Suppose the graph of f is given. Write equations for the graphs that are obtained from the graph of f as follows. (a) Shift 3 units upward. (b) Shift 3 units downward. (c) Shift 3 units to the right. (d) Shift 3 units to the left. (e) Reflect about the x-axis. (f) Reflect about the y-axis. (g) Stretch vertically by a factor of 3. (h) Shrink vertically by a factor of 3.
Read moreTable of Contents
Appendix A
Numbers, Inequalities, and Absolute Values
Appendix B
Coordinate Geometry and Lines
Appendix C
Graphs of Second-Degree Equations
Appendix D
Trigonometry
Appendix E
Sigma Notation
Appendix F
Proofs of Theorems
Appendix G
The Logarithm Defined as an Integral
Appendix H
Complex Numbers
Appendix I
Answers to Odd-Numbered Exercises
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
Derivatives and Rates of Change
2.8
The Derivative as a Function
3
Differentiation Rules
3.1
Derivatives of Polynomials and Exponential Functions
3.10
Linear Approximations and Differentials
3.11
Hyperbolic Functions
3.2
The Product and Quotient Rules
3.3
Derivatives of Trigonometric Functions
3.4
The Chain Rule
3.5
Implicit Differentiation
3.6
Derivatives of Logarithmic Functions
3.7
Rates of Change in the Natural and Social Sciences
3.8
Exponential Growth and Decay
3.9
Related Rates
4
Applications of Differentiation
4.1
Maximum and Minimum Values
4.2
The Mean Value Theorem
4.3
How Derivatives Affect the Shape of a Graph
4.4
Indeterminate Forms and l’Hospital’s Rule
4.5
Summary of Curve Sketching
4.6
Graphing with Calculus and Calculators
4.7
Optimization Problems
4.8
Newton’s Method
4.9
Antiderivatives
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
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 Euler’s Method
9.3
Separable Equations
9.4
Models for Population Growth
9.5
Linear Equations
9.6
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
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
Textbook Solutions for Single Variable Calculus: Early Transcendentals,
Chapter 1.3 Problem 1
Question
Suppose the graph of f is given. Write equations for the graphs that are obtained from the graph of f as follows.
(a) Shift 3 units upward.
(b) Shift 3 units downward.
(c) Shift 3 units to the right.
(d) Shift 3 units to the left.
(e) Reflect about the x-axis.
(f) Reflect about the y-axis.
(g) Stretch vertically by a factor of 3.
(h) Shrink vertically by a factor of 3.
Solution
The first step in solving 1.3 problem number trying to solve the problem we have to refer to the textbook question: Suppose the graph of f is given. Write equations for the graphs that are obtained from the graph of f as follows. (a) Shift 3 units upward. (b) Shift 3 units downward. (c) Shift 3 units to the right. (d) Shift 3 units to the left. (e) Reflect about the x-axis. (f) Reflect about the y-axis. (g) Stretch vertically by a factor of 3. (h) Shrink vertically by a factor of 3.
From the textbook chapter New Functions from Old Functions you will find a few key concepts needed to solve this.
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full solution
Title
Single Variable Calculus: Early Transcendentals, 7
Author
James Stewart
ISBN
9780538498678
?Suppose the graph of f is given. Write equations for the graphs that are obtained from
Chapter 1.3 textbook questions
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Chapter 1: Problem 1 Single Variable Calculus: Early Transcendentals, 7
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Chapter 1: Problem 2 Single Variable Calculus: Early Transcendentals, 7
Explain how each graph is obtained from the graph of y = f(x). 1. y = f(x) + 8 2. y = f(x + 8) 3. y = 8f(x) 4. y = f(8x) 5. y = - f(x) - 1 6. y = 8f(x)
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Chapter 1: Problem 3 Single Variable Calculus: Early Transcendentals, 7
The graph of y = f(x) is given. Match each equation with its graph and give reasons for your choices.
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Chapter 1: Problem 4 Single Variable Calculus: Early Transcendentals, 7
The graph of f is given. Draw the graphs of the following functions.
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Chapter 1: Problem 5 Single Variable Calculus: Early Transcendentals, 7
The graph of f is given. Use it to graph the following functions.
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Chapter 1: Problem 6 Single Variable Calculus: Early Transcendentals, 7
6–7 The graph of y = is given. Use transformations to create a function whose graph is as shown.
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Chapter 1: Problem 7 Single Variable Calculus: Early Transcendentals, 7
6–7 The graph of y = is given. Use transformations to create a function whose graph is as shown.
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Chapter 1: Problem 8 Single Variable Calculus: Early Transcendentals, 7
(a) How is the graph of related to the graph of ? Use your answer and Figure 6 to sketch the graph of . (b) How is the graph of related to the graph of ? Use your answer and Figure 4(a) to sketch the graph of .
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Chapter 1: Problem 9 Single Variable Calculus: Early Transcendentals, 7
9–24 Graph the function by hand, not by plotting points, but by starting with the graph of one of the standard functions given in Section 1.2, and then applying the appropriate transformations. 9.
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Chapter 1: Problem 10 Single Variable Calculus: Early Transcendentals, 7
9–24 Graph the function by hand, not by plotting points, but by starting with the graph of one of the standard functions given in Section 1.2, and then applying the appropriate transformations. 10.
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Chapter 1: Problem 11 Single Variable Calculus: Early Transcendentals, 7
9–24 Graph the function by hand, not by plotting points, but by starting with the graph of one of the standard functions given in Section 1.2, and then applying the appropriate transformations. 11.
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Chapter 1: Problem 12 Single Variable Calculus: Early Transcendentals, 7
9–24 Graph the function by hand, not by plotting points, but by starting with the graph of one of the standard functions given in Section 1.2, and then applying the appropriate transformations. 12.
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Chapter 1: Problem 13 Single Variable Calculus: Early Transcendentals, 7
9–24 Graph the function by hand, not by plotting points, but by starting with the graph of one of the standard functions given in Section 1.2, and then applying the appropriate transformations. 13. - 1
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Chapter 1: Problem 14 Single Variable Calculus: Early Transcendentals, 7
9–24 Graph the function by hand, not by plotting points, but by starting with the graph of one of the standard functions given in Section 1.2, and then applying the appropriate transformations. 14.
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Chapter 1: Problem 15 Single Variable Calculus: Early Transcendentals, 7
9–24 Graph the function by hand, not by plotting points, but by starting with the graph of one of the standard functions given in Section 1.2, and then applying the appropriate transformations.
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Chapter 1: Problem 16 Single Variable Calculus: Early Transcendentals, 7
9–24 Graph the function by hand, not by plotting points, but by starting with the graph of one of the standard functions given in Section 1.2, and then applying the appropriate transformations.
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Chapter 1: Problem 17 Single Variable Calculus: Early Transcendentals, 7
9–24 Graph the function by hand, not by plotting points, but by starting with the graph of one of the standard functions given in Section 1.2, and then applying the appropriate transformations. 17.
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Chapter 1: Problem 18 Single Variable Calculus: Early Transcendentals, 7
9–24 Graph the function by hand, not by plotting points, but by starting with the graph of one of the standard functions given in Section 1.2, and then applying the appropriate transformations.
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Chapter 1: Problem 19 Single Variable Calculus: Early Transcendentals, 7
9–24 Graph the function by hand, not by plotting points, but by starting with the graph of one of the standard functions given in Section 1.2, and then applying the appropriate transformations.
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Chapter 1: Problem 20 Single Variable Calculus: Early Transcendentals, 7
9–24 Graph the function by hand, not by plotting points, but by starting with the graph of one of the standard functions given in Section 1.2, and then applying the appropriate transformations.
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Chapter 1: Problem 21 Single Variable Calculus: Early Transcendentals, 7
9–24 Graph the function by hand, not by plotting points, but by starting with the graph of one of the standard functions given in Section 1.2, and then applying the appropriate transformations.
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Chapter 1: Problem 22 Single Variable Calculus: Early Transcendentals, 7
9–24 Graph the function by hand, not by plotting points, but by starting with the graph of one of the standard functions given in Section 1.2, and then applying the appropriate transformations.
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Chapter 1: Problem 23 Single Variable Calculus: Early Transcendentals, 7
9–24 Graph the function by hand, not by plotting points, but by starting with the graph of one of the standard functions given in Section 1.2, and then applying the appropriate transformations.
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Chapter 1: Problem 24 Single Variable Calculus: Early Transcendentals, 7
9–24 Graph the function by hand, not by plotting points, but by starting with the graph of one of the standard functions given in Section 1.2, and then applying the appropriate transformations.
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Chapter 1: Problem 25 Single Variable Calculus: Early Transcendentals, 7
The city of New Orleans is located at latitude 30°N . Use Figure 9 to find a function that models the number of hours of daylight at New Orleans as a function of the time of year. To check the accuracy of your model, use the fact that on March 31 the sun rises at 5:51AM and sets at 6:18 PM in New Orleans.
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Chapter 1: Problem 26 Single Variable Calculus: Early Transcendentals, 7
A variable star is one whose brightness alternately increases and decreases. For the most visible variable star, Delta Cephei, the time between periods of maximum brightness is 5.4 days, the average brightness (or magnitude) of the star is 4.0, and its brightness varies by ± 0.35 magnitude. Find a function that models the brightness of Delta Cephei as a function of time.
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Chapter 1: Problem 27 Single Variable Calculus: Early Transcendentals, 7
(a) How is the graph of related to the graph of ? (b) Sketch the graph of . (c) Sketch the graph of .
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Chapter 1: Problem 28 Single Variable Calculus: Early Transcendentals, 7
Use the given graph of to sketch the graph of .Which features of are the most important in sketching ? Explain how they are used
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Chapter 1: Problem 29 Single Variable Calculus: Early Transcendentals, 7
29–30 Find , , , and and state their domains.
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Chapter 1: Problem 30 Single Variable Calculus: Early Transcendentals, 7
29–30 Find , , , and and state their domains.
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Chapter 1: Problem 31 Single Variable Calculus: Early Transcendentals, 7
31–36 Find the functions , , , and and their domains.
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Chapter 1: Problem 32 Single Variable Calculus: Early Transcendentals, 7
31–36 Find the functions , , , and and their domains.
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Chapter 1: Problem 33 Single Variable Calculus: Early Transcendentals, 7
31–36 Find the functions , , , and and their domains. f(x)=1 - 3x, g(x)= cos
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Chapter 1: Problem 34 Single Variable Calculus: Early Transcendentals, 7
31–36 Find the functions , , , and and their domains.
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Chapter 1: Problem 35 Single Variable Calculus: Early Transcendentals, 7
31–36 Find the functions , , , and and their domains.
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Chapter 1: Problem 36 Single Variable Calculus: Early Transcendentals, 7
31–36 Find the functions , , , and and their domains.
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Chapter 1: Problem 41 Single Variable Calculus: Early Transcendentals, 7
41–46 Express the function in the form
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Chapter 1: Problem 42 Single Variable Calculus: Early Transcendentals, 7
41–46 Express the function in the form
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Chapter 1: Problem 43 Single Variable Calculus: Early Transcendentals, 7
41–46 Express the function in the form
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Chapter 1: Problem 44 Single Variable Calculus: Early Transcendentals, 7
41–46 Express the function in the form
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Chapter 1: Problem 45 Single Variable Calculus: Early Transcendentals, 7
41–46 Express the function in the form
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Chapter 1: Problem 46 Single Variable Calculus: Early Transcendentals, 7
41–46 Express the function in the form
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Chapter 1: Problem 47 Single Variable Calculus: Early Transcendentals, 7
47–49 Express the function in the form .
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Chapter 1: Problem 48 Single Variable Calculus: Early Transcendentals, 7
47–49 Express the function in the form .
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Chapter 1: Problem 49 Single Variable Calculus: Early Transcendentals, 7
47–49 Express the function in the form .
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Chapter 1: Problem 50 Single Variable Calculus: Early Transcendentals, 7
Use the table to evaluate each expression. a. f(g(1)) b. g(f(1)) c. f(f(1)) d. g(g(1)) e. f.
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Chapter 1: Problem 51 Single Variable Calculus: Early Transcendentals, 7
Use the given graphs of and to evaluate each expression, or explain why it is undefined. 1. 2. 3. 4. 5. 6.
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Chapter 1: Problem 52 Single Variable Calculus: Early Transcendentals, 7
Use the given graphs of and to estimate the value of for . Use these estimates to sketch a rough graph of .
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Chapter 1: Problem 53 Single Variable Calculus: Early Transcendentals, 7
A stone is dropped into a lake, creating a circular ripple that travels outward at a speed of 60 cm/s. (a) Express the radius of this circle as a function of the time ( in seconds). (b) If A is the area of this circle as a function of the radius, find and interpret it.
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Chapter 1: Problem 54 Single Variable Calculus: Early Transcendentals, 7
A spherical balloon is being inflated and the radius of the balloon is increasing at a rate of 2 cm/s. (a) Express the radius of the balloon as a function of the time ( in seconds). (b) If V is the volume of the balloon as a function of the radius, find, find and interpret it.
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Chapter 1: Problem 55 Single Variable Calculus: Early Transcendentals, 7
A ship is moving at a speed of 30 cm/s parallel to a straight shoreline. The ship is 6 km from shore and it passes a lighthouse at noon. (a) Express the distance s between the lighthouse and the ship as a function of d, the distance the ship has traveled since noon; that is, find f so that . (b) Express d as a function of t , the time elapsed since noon; that is, find g so that . (c) Find . What does this function represent?
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Chapter 1: Problem 56 Single Variable Calculus: Early Transcendentals, 7
An airplane is flying at a speed of 350 mi/h at an altitude of one mile and passes directly over a radar station at time . (a) Express the horizontal distance d ( in miles) that the plane has flown as a function of t . (b) Express the distance s between the plane and the radar station as a function of d . (c) Use composition to express s as a function of t .
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Chapter 1: Problem 57 Single Variable Calculus: Early Transcendentals, 7
The Heaviside function H is defined by It is used in the study of electric circuits to represent the sudden surge of electric current, or voltage, when a switch is instantaneously turned on. (a) Sketch the graph of the Heaviside function. (b) Sketch the graph of the voltage in a circuit if the switch is turned on at time and 120 volts are applied instantaneously to the circuit. Write a formula for in terms of . (c) Sketch the graph of the voltage in a circuit if the switch is turned on at time seconds and 240 volts are applied instantaneously to the circuit. Write a formula for in terms of . (Note that starting at corresponds to a translation.)
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Chapter 1: Problem 58 Single Variable Calculus: Early Transcendentals, 7
The Heaviside function defined in Exercise 57 can also be used to define the ramp function , which represents a gradual increase in voltage or current in a circuit. (a) Sketch the graph of the ramp function . (b) Sketch the graph of the voltage in a circuit if the switch is turned on at time and the voltage is gradually increased to 120 volts over a 60-second time interval. Write a formula for in terms of for . (c) Sketch the graph of the voltage in a circuit if the switch is turned on at time seconds and the voltage is gradually increased to 100 volts over a period of 25 seconds. Write a formula for in terms of for .
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Chapter 1: Problem 59 Single Variable Calculus: Early Transcendentals, 7
Let and be linear functions with equations and . Is also a linear function? If so, what is the slope of its graph?
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Chapter 1: Problem 60 Single Variable Calculus: Early Transcendentals, 7
If you invest dollars at 4% interest compounded annually, then the amount of the investment after one year is . Find , ,and . What do these compositions represent? Find a formula for the composition of copies of .
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Chapter 1: Problem 61 Single Variable Calculus: Early Transcendentals, 7
(a) If and , find a function such that . (Think about what operations you would have to perform on the formula for to end up with the formula for h.) (b) If and , find a function such that .
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Chapter 1: Problem 62 Single Variable Calculus: Early Transcendentals, 7
If and , find a function such that .
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Chapter 1: Problem 63 Single Variable Calculus: Early Transcendentals, 7
Suppose is an even function and let . Is always an even function?
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Chapter 1: Problem 64 Single Variable Calculus: Early Transcendentals, 7
Suppose is an even function and let . Is always an odd function? What if is odd? What if is even?
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