Classify each function as a power function, root function, polynomial (state its degree), rational function, algebraic function, trigonometric function, exponential function, or logarithmic function.
Read moreTable of Contents
1
Functions and Models
1.1
Four Ways to Represent a Function
1.2
Mathematical Models: A Catalog of Essential Functions
1.3
New Functions from Old Functions
1.4
Graphing Calculators and Computers
1.5
Exponential Functions
1.6
Inverse Functions and Logarithms
2
Limits and Derivatives
2.1
The Tangent and Velocity Problems
2.2
The Limit of a Function
2.3
Calculating Limits Using the Limit Laws
2.4
The Precise Definition of a Limit
2.5
Continuity
2.6
Limits at Infinity; Horizontal Asymptotes
2.7
Tangents, Velocities, and Other Rates of Change
2.8
Derivatives
2.9
The Derivative as a Function
3
Differentiation Rules
3.1
Derivatives of Polynomials and Exponential Functions
3.10
Related Rates
3.11
Linear Approximations and Differentials
3.3
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3.4
Derivatives of Trigonometric Functions
3.5
The Chain Rule
3.6
Implicit Differentiation
3.7
Higher Derivatives
3.8
Derivatives of Logarithmic Functions
3.9
Hyperbolic Functions
4
Applications of Differentiation
4.1
Maximum and Minimum Values
4.10
Antiderivatives
4.2
The Mean Value Theorem
4.3
How Derivatives Affect the Shape of a Graph
4.4
Indeterminate Forms and LHospitals Rule
4.5
Summary of Curve Sketching
4.6
Graphing with Calculus and Calculators
4.7
Optimization Problems
4.8
Applications to Business and Economics
4.9
Newtons Method
5
Integrals
5.1
Areas and Distances
5.2
The Definite Integral
5.3
The Fundamental Theorem of Calculus
5.4
Indefinite Integrals and the Net Change Theorem
5.5
The Substitution Rule
5.6
The Logarithm Defined as an Integral
6
Applications of Integration
6.1
Areas between Curves
6.2
Volumes
6.3
Volumes by Cylindrical Shells
6.4
Work
6.5
Average Value of a Function
7
Techniques of Integration
7.1
Integration by Parts
7.2
Trigonometric Integrals
7.3
Trigonometric Substitution
7.4
Integration of Rational Functions by Partial Fractions
7.5
Strategy for Integration
7.6
Integration Using Tables and Computer Algebra Systems
7.7
Approximate Integration
7.8
Improper Integrals
8
Further Applications of Integration
8.1
Arc Length
8.2
Area of a Surface of Revolution
8.3
Applications to Physics and Engineering
8.4
Applications to Economics and Biology
8.5
Probability
9
Differential Equations
9.1
Modeling with Differential Equations
9.2
Direction Fields and Eulers Method
9.3
Separable Equations
9.4
Exponential Growth and Decay
9.5
The Logistic Equation
9.6
Linear Equations
9.7
Predator-Prey Systems
10
Parametric Equations and Polar Coordinates
10.1
Curves Defined by Parametric Equations
10.2
Calculus with Parametric Curves
10.3
Polar Coordinates
10.4
Areas and Lengths in Polar Coordinates
10.5
Conic Sections
10.6
Conic Sections in Polar Coordinates
11
Infinite Sequences and Series
11.1
Sequences
11.10
Taylor and Maclaurin Series
11.11
The Binomial Series
11.12
Applications of Taylor Polynomials
11.2
Series
11.3
The Integral Test and Estimates of Sums
11.4
The Comparison Tests
11.5
Alternating Series
11.6
Absolute Convergence and the Ratio and Root Tests
11.7
Strategy for Testing Series
11.8
Power Series
11.9
Representations of Functions as Power Series
12
Vectors and the Geometry of Space
12.1
Three-Dimensional Coordinate Systems
12.2
Vectors
12.3
The Dot Product
12.4
The Cross Product
12.5
Equations of Lines and Planes
12.6
Cylinders and Quadric Surfaces
12.7
Cylindrical and Spherical Coordinates
13
Vector Functions
13.1
Vector Functions and Space Curves
13.2
Derivatives and Integrals of Vector Functions
13.3
Arc Length and Curvature
13.4
Motion in Space: Velocity and Acceleration
14
Partial Derivatives
14.1
Functions of Several Variables
14.2
Limits and Continuity
14.3
Partial Derivatives
14.4
Tangent Planes and Linear Approximations
14.5
The Chain Rule
14.6
Directional Derivatives and the Gradient Vector
14.7
Maximum and Minimum Values
14.8
Lagrange Multipliers
15
Multiple Integrals
15.1
Double Integrals over Rectangles
15.2
Iterated Integrals
15.3
Double Integrals over General Regions
15.4
Double Integrals in Polar Coordinates
15.5
Applications of Double Integrals
15.6
Surface Area
15.7
Triple Integrals
15.8
Triple Integrals in Cylindrical and Spherical Coordinates
15.9
Change of Variables in Multiple Integrals
16
Vector Calculus
16.1
Vector Fields
16.2
Line Integrals
16.3
The Fundamental Theorem for Line Integrals
16.4
The Fundamental Theorem for Line Integrals
16.5
Curl and Divergence
16.6
Parametric Surfaces and Their Areas
16.7
Surface Integrals
16.8
Stokes Theorem
16.9
The Divergence Theorem
17
Second-Order Differential Equations
17.1
Second-Order Linear Equations
17.2
Nonhomogeneous Linear Equations
17.3
Applications of Second-Order Differential Equations
17.4
Series Solutions
Textbook Solutions for Calculus,
Chapter 1.2 Problem 22
Question
The table shows the mean (average) distances d of the planets from the Sun (taking the unit of measurement to be the distance from Earth to the Sun) and their periods T (time ofrevolution in years). a) Fit a power model to the data.(b) Keplers Third Law of Planetary Motion states that Thesquare of the period of revolution of a planet is proportionalto the cube of its mean distance from the Sun. Does yourmodel corroborate Keplers Third Law?
Solution
Problem 22
The table shows the mean (average) distances d of the planets from the Sun (taking the unit of measurement to be the distance from 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?
Step by step solution
Step 1 of 6
Consider the data
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full solution
Title
Calculus, 5
Author
James Stewart
ISBN
9780534393397
The table shows the mean (average) distances d of the planets from the Sun (taking the
Chapter 1.2 textbook questions
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Chapter 1: Problem 1 Calculus, 5
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Chapter 1: Problem 2 Calculus, 5
Classify each function as a power function, root function, polynomial (state its degree), rational function, algebraic function, trigonometric function, exponential function, or logarithmic function.
Read more -
Chapter 1: Problem 3 Calculus, 5
Match each equation with its graph. Explain your choices. (Dont use a computer or graphing calculator.)
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Chapter 1: Problem 4 Calculus, 5
Match each equation with its graph. Explain your choices. (Dont use a computer or graphing calculator.)
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Chapter 1: Problem 5 Calculus, 5
(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 and sketch several members of the family. (c) Which function belongs to both families?
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Chapter 1: Problem 6 Calculus, 5
What do all members of the family of linear functions have in common? Sketch several members of the family.
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Chapter 1: Problem 7 Calculus, 5
What do all members of the family of linear functions have in common? Sketch several members of the family.
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Chapter 1: Problem 8 Calculus, 5
The manager of a weekend flea market knows from past experience that if he charges dollars for a rental space at the flea market, then the number of spaces he can rent is given by the equation . (a) Sketch a graph of this linear function. (Remember that the rental charge per space and the number of spaces rented cant be negative quantities.) (b) What do the slope, the y-intercept, and the x-intercept of the graph represent?
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Chapter 1: Problem 9 Calculus, 5
The relationship between the Fahrenheit and Celsius temperature scales is given by the linear function . (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?
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Chapter 1: Problem 10 Calculus, 5
Jason leaves Detroit at 2:00 P.M. and drives at a constant speed west along I-96. He passes Ann Arbor, 40 mi from Detroit, at 2:50 P.M. (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?
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Chapter 1: Problem 11 Calculus, 5
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 and 173 chirps per minute at . (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
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Chapter 1: Problem 12 Calculus, 5
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?
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Chapter 1: Problem 13 Calculus, 5
At the surface of the ocean, the water pressure is the same as the air pressure above the water, . Below the surface, the water pressure increases by 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 ?
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Chapter 1: Problem 14 Calculus, 5
The monthly cost of driving a car depends on the number of miles driven. Lynn found that in May it cost her $380 to drive 480 mi and in June it cost her $460 to drive 800 mi. (a) Express the monthly cost as a function of the distance driven 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 y-intercept represent? (e) Why does a linear function give a suitable model in this situation?
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Chapter 1: Problem 15 Calculus, 5
For each scatter plot, decide what type of function you might choose as a model for the data. Explain your choices.
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Chapter 1: Problem 16 Calculus, 5
For each scatter plot, decide what type of function you might choose as a model for the data. Explain your choices.
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Chapter 1: Problem 17 Calculus, 5
The table shows (lifetime) peptic ulcer rates (per 100 population) for various family incomes as reported by the 1989 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. (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?
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Chapter 1: Problem 18 Calculus, 5
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 .
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Chapter 1: Problem 19 Calculus, 5
The table gives the winning heights for the Olympic pole vault competitions in the 20th century. (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 predict the height of the winning pole vault at the 2000 Olympics and compare with the winning height of 19.36 feet. (d) Is it reasonable to use the model to predict the winning height at the 2100 Olympics?
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Chapter 1: Problem 20 Calculus, 5
A study by the U. S. Office of Science and Technology in 1972 estimated the cost (in 1972 dollars) to reduce automobile emissions by certain percentages: Find a model that captures the diminishing returns trend of these data.
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Chapter 1: Problem 21 Calculus, 5
Use the data in the table to model the population of the world in the 20th century by a cubic function. Then use your model to estimate the population in the year 1925.
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Chapter 1: Problem 22 Calculus, 5
The table shows the mean (average) distances d of the planets from the Sun (taking the unit of measurement to be the distance from Earth to the Sun) and their periods T (time of revolution in years). a) Fit a power model to the data. (b) Keplers 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 Keplers Third Law?
Read more