Calculus

Dale Varberg, Edwin Purcell deceased, Steve Rigdon

Chapter 2

The Derivative - all with Video Answers

Educators

Section 1

Two Problems with One Theme

Problem 1

A tangent line to a curve is drawn. Estimate its slope (slope$=$rise/run ). Be careful to note the difference in scales on the two axes.

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Chris Bolognese

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Problem 2

A tangent line to a curve is drawn. Estimate its slope (slope$=$rise/run ). Be careful to note the difference in scales on the two axes.

Chris Bolognese

Chris Bolognese

Numerade Educator

Problem 3

Draw the tangent line to the curve through the indicated point and estimate its slope.

Chris Bolognese

Chris Bolognese

Numerade Educator

Problem 4

Draw the tangent line to the curve through the indicated point and estimate its slope.

Chris Bolognese

Chris Bolognese

Numerade Educator

Problem 5

Draw the tangent line to the curve through the indicated point and estimate its slope.

Chris Bolognese

Chris Bolognese

Numerade Educator

Problem 6

Draw the tangent line to the curve through the indicated point and estimate its slope.

Chris Bolognese

Chris Bolognese

Numerade Educator

Problem 7

Consider $y=x^{2}+1$.
(a) Sketch its graph as carefully as you can.
(b) Draw the tangent line at (1,2).
(c) Estimate the slope of this tangent line.
(d) Calculate the slope of the secant line through (1,2) and $\left(1.01,(1.01)^{2}+1.0\right)$.
(e) Find by the limit process (see Example 1 ) the slope of the tangent line at (1,2).

Chris Bolognese

Chris Bolognese

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Problem 8

Consider $y=x^{3}-1$.
(a) Sketch its graph as carefully as you can.
(b) Draw the tangent line at (2,7).
(c) Estimate the slope of this tangent line.
(d) Calculate the slope of the secant line through (2,7) and $\left(2.01,(2.01)^{3}-1.0\right)$.
(e) Find by the limit process (see Example 1 ) the slope of the tangent line at (2,7).

Chris Bolognese

Chris Bolognese

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Problem 9

Find the slopes of the tangent lines to the curve $y=x^{2}-1$ at the points where $x=-2,-1,0,1,2$ (see Example 2).

Chris Bolognese

Chris Bolognese

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Problem 10

Find the slopes of the tangent lines to the curve $y=x^{3}-3 x$ at the points where $x=-2,-1,0,1,2$.

Chris Bolognese

Chris Bolognese

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Problem 11

Sketch the graph of $y=1 /(x+1)$ and then find the equation of the tangent line at $\left(1, \frac{1}{2}\right)$ (see Example 3 ).

Chris Bolognese

Chris Bolognese

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Problem 12

Find the equation of the tangent line to $y=1 /(x-1)$ at (0,-1).

Chris Bolognese

Chris Bolognese

Numerade Educator

Problem 13

Experiment suggests that a falling body will fall approximately $16 t^{2}$ feet in $t$ seconds.
(a) How far will it fall between $t=0$ and $t=1 ?$
(b) How far will it fall between $t=1$ and $t=2 ?$
(c) What is its average velocity on the interval $2 \leq t \leq 3 ?$

Chris Bolognese

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Problem 14

An object travels along a line so that its position $s$ is $s=t^{2}+1$ meters after $t$ seconds.
(a) What is its average velocity on the interval $2 \leq t \leq 3 ?$
(b) What is its average velocity on the interval $2 \leq t \leq 2.003 ?$
(c) What is its average velocity on the interval $2 \leq t \leq 2+h ?$

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Chris Bolognese

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Problem 15

Suppose that an object moves along a coordinate line so that its directed distance from the origin after $t$ seconds is $\sqrt{2 t+1}$ feet.
(a) Find its instantaneous velocity at $t=\alpha, \alpha>0$.
(b) When will it reach a velocity of $\frac{1}{2}$ foot per second? (see Example 5.)

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Problem 16

If a particle moves along a coordinate line so that its directed distance from the origin after $t$ seconds is $\left(-t^{2}+4 t\right)$ feet, when did the particle come to a momentary stop (i.e., when did its instantaneous velocity become zero)?

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Problem 17

A certain bacterial culture is growing so that it has a mass of $\frac{1}{2} t^{2}+1$ grams after $t$ hours.
(a) How much did it grow during the interval $2 \leq t \leq 2.01 ?$
(b) What was its average growth rate during the interval $2 \leq t \leq 2.01 ?$

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Problem 18

A business is prospering in such a way that its total (accumulated) profit after $t$ years is $1000 t^{2}$ dollars.
(a) How much did the business make during the third year (between $t=2$ and $t=3$ )?
(b) What was its average rate of profit during the first half of the third year, between $t=2$ and $t=2.5 ?$ (The rate will be in dollars per year.)
(c) What was its instantaneous rate of profit at $t=2 ?$

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Chris Bolognese

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Problem 19

A wire of length 8 centimeters is such that the mass between its left end and a point $x$ centimeters to the right is $x^{3}$ grams (Figure 12).
(a) What is the average density of the middle 2 -centimeter segment of this wire? Note: Average density equals mass/length.
(b) What is the actual density at the point 3 centimeters from the left end?

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Problem 20

Suppose that the revenue $R(n)$ in dollars from producing $n$ computers is given by $R(n)=0.4 n-0.001 n^{2} .$ Find the instantaneous rates of change of revenue when $n=10$ and $n=100$ (The instantaneous rate of change of revenue with respect to the amount of product produced is called the marginal revenue.)

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Problem 21

The rate of change of velocity with respect to time is called acceleration. Suppose that the velocity at time $t$ of a particle is given by $v(t)=2 t^{2} .$ Find the instantaneous acceleration when $t=1$ second.

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Problem 22

A city is hit by an Asian flu epidemic. Officials estimate that $t$ days after the beginning of the epidemic the number of persons sick with the flu is given by $p(t)=120 t^{2}-2 t^{3},$ when $0 \leq t \leq 40 .$ At what rate is the flu spreading at time $t=10 ; t=20 ; t=40 ?$

Chris Bolognese

Chris Bolognese

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Problem 23

The graph in Figure 13 shows the amount of water in a city water tank during one day when no water was pumped into the tank. What was the average rate of water usage during the day? How fast was water being used at 8 A.M.?

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Problem 24

Passengers board an elevator at the ground floor (i.e., the zeroth floor) and take it to the seventh floor, which is 84 feet above the ground floor. The elevator's position $s$ as a function of time $t$ (measured in seconds) is shown in Figure 14.
(a) What is the average velocity of the elevator from the time the elevator began moving until the time that it reached the seventh floor?
(b) What was the elevator's approximate velocity at time $t=20 ?$
(c) How many stops did the elevator make between the ground floor and the seventh floor (excluding the ground and seventh floors)? On which floors do you think the elevator stopped?

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Problem 25

Figure 15 shows the normal high temperature for $\mathrm{St}$. Louis, Missouri, as a function of time (measured in days beginning January 1).
(a) What is the approximate rate of change in the normal high temperature on March 2 (i.e., on day number 61)? What are the units of this rate of change?
(b) What is the approximate rate of change in the normal high temperature on July 10 (i.e., on day number 191 )?
(c) In what months is there a moment when the rate of change is equal to $0 ?$
(d) In what months is the absolute value of the rate of change the greatest?

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Problem 26

Figure 16 shows the population in millions of a developing country for the years 1900 to $1999 .$ What is the approximate rate of change of the population in $1930 ?$ In $1990 ?$ The percentage growth is often a more appropriate measure of population growth. This is the rate of growth divided by the population size at that time. For this population, what was the approximate percentage growth in 1930? In 1990?

Chris Bolognese

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Problem 27

Figures $17 \mathrm{a}$ and $17 \mathrm{b}$ show the position $s$ as a function of time $t$ for two particles that are moving along a line. For each particle, is the velocity increasing or decreasing? Explain.
(a)
(b)

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Problem 28

The rate of change of electric charge with respect to time is called current. Suppose that $\frac{1}{3} t^{3}+t$ coulombs of charge flow through a wire in $t$ seconds. Find the current in amperes (coulombs per second) after 3 seconds. When will a 20 -ampere fuse in the line blow?

Chris Bolognese

Chris Bolognese

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Problem 29

The radius of a circular oil spill is growing at a constant rate of 2 kilometers per day. At what rate is the area of the spill growing 3 days after it began?

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Chris Bolognese

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Problem 30

The radius of a spherical balloon is increasing at the rate of 0.25 inch per second. If the radius is 0 at time $t=0,$ find the rate of change in the volume at time $t=3$.

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Problem 31

Use a graphing calculator or a CAS.
Draw the graph of $y=f(x)=x^{3}-2 x^{2}+1 .$ Then find the slope of the tangent line at
$\begin{array}{llll}\text { (a) }-1 & \text { (b) } 0 & \text { (c) } 1 & \text { (d) } 3.2\end{array}$

Chris Bolognese

Chris Bolognese

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Problem 32

Use a graphing calculator or a CAS.
Draw the graph of $y=f(x)=\sin x \sin ^{2} 2 x$. Then find the slope of the tangent line at
(a) $\pi / 3$
(b) 2.8
(c) $\pi$
(d) 4.2

Chris Bolognese

Chris Bolognese

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Problem 33

Use a graphing calculator or a CAS.
If a point moves along a line so that its distance $s$ (in feet) from 0 is given by $s=t+t \cos ^{2} t$ at time $t$ seconds, find its instantaneous velocity at $t=3$

Chris Bolognese

Chris Bolognese

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Problem 34

Use a graphing calculator or a CAS.
If a point moves along a line so that its distance $s$ (in meters) from 0 is given by $s=(t+1)^{3} /(t+2)$ at time $t$ minutes, find its instantancous velocity at $t=1.6$

Chris Bolognese

Chris Bolognese

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