $y=k x^{n},$ for small changes, the percent change in output $y$ is approximately $n$ times the percent change in input $x$. (See Problem 88.)
An error of $5 \%$ in the measurement of the radius $r$ of a
circle leads to what percent error in the area $A ?$

Kaitlin Coad

Numerade Educator

Problem 90

Use the fact that for a power function $y=k x^{n},$ for small changes, the percent change in output $y$ is approximately $n$ times the percent change in input $x$. (See Problem 88.)
If we want to measure the volume $V$ of a sphere accurate to $3 \%,$ how accurately must we measure the radius $r ?$

Kaitlin Coad

Numerade Educator

Problem 91

Use the fact that for a power function $y=k x^{n},$ for small changes, the percent change in output $y$ is approximately $n$ times the percent change in input $x$. (See Problem 88.)
The stopping distance $s,$ in feet, of a car traveling $v$ mph is $s=v^{2} / 20 .$ An increase of $10 \%$ in the speed of the car leads to what percent increase in the stopping distance?

Kaitlin Coad

Numerade Educator

Problem 92

Use the fact that for a power function $y=k x^{n},$ for small changes, the percent change in output $y$ is approximately $n$ times the percent change in input $x$. (See Problem 88.)
The average wind speed in Hyannis, MA, in August is 9 mph. In nearby Nantucket, it is 10 mph. What percent increase in power, $P,$ is there for a wind turbine in Nantucket compared to Hyannis in August? Assume $P=k v^{3},$ where $v$ is wind speed.

Kaitlin Coad

Numerade Educator

Problem 93

The depth, $h$ (in $\mathrm{mm}$ ), of the water runoff down a slope during steady rain $^{3}$ is a function of the distance, $x$ (in meters), from the top of the slope, $h=f(x)=0.07 x^{2 / 3}$
(a) Find $f^{\prime}(x)$.
(b) Find $f^{\prime}(30) .$ Include units.
(c) Explain how you can use your answer to part (b) to estimate the difference in runoff depths between a point 30 meters down the slope and a point 6 meters farther down.

Kian Manafi

Numerade Educator

Problem 94

If $M$ is the mass of the earth and $G$ is a constant, the acceleration due to gravity, $g,$ at a distance $r$ from the center of the earth is given by $$g=\frac{G M}{r^{2}}$$,
(a) Find $d g / d r$
(b) What is the practical interpretation (in terms of acceleration) of $d g / d r ?$ Why would you expect it to be negative?
(c) You are told that $M=6 \cdot 10^{24}$ and $G=6.67 \cdot 10^{-20}$ where $M$ is in kilograms, $r$ in kilometers, and $g$ in $\mathrm{km} / \mathrm{sec}^{2} .$ What is the value of $d g / d r$ at the surface of the earth $(r=6400 \mathrm{km}) ?$ Include units.
(d) What does this tell you about whether or not it is reasonable to assume $g$ is constant near the surface of the earth?

Kaitlin Coad

Numerade Educator

Problem 95

The period, $T,$ of a pendulum is given in terms of its length, $l,$ by $$T=2 \pi \sqrt{\frac{l}{g}}$$, where $g$ is the acceleration due to gravity (a constant).
(a) Find $d T / d l$
(b) What is the sign of $d T / d l ?$ What does this tell you about the period of pendulums?

Kian Manafi

Numerade Educator

Problem 96

(a) Use the formula for the area of a circle of radius $r$, $A=\pi r^{2},$ to find $d A / d r$
(b) The result from part (a) should look familiar. What does $d A / d r$ represent geometrically?
(c) Use the difference quotient to explain the observation you made in part (b).

Adam Dehollander

Adam Dehollander

Numerade Educator

Problem 97

Show that for any power function $f(x)=x^{n},$ we have $f^{\prime}(1)=n$.

Yaw Asomani

Numerade Educator

Problem 98

Suppose $W$ is proportional to $r^{3} .$ The derivative $d W / d r$ is proportional to what power of $r ?$

Yaw Asomani

Numerade Educator

Problem 99

Given a power function of the form $f(x)=a x^{n},$ with $f^{\prime}(2)=3$ and $f^{\prime}(4)=24,$ find $n$ and $a$.

Gregory Higby

Numerade Educator

Problem 100

(a) Find the value of $a$ making $f(x)$ continuous at $x=1:$ f(x)=\left\{\begin{array}{ll}
a x & 0 \leq x \leq 1 \\x^{2}+3 & 1<x \leq 2\end{array}\right.
(b) With the value of $a$ you found in part (a), does $f(x)$ have a derivative at every point in $0<x<2 ?$ Explain.

Gregory Higby

Numerade Educator

Problem 101

Let $f(x)=x^{3}+3 x^{2}-2 x+1$ and $g(x)=x^{3}+3 x^{2}-2 x-4$
(a) Show that the derivatives of $f(x)$ and $g(x)$ are the
same.
(b) Use the graphs of $f(x)$ and $g(x)$ to explain why their derivatives are the same.
(c) Are there other functions which share the same derivative as $f(x)$ and $g(x) ?$

Gregory Higby

Numerade Educator

Problem 102

Let $f(x)=x^{4}-3 x^{2}+1$
(a) Show that $f(x)$ is an even function.
(b) Show that $f^{\prime}(x)$ is an odd function.
(c) If $g(x)$ is a polynomial that is an even function, will its derivative always be an odd function?

Kian Manafi

Numerade Educator

Problem 103

Explain what is wrong with the statement.
The only function that has derivative $2 x$ is $x^{2}$.

Gregory Higby

Numerade Educator

Problem 104

Explain what is wrong with the statement.
The derivative of $f(x)=1 / x^{2}$ is $f^{\prime}(x)=1 /(2 x)$.

Gregory Higby

Numerade Educator

Problem 105

Give an example of:
Two functions $f(x)$ and $g(x)$ such that $$\frac{d}{d x}(f(x)+g(x))=2 x+3$$.

Gregory Higby

Numerade Educator

Problem 106

Give an example of:
A function whose derivative is $g^{\prime}(x)=2 x$ and whose graph has no $x$ -intercepts.

Gregory Higby

Numerade Educator

Problem 107

Give an example of:
A function which has second derivative equal to 6 everywhere.

Gregory Higby

Numerade Educator

Problem 108

Is the statement true or false? Give an explanation for your answer.
The derivative of a polynomial is always a polynomial.

Kian Manafi

Numerade Educator

Problem 109

Is the statement true or false? Give an explanation for your answer.
The derivative of $\pi / x^{2}$ is $-\pi / x$.

Gregory Higby

Numerade Educator

Problem 110

Is the statement true or false? Give an explanation for your answer.
If $f^{\prime}(2)=3.1$ and $g^{\prime}(2)=7.3,$ then the graph of $f(x)+g(x)$ has slope 10.4 at $x=2$.

Gregory Higby

Numerade Educator

Problem 111

Is the statement true or false? You are told that $f^{\prime \prime}$ and $g^{\prime \prime}$ exist and that $f$ and $g$ are concave up for all $x .$ If a statement is true, explain how you know. If a statement is false, give a counterexample.
$f(x)+g(x)$ is concave up for all $x$.

Gregory Higby

Numerade Educator

Problem 112

Is the statement true or false? You are told that $f^{\prime \prime}$ and $g^{\prime \prime}$ exist and that $f$ and $g$ are concave up for all $x .$ If a statement is true, explain how you know. If a statement is false, give a counterexample.
$f(x)-g(x)$ cannot be concave up for all $x$.

Gregory Higby

Numerade Educator