Calculus A New Horizon

Howard Anton

Chapter 3

The Derivative - all with Video Answers

Educators

Section 1

Tangent Lines and Rates of Change

Problem 1

A function $y=f(x)$ and values of $x_{0}$ and $x_{1}$ are given.
(a) Find the average rate of change of $y$ with respect to $x$ over the interval $\left[x_{0}, x_{1}\right]$
(b) Find the instantaneous rate of change of $y$ with respect to $x$ at the given value of $x_{0}$.
(c) Find the instantaneous rate of change of $y$ with respect to $x$ at a general point $x_{0}$.
(d) Sketch the graph of $y=f(x)$ together with the secant and tangent lines whose slopes are given by the results in parts (a) and (b).
$$y=\frac{1}{2} x^{2} ; x_{0}=3, x_{1}=4$$

Carson Merrill

Carson Merrill

Numerade Educator

Problem 2

A function $y=f(x)$ and values of $x_{0}$ and $x_{1}$ are given.
(a) Find the average rate of change of $y$ with respect to $x$ over the interval $\left[x_{0}, x_{1}\right]$
(b) Find the instantaneous rate of change of $y$ with respect to $x$ at the given value of $x_{0}$.
(c) Find the instantaneous rate of change of $y$ with respect to $x$ at a general point $x_{0}$.
(d) Sketch the graph of $y=f(x)$ together with the secant and tangent lines whose slopes are given by the results in parts (a) and (b).
$$y=x^{3} ; x_{0}=1, x_{1}=2$$

Carson Merrill

Numerade Educator

Problem 3

A function $y=f(x)$ and values of $x_{0}$ and $x_{1}$ are given.
(a) Find the average rate of change of $y$ with respect to $x$ over the interval $\left[x_{0}, x_{1}\right]$
(b) Find the instantaneous rate of change of $y$ with respect to $x$ at the given value of $x_{0}$.
(c) Find the instantaneous rate of change of $y$ with respect to $x$ at a general point $x_{0}$.
(d) Sketch the graph of $y=f(x)$ together with the secant and tangent lines whose slopes are given by the results in parts (a) and (b).
$$y=1 / x ; x_{0}=2, x_{1}=3$$

Carson Merrill

Numerade Educator

Problem 4

A function $y=f(x)$ and values of $x_{0}$ and $x_{1}$ are given.
(a) Find the average rate of change of $y$ with respect to $x$ over the interval $\left[x_{0}, x_{1}\right]$
(b) Find the instantaneous rate of change of $y$ with respect to $x$ at the given value of $x_{0}$.
(c) Find the instantaneous rate of change of $y$ with respect to $x$ at a general point $x_{0}$.
(d) Sketch the graph of $y=f(x)$ together with the secant and tangent lines whose slopes are given by the results in parts (a) and (b).
$$y=1 / x^{2} ; x_{0}=1, x_{1}=2$$

Carson Merrill

Numerade Educator

Problem 5

A function $f$ and a value of $x_{0}$ are given.
(a) Find the slope of the tangent to the graph of $f$ at a general point $x_{0}$
(b) Use the result in part (a) to find the slope of the tangent line at the given value of $x_{0}$
$$f(x)=x^{2}+1 ; x_{0}=2$$

Carson Merrill

Numerade Educator

Problem 6

A function $f$ and a value of $x_{0}$ are given.
(a) Find the slope of the tangent to the graph of $f$ at a general point $x_{0}$
(b) Use the result in part (a) to find the slope of the tangent line at the given value of $x_{0}$
$$f(x)=x^{2}+3 x+2 ; x_{0}=2$$

Carson Merrill

Numerade Educator

Problem 7

A function $f$ and a value of $x_{0}$ are given.
(a) Find the slope of the tangent to the graph of $f$ at a general point $x_{0}$
(b) Use the result in part (a) to find the slope of the tangent line at the given value of $x_{0}$
$$f(x)=\sqrt{x} ; x_{0}=1$$

Carson Merrill

Numerade Educator

Problem 8

A function $f$ and a value of $x_{0}$ are given.
(a) Find the slope of the tangent to the graph of $f$ at a general point $x_{0}$
(b) Use the result in part (a) to find the slope of the tangent line at the given value of $x_{0}$
$$f(x)=1 / \sqrt{x} ; x_{0}=4$$

Carson Merrill

Numerade Educator

Problem 9

The accompanying figure shows the position versus time curve for an elevator that moves upward a distance of $60 \mathrm{m}$ and then discharges its passengers. (FIGURE CAN'T COPY)
(a) Estimate the instantaneous velocity of the elevator at $t=10 \mathrm{s}$
(b) Sketch a velocity versus time curve for the motion of the elevator for $0 \leq t \leq 20$

Suman Saurav Thakur

Suman Saurav Thakur

Numerade Educator

Problem 10

The accompanying figure shows the position versus time curve for a certain particle moving along a straight line. Estimate each of the following from the graph: (FIGURE CAN'T COPY)
(a) the average velocity over the interval $0 \leq t \leq 3$
(b) the values of $t$ at which the instantaneous velocity is zero
(c) the values of $t$ at which the instantaneous velocity is either a maximum or a minimum
(d) the instantaneous velocity when $t=3 \mathrm{s}$

Carson Merrill

Numerade Educator

Problem 11

The accompanying figure shows the position versus time curve for a certain particle moving on a straight line. (FIGURE CAN'T COPY)
(a) Is the particle moving faster at time $t_{0}$ or time $t_{2} ?$ Explain.
(b) At the origin, the tangent is horizontal. What does this tell us about the initial velocity of the particle?
(c) Is the particle speeding up or slowing down in the interval $\left[t_{0}, t_{1}\right] ?$ Explain.
(d) Is the particle speeding up or slowing down in the interval $\left[t_{1}, t_{2}\right] ?$ Explain.

Carson Merrill

Numerade Educator

Problem 12

An automobile, initially at rest, begins to move along a straight track. The velocity increases steadily until suddenly the driver sees a concrete barrier in the road and applies the brakes sharply at time $t_{0} .$ The car decelerates rapidly, but it is too late the car crashes into the barrier at time $t_{1}$ and instantaneously comes to rest. Sketch a position versus time curve that might represent the motion of the car.

Carson Merrill

Numerade Educator

Problem 13

If a particle moves at constant velocity, what can you say about its position versus time curve?

Joseph Lentino

Numerade Educator

Problem 14

The accompanying figure shows the position versus time curves of four different particles moving on a straight line. For each particle, determine whether its instantaneous velocity is increasing or decreasing with time. (FIGURE CAN'T COPY)

Joseph Lentino

Numerade Educator

Problem 15

Suppose that the outside temperature versus time curve over a 24 -hour period is as shown in the accompanying figure. (FIGURE CAN'T COPY)
(a) Estimate the maximum temperature and the time at which it occurs.
(b) The temperature rise is fairly linear from 8 A.M. to 2 P.M. Estimate the rate at which the temperature is increasing during this time period.
(c) Estimate the time at which the temperature is decreasing most rapidly. Estimate the instantaneous rate of change of temperature with respect to time at this instant.

Joseph Lentino

Numerade Educator

Problem 16

The accompanying figure shows the graph of the pressure $p$ in atmospheres (atm) versus the volume $V$ in liters (L) of 1 mole of an ideal gas at a constant temperature of $300 \mathrm{K}$ (kelvins). Use the tangent lines shown in the figure to estimate the rate of change of pressure with respect to volume at the points where $V=10 \mathrm{L}$ and $V=25 \mathrm{L}$ (FIGURE CAN'T COPY)

Carson Merrill

Numerade Educator

Problem 17

The accompanying figure shows the graph of the height $h$ in centimeters versus the age $t$ in years of an individual from birth to age 20. (FIGURE CAN'T COPY)
(a) When is the growth rate greatest?
(b) Estimate the growth rate at age 5
(c) At approximately what age between 10 and 20 is the growth rate greatest? Estimate the growth rate at this age.
(d) Draw a rough graph of the growth rate versus age.

Suzanne W.

Numerade Educator

Problem 18

Use Formulas (2) and (3) to find the average and instantaneous velocity.
A rock is dropped from a height of $576 \mathrm{ft}$ and falls toward Earth in a straight line. In $t$ seconds the rock drops a distance of $s=16 t^{2} \mathrm{ft}$
(a) How many seconds after release does the rock hit the ground?
(b) What is the average velocity of the rock during the time it is falling?
(c) What is the average velocity of the rock for the first 3 s?
(d) What is the instantaneous velocity of the rock when it hits the ground?

Carson Merrill

Numerade Educator

Problem 19

Use Formulas (2) and (3) to find the average and instantaneous velocity.
During the first 40 s of a rocket flight, the rocket is propelled straight up so that in $t$ seconds it reaches a height of $s=5 t^{3} \mathrm{ft}$
(a) How high does the rocket travel in 40 s?
(b) What is the average velocity of the rocket during the first $40 \mathrm{s} ?$
(c) What is the average velocity of the rocket during the first $135 \mathrm{ft}$ of its flight?
(d) What is the instantaneous velocity of the rocket at the end of $40 \mathrm{s} ?$

Carson Merrill

Numerade Educator

Problem 20

Use Formulas (2) and (3) to find the average and instantaneous velocity.
A particle moves on a line away from its initial position so that after $t$ hours it is $s=3 t^{2}+t$ miles from its initial position.
(a) Find the average velocity of the particle over the interval [1,3]
(b) Find the instantaneous velocity at $t=1$

Carson Merrill

Numerade Educator

Problem 21

Use Formulas (2) and (3) to find the average and instantaneous velocity.
A particle moves in the positive direction along a straight line so that after $t$ minutes its distance is $s=6 t^{4}$ feet from the origin.
(a) Find the average velocity of the particle over the interval [2,4]
(b) Find the instantaneous velocity at $t=2$

Carson Merrill

Numerade Educator