Calculus: Early Transcendentals

James Stewart

Chapter 1

Functions and Models - all with Video Answers

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Section 2

Mathematical Models: A Catalog of Essential Functions

Problem 1

Classify each function as a power function, root function, polynomial (state its degree), rational function, algebraic function, trigonometric function, exponential function, or logarithmic function.

(a) $ f(x) = \log_2 x $
(b) $ g(x) = \sqrt[4]{x} $
(c) $ h(x) = \frac{2x^3}{1 - x^2} $
(d) $ u(t) = 1-1.1t + 2.54t^2 $
(e) $ v(t) = 5^t $
(f) $ w(\theta) = \sin \theta \cos^2 \theta $

Heather Zimmers

Heather Zimmers

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Problem 2

Classify each function as a power function, root function, polynomial (state its degree), rational function, algebraic function, trigonometric function, exponential function, or logarithmic function.

(a) $ y = \pi^x $
(b) $ y = x^\pi $
(c) $ y = x^2 (2 - x^3) $
(d) $ y = \tan t - \cos t $
(e) $ y = \frac{s}{1 + s} $
(f) $ y = \frac{\sqrt {x^3 - 1}}{1 + \sqrt[3]{x}} $

Carson Merrill

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Problem 3

Match each equation with its graph. Explain your choices. (Don't use a computer or graphing calculator.)

(a) $ y = x^2 $
(b) $ y = x^5 $
(c) $ y = x^8 $

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Heather Zimmers

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Problem 4

Match each equation with its graph. Explain your choices. (Don't use a computer or graphing calculator.)

(a) $ y = 3x $
(b) $ y = 3^x $
(c) $ y = x^3 $
(d) $ y = \sqrt[3]{x} $

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Heather Zimmers

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Problem 5

Find the domain of the function.

$ f(x) = \dfrac{\cos x}{1 - \sin x} $

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Heather Zimmers

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Problem 6

Find the domain of the function.

$ g(x) = \dfrac{1}{1 - \tan x} $

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Heather Zimmers

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Problem 7

(a) Find an equation for the family of linear functions with slope 2 and sketch several members of the family.
(b) Find an equation for the family of linear functions such that $ f(2) = 1 $ and sketch several members of the family.
(c) Which function belongs to both families?

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Heather Zimmers

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Problem 8

What do all members of the family of linear functions $ f(x) = 1 + m(x + 3) $ have in common? Sketch several members of the family.

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Heather Zimmers

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Problem 9

What do all members of the family of linear functions $ f(x) = c - x $ have in common? Sketch several members of the family.

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Heather Zimmers

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Problem 10

Find expressions for the quadratic functions whose graphs are shown.

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Heather Zimmers

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Problem 11

Find an expression for a cubic function $ f $ if $ f(1) = 6 $ and $ f(-1) = f(0) = f(2) = 0 $.

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Heather Zimmers

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Problem 12

Recent studies indicate that the average surface temperature of the earth has been rising steadily. Some scientists have modeled the temperature by the linear function $ T = 0.02t 1 8.50 $ where $ T $ is temperature in $ ^{\circ} C $ and $ t $ represents years since 1900.

(a) What do the slope and T-intercept represent?
(b) Use the equation to predict the average global surface temperature in 2100.

Heather Zimmers

Heather Zimmers

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Problem 13

If the recommended adult dosage for a drug is $ D $ (in mg), then to determine the appropriate dosage $ c $ for a child of age $ a $ , pharmacists use the equation $ c = 0.0417D ( a + 1) $. Suppose the dosage for an adult is 200 mg.

(a) Find the slope of the graph of $ c $. What does it represent?
(b) What is the dosage for a newborn?

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Heather Zimmers

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Problem 14

The manager of a weekend flea market knows from past experience that if he charges $ x $ dollars for a rental space at the market, then the number $ y $ of spaces he can rent is given by the equation
$ y = 200 -4x $.

(a) Sketch a graph of the linear function. (Remember that the rental charge per space and the number of spaces rented can't be negative quantities.)
(b) What do the slope, the y-intercept, and the x-intercept of the graph represent.

Heather Zimmers

Heather Zimmers

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Problem 15

The relationship between the Fahrenheit $ (F) $ and Celsius $ (C) $ temperature scales is given by the linear function $ F = \frac{9}{5} C + 32 $.

(a) Sketch a graph of this function.
(b) What is the slope of the graph and what does it represent? What is the F-intercept and what does it represent?

Heather Zimmers

Heather Zimmers

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Problem 16

Jason leaves Detroit at 2: 00 PM and drives at a constant speed west along $\mathrm{I}-94 .$ He passes Ann Arbor, $40 \mathrm{mi}$ from Detroit, at 2: 50 PM.
(a) Express the distance traveled in terms of the time elapsed.
(b) Draw the graph of the equation in part (a).
(c) What is the slope of this line? What does it represent?

Mary Wakumoto

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Problem 17

Biologists have noticed that the chirping rate of crickets of a certain species is related to temperature, and the relationship appears to be very nearly linear. A cricket produces 113 chirps per minute at 70 $ ^{\circ} F $ and 173 chirps per minute at 80 $ ^{\circ} F $.

(a) Find a linear equation that models the temperature $ T $ as a function of the number of chirps per minute $ N $.
(b) What is the slope of the graph? What does it represent?
(c) If the crickets are chirping at 150 chirps per minute, estimate the temperature.

Mary Wakumoto

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Problem 18

The manager of a furniture factory finds that it costs\ $2200 to manufacture 100 chairs in one day and \$4800 to produce 300 chairs in one day.
(a) Express the cost as a function of the number of chairs produced, assuming that it is linear. Then sketch the graph.
(b) What is the slope of the graph and what does it represent?
(c) What is the y-intercept of the graph and what does it represent?

Heather Zimmers

Heather Zimmers

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Problem 19

At the surface of the ocean, the water pressure is the same as the air pressure above the water, 15 $ lb/in^2 $. Below the surface, the water pressure increases by 4.34 $ lb/in^2 $ for every 10 ft of descent.

(a) Express the water pressure as a function of the depth below the ocean surface.
(b) At what depth is the pressure 100 $ lb/in^2 $?

Heather Zimmers

Heather Zimmers

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Problem 20

The monthly cost of driving a car depends on the number of miles driven. Lynn found that in May it cost her 380 dollars to drive 480 mi and in June it cost her 460 dollars to drive 800 mi.

(a) Express the monthly cost $ C $ as a function of the distance driven $ d $, assuming that a linear relationship gives a suitable model.
(b) Use part (a) to predict the cost of driving 1500 miles per month.
(c) Draw the graph of the linear function. What does the slope represent?
(d) What does the C-intercept represent?
(e) Why does a linear function give a suitable model in this situation?

Heather Zimmers

Heather Zimmers

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Problem 21

For each scatter plot, decide what type of function you might choose as a model for the data. Explain your choices.

Heather Zimmers

Heather Zimmers

Numerade Educator

Problem 22

For each scatter plot, decide what type of function you might choose as a model for the data. Explain your choices.

Heather Zimmers

Heather Zimmers

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Problem 23

The table shows (lifetime) peptic ulcer rates (per 100 population) for various family incomes as reported by the National Health Interview Survey.

(a) Make a scatter plot of these data and decide whether a linear model is appropriate.
(b) Find and graph a linear model using the first and last data points.
(c) Find and graph the least squares regression line.
(d) Use the linear model in part (c) to estimate the ulcer rate for an income of \$25,000 dollars.
(e) According to the model, how likely is someone with an income of \$80,000 to suffer from peptic ulcers?
(f) Do you think it would be reasonable to apply the model to someone with an income of \$200,000?

Heather Zimmers

Heather Zimmers

Numerade Educator

Problem 24

Biologists have observed that the chirping rate of crickets of a certain species appears to be related to temperature. The table shows the chirping rates for various temperatures.

(a) Make a scatter plot of the data.
(b) Find and graph the regression line.
(c) Use the linear model in part (b) to estimate the chirping rate at 100 $ ^{\circ} F $.

Heather Zimmers

Heather Zimmers

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Problem 25

Anthropologists use a linear model that relates human femur (thighbone) length to height. The model allows an anthropologist to determine the height of an individual when only a partial skeleton (including the femur) is found. Here we find the model by analyzing the data on femur length and height for the eight males given in the following table.

(a) Make a scatter plot of the data.
(b) Find and graph the regression line that models the data.
(c) An anthropologist finds a human femur of length 53 cm. How tall was the person?

Heather Zimmers

Heather Zimmers

Numerade Educator

Problem 26

When laboratory rats are exposed to asbestos fibers, some of them develop lung tumors. The table lists the results of several experiments by different scientists.

(a) Find the regression line for the data.
(b) Make a scatter plot and graph the regression line. Does the regression line appear to be a suitable model for the data?
(c) What does the y-intercept of the regression line represent?

Heather Zimmers

Heather Zimmers

Numerade Educator

Problem 27

The table shows world average daily oil consumption from 1985 to 2010 measured in thousands of barrels per day.

(a) Make a scatter plot and decide whether a linear model is appropriate.
(b) Find and graph the regression line.
(c) Use the linear model to estimate the oil consumption in 2002 and 2012.

Heather Zimmers

Heather Zimmers

Numerade Educator

Problem 28

The table shows average US retail residential prices of electricity from 2000 to 2012, measured in cents per kilowatt hour.

(a) Make a scatter plot. Is a linear model appropriate?
(b) Find and graph the regression line.
(c) Use your linear model from part (b) to estimate the average retail price of electricity in 2005 and 2013.

Heather Zimmers

Heather Zimmers

Numerade Educator

Problem 29

Many physical quantities are connected by inverse square laws, that is, by power functions of the form
$ f(x) = kx^{-2} $. In particular, the illumination of an object by a light source is inversely proportional to the square of the distance from the source. Suppose that after dark you are in a room with just one lamp and you are trying to read a book. The light is too dim and so you move halfway to the lamp. How much
brighter is the light?

Heather Zimmers

Heather Zimmers

Numerade Educator

Problem 30

It makes sense that the larger the area of a region, the larger the number of species that inhabit the region. Many ecologists have modeled the species-area relation with a power function and, in particular, the number of species $ S $ of bats living in caves in central Mexico has been related to the surface area $ A $ of the caves by the equation $ S = 0.7A^ {0.3} $.

(a) The cave called Mision Imposible near Puebla, Mexico, has a surface area of $ A = 60 m^2 $ . How many species of bats would you expect to find in that cave?
(b) If you discover that four species of bats live in a cave, estimate the area of the cave.

Heather Zimmers

Heather Zimmers

Numerade Educator

Problem 31

The table shows the number $ N $ of species of reptiles and amphibians inhabiting Caribbean islands and the area $ A $ of the island in square miles.

(a) Use a power function to model N as a function of $ A $.
(b) The Caribbean island of Dominica has area $ 291 mi^2 $. How many species of reptiles and amphibians would you expect to find on Dominica?

Carson Merrill

Numerade Educator

Problem 32

The table shows the mean (average) distances $ d $ of the planets from the sun (taking the unit of measurement to be the distance from the earth to the sun) and their periods $ T $ (time of revolution in years).

(a) Fit a power model to the data.
(b) Kepler's Third Law of Planetary Motion states that " The square of the period of revolution of a planet
is proportional to the cube of its mean distance from the sun."
Does your model corroborate Kepler's Third Law?

Heather Zimmers

Heather Zimmers

Numerade Educator