In Exercises 1-4. determine whether fix) approaches oc or -oc as .V approaches -2 from the left and from the right. fix)
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Table of Contents
P
Preparation for Calculus
P.1
Graphs and Models
P.2
Linear Models and Rates of Change
P.3
Functions and Their Graph's
P.4
Fitting Models to Data
1
Limits and Their Properties
1.1
A Preview of Calculus
1.2
Finding Limits Graphically and Numerically
1.3
Evaluating Limits Analytically
1.4
Continuity and One-Sided Limits
1.5
Infinite Limits
2
Differentiation
2.1
The Derivative and the Tangent Line Problem
2.2
Basic Differentiation Rules and Rates of Change
2.3
The Product and Quotient Rules and Higher-Order Derivatives
2.4
The Chain Rule
2.5
Implicit Differentation
2.6
Related Rates
3
Applications of Differentation
3.1
Extrema on an Interval
3.2
Rolle's Theorem and the Mean Value Theorem
3.3
Increasmg and Decreasing Functions and the First Derivative Test
3.4
Concavity and the Second Derivative Test
3.5
Limits at Infinity
3.6
A Summary of Curve Sketching
3.7
Optimization Problems
3.8
Newton's Method
3.9
Differentials
4
Integration
4.1
Antiderivatives and Indefinite Integration
4.2
Area
4.3
Riemann Sums and Definite Integrals
4.4
The Fimdamental Theorem of Calculus
4.5
Integration by Substitution
4.6
Numerical Integration
5
Logaritliniic, Exponential, and Other TianscenUcntal Functions
5.1
The Natural Logarithmic Function: Differentiation
5.2
The Natural Logarithmic Function: Integration
5.3
Inverse Fimctions
5.4
Exponential Functions: Differentiation and Integration
5.5
Bases Other than e and Applications
5.6
Differential Equations: Growth and Decay
5.7
Differential Equations: Separation of Variables
5.8
Inverse Trigonometric Functions: Differentiation
5.9
Inverse Trigonometric Functions: Integration
6
Applications of Integration
6.1
Area of a Region Between Two Curves
6.2
Volume: The Disk Method
6.3
Volume: The Shell Method
6.4
Arc Lencth and Surfaces of Ro\nliition
6.5
Work
6.6
Monicnls. Centers of Mass, and Centioids
6.7
Fluid Pressure and Fluid Force
7
Integraticm Techniques, L^Hopital's Rule, and Improper Integrals
7.1
BLisic liilesjration Rules
7.2
lntegralion by Parts
7.3
Trigonometric Integrals
7.4
Trigonometric Substitution
7.5
Partial Fractions
7.6
Integration by Tables and Other Integration Techniques
7.7
Indeterminate Forms and l/Hopital's Rule
7.8
Improper Integrals
8
Infinite Series
8.1
Sequences
8.10
Tarjrior iwia Maclaurin Series
8.2
Series and Convergence
8.3
The Integral Test and Series
8.4
Comparisons of Series
8.5
Alternating Series
8.6
The Ratio and Root Tests
8.7
Taylor Polynomials and ApprOxiniatlons
8.8
Power Series
8.9
Representation of Functions by Power Series
9
Conies. Parametric Equations, and Polar Coordinates
9.1
Conics and Calculus
9.2
Plane Curves and Parametric Equations
9.3
Parametric Equations and Calculus
9.4
Polar Cqordmates and Polar Graphs
9.5
Area andArc Length in Polar Coordinates
9.6
Polar Equations of Conies and Kepler's Laws
Textbook Solutions for Calculus of A Single Variable
Chapter 1.5 Problem 5
Question
Sunurical and Graphical Analysis In Exercises 5-8. determine whether fix) approaches oc or oc as x approaches -3 from the left and from the right h\ completing (he tahle. I se a graphing utility to graph the function and confirm your answer. fix)
Solution
The first step in solving 1.5 problem number 5 trying to solve the problem we have to refer to the textbook question: Sunurical and Graphical Analysis In Exercises 5-8. determine whether fix) approaches oc or oc as x approaches -3 from the left and from the right h\ completing (he tahle. I se a graphing utility to graph the function and confirm your answer. fix)
From the textbook chapter Infinite Limits you will find a few key concepts needed to solve this.
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Title
Calculus of A Single Variable 7
Author
Ron Larson, Robert P. Hostetler, Bruce H. Edwards, David E. Heyd
ISBN
9780618149162
Sunurical and Graphical Analysis In Exercises 5-8. determine whether fix) approaches oc
Chapter 1.5 textbook questions
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Chapter 1: Problem 1 Calculus of A Single Variable 7
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Chapter 1: Problem 2 Calculus of A Single Variable 7
In Exercises 1-4. determine whether fix) approaches oc or -oc as .V approaches -2 from the left and from the right. fix) X + 2
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Chapter 1: Problem 3 Calculus of A Single Variable 7
In Exercises 1-4. determine whether fix) approaches oc or -oc as .V approaches -2 from the left and from the right. fix) = tan
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Chapter 1: Problem 4 Calculus of A Single Variable 7
In Exercises 1-4. determine whether fix) approaches oc or -oc as .V approaches -2 from the left and from the right. fix) = sec
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Chapter 1: Problem 5 Calculus of A Single Variable 7
Sunurical and Graphical Analysis In Exercises 5-8. determine whether fix) approaches oc or oc as x approaches -3 from the left and from the right h\ completing (he tahle. I se a graphing utility to graph the function and confirm your answer. fix)
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Chapter 1: Problem 6 Calculus of A Single Variable 7
Sunurical and Graphical Analysis In Exercises 5-8. determine whether fix) approaches oc or oc as x approaches -3 from the left and from the right h\ completing (he tahle. I se a graphing utility to graph the function and confirm your answer. fix) I- 9
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Chapter 1: Problem 7 Calculus of A Single Variable 7
Sunurical and Graphical Analysis In Exercises 5-8. determine whether fix) approaches oc or oc as x approaches -3 from the left and from the right h\ completing (he tahle. I se a graphing utility to graph the function and confirm your answer. /(a)
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Chapter 1: Problem 8 Calculus of A Single Variable 7
Sunurical and Graphical Analysis In Exercises 5-8. determine whether fix) approaches oc or oc as x approaches -3 from the left and from the right h\ completing (he tahle. I se a graphing utility to graph the function and confirm your answer.fix) = sec n
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Chapter 1: Problem 9 Calculus of A Single Variable 7
In Exercises 9-28. find the vertical asymptotes (if any) of the function. A.v)
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Chapter 1: Problem 10 Calculus of A Single Variable 7
In Exercises 9-28. find the vertical asymptotes (if any) of the function. /Iv) = 4 (a - 2)-'
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Chapter 1: Problem 11 Calculus of A Single Variable 7
In Exercises 9-28. find the vertical asymptotes (if any) of the function. /Iv) = 4 (a - 2)-'
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Chapter 1: Problem 12 Calculus of A Single Variable 7
In Exercises 9-28. find the vertical asymptotes (if any) of the function. .?(-v) = 2 + X A-( 1 -
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Chapter 1: Problem 13 Calculus of A Single Variable 7
In Exercises 9-28. find the vertical asymptotes (if any) of the function. fix)
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Chapter 1: Problem 14 Calculus of A Single Variable 7
In Exercises 9-28. find the vertical asymptotes (if any) of the function. /(.v) = -4a A- + 4
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Chapter 1: Problem 15 Calculus of A Single Variable 7
In Exercises 9-28. find the vertical asymptotes (if any) of the function. git) = -^T^^ * /- -t- 1
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Chapter 1: Problem 16 Calculus of A Single Variable 7
In Exercises 9-28. find the vertical asymptotes (if any) of the function./,lvl 2.V - 3
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Chapter 1: Problem 17 Calculus of A Single Variable 7
In Exercises 9-28. find the vertical asymptotes (if any) of the function. fix) = tan 2a
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Chapter 1: Problem 18 Calculus of A Single Variable 7
In Exercises 9-28. find the vertical asymptotes (if any) of the function. fix) = sec 7TA
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Chapter 1: Problem 19 Calculus of A Single Variable 7
In Exercises 9-28. find the vertical asymptotes (if any) of the function. Til) ^ 1 - i 20. 1
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Chapter 1: Problem 20 Calculus of A Single Variable 7
In Exercises 9-28. find the vertical asymptotes (if any) of the function. '^'^' = 3a^ - 6a -
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Chapter 1: Problem 21 Calculus of A Single Variable 7
In Exercises 9-28. find the vertical asymptotes (if any) of the function. fix) X X- + X - 2
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Chapter 1: Problem 22 Calculus of A Single Variable 7
In Exercises 9-28. find the vertical asymptotes (if any) of the function./'(-v) 4x- + 4v - 24 A^ - 2a' - 9a- + ISv
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Chapter 1: Problem 23 Calculus of A Single Variable 7
In Exercises 9-28. find the vertical asymptotes (if any) of the function. ,dA) _ A-' + 1 ;,( .1 '" ^ "^ A + 1
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Chapter 1: Problem 24 Calculus of A Single Variable 7
In Exercises 9-28. find the vertical asymptotes (if any) of the function.;,( .1 '" ^ "^ A + 1 - a' + 2a- -1- a -1- 2
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Chapter 1: Problem 25 Calculus of A Single Variable 7
In Exercises 9-28. find the vertical asymptotes (if any) of the function.fix) v^ - 2x -\5 1,1,) '' ~ -' A^ - 5.v^ + A - 5
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Chapter 1: Problem 26 Calculus of A Single Variable 7
In Exercises 9-28. find the vertical asymptotes (if any) of the function.1,1,) '' ~ -' A^ - 5.v^ + A - 5 -^ t^ -
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Chapter 1: Problem 27 Calculus of A Single Variable 7
In Exercises 9-28. find the vertical asymptotes (if any) of the function.il) 28. sin f
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Chapter 1: Problem 28 Calculus of A Single Variable 7
In Exercises 9-28. find the vertical asymptotes (if any) of the function. '-'-r
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Chapter 1: Problem 29 Calculus of A Single Variable 7
111 Kxercises 29-32. determine whether the function has a vertical asymptote or a removahle discontinuity at x = I. Graph the function using a graphing utility to confirm your answer. fix) = 31. fix) = 1 A + 1
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Chapter 1: Problem 30 Calculus of A Single Variable 7
111 Kxercises 29-32. determine whether the function has a vertical asymptote or a removahle discontinuity at x = I. Graph the function using a graphing utility to confirm your answer. /(a) 32. /Iv) A- 6a - 7 .V + 1
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Chapter 1: Problem 31 Calculus of A Single Variable 7
111 Kxercises 29-32. determine whether the function has a vertical asymptote or a removahle discontinuity at x = I. Graph the function using a graphing utility to confirm your answer.fix) = 1 A + 1 A- -I- 1 A -I- 1
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Chapter 1: Problem 32 Calculus of A Single Variable 7
111 Kxercises 29-32. determine whether the function has a vertical asymptote or a removahle discontinuity at x = I. Graph the function using a graphing utility to confirm your answer./Iv) A- 6a - 7 .V + 1 sinl.v -I- 1 ) .V -I- 1
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Chapter 1: Problem 33 Calculus of A Single Variable 7
In Exercises .13-48. find the limit. Iini
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Chapter 1: Problem 34 Calculus of A Single Variable 7
In Exercises .13-48. find the limit. lull 35. lim .V.v^l- 1 -A
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Chapter 1: Problem 35 Calculus of A Single Variable 7
In Exercises .13-48. find the limit.lim .V.v^l- 1 -A A37. 1 ! A- 9
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Chapter 1: Problem 36 Calculus of A Single Variable 7
In Exercises .13-48. find the limit.lini -T im 39. lim 7-T ! .V- + .V - 6 ,V- .A .-I (a- + \)ix - 1) I 38. Iim 40. 1 1 4 A- + 16
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Chapter 1: Problem 37 Calculus of A Single Variable 7
In Exercises .13-48. find the limit. im 39. lim 7-T ! .V- + .V - 6
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Chapter 1: Problem 38 Calculus of A Single Variable 7
In Exercises .13-48. find the limit.Iim 40. 1 1 4 A- + 16 6a- -F .V - 1 4v- 4a - 3
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Chapter 1: Problem 39 Calculus of A Single Variable 7
In Exercises .13-48. find the limit.lim 7-T ! .V- + .V - 6 ,V- .A .-I (a- + \)ix - 1)
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Chapter 1: Problem 40 Calculus of A Single Variable 7
In Exercises .13-48. find the limit.lim x - 2
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Chapter 1: Problem 41 Calculus of A Single Variable 7
In Exercises .13-48. find the limit.41. lim 1 +
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Chapter 1: Problem 42 Calculus of A Single Variable 7
In Exercises .13-48. find the limit. lim x2 - 1 x
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Chapter 1: Problem 43 Calculus of A Single Variable 7
In Exercises .13-48. find the limit.lim - ,--.n- sin
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Chapter 1: Problem 44 Calculus of A Single Variable 7
In Exercises .13-48. find the limit.1 (rr/2) ' cos A
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Chapter 1: Problem 45 Calculus of A Single Variable 7
In Exercises .13-48. find the limit.Inn
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Chapter 1: Problem 46 Calculus of A Single Variable 7
In Exercises .13-48. find the limit.lim A
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Chapter 1: Problem 47 Calculus of A Single Variable 7
In Exercises .13-48. find the limit.lim .V sec a
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Chapter 1: Problem 48 Calculus of A Single Variable 7
In Exercises .13-48. find the limit.lim A- tan ttx
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Chapter 1: Problem 49 Calculus of A Single Variable 7
In Exercises 49-52. use a graphing utility to graph the function and determine the one-sided limit. ,/(a) A- 4- A -I- 1 A' - 1 50. /lA a' - 1 lim fix)
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Chapter 1: Problem 50 Calculus of A Single Variable 7
In Exercises 49-52. use a graphing utility to graph the function and determine the one-sided limit. /lA a' - 1 lim fix) V- + .V + 1 lim ,/(.v)
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Chapter 1: Problem 51 Calculus of A Single Variable 7
In Exercises 49-52. use a graphing utility to graph the function and determine the one-sided limit.fix 1 V- 25 lim fix)
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Chapter 1: Problem 52 Calculus of A Single Variable 7
In Exercises 49-52. use a graphing utility to graph the function and determine the one-sided limit.fix) = sec lim fix
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Chapter 1: Problem 53 Calculus of A Single Variable 7
In your own words, describe tlie meaning ol an mtniite limit. Is oc a real nnniber?
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Chapter 1: Problem 54 Calculus of A Single Variable 7
In your own words, describe wliat is meant by an asymptote of a grapli.
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Chapter 1: Problem 55 Calculus of A Single Variable 7
Write a rational function with \ertical asymptotes at .V = 6 and A = -2, and wuh a zero at .v = -^
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Chapter 1: Problem 56 Calculus of A Single Variable 7
Does e\er\ rational liuiclion ha\e a \ertical asymptote? Explain.
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Chapter 1: Problem 57 Calculus of A Single Variable 7
Use the graph of the fmiction / (see llgure) to sketch the graph of vfv) = l//(.v) on the mter\al [-2. 3]. To print an enlarged copy of the graph, go to the website \\)\\\:iihilli:^niplis.ciiin.
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Chapter 1: Problem 58 Calculus of A Single Variable 7
Hoyle's Law For a i|iiantil\ of gas at a constant temperature, the prcsstire F is in\ersely proportional to the \(Mume \'. Fnid the Imin of /' as V>() '
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Chapter 1: Problem 59 Calculus of A Single Variable 7
A gi\en simi .S' is inversely proportional to I ;. where < /| < I . Find the limit of S as r^ I
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Chapter 1: Problem 60 Calculus of A Single Variable 7
Rate of Change A patrol car is parked 5() feet from a long warehouse (see Figuiel The levohing light on top of the car turns at a rate of 3 revolution per second. The rate at which the light beam moves along the wall is ) = ,S{)7rsec- f ft/sec. (a) Find the rate ; when H is tt/6. (h) Find the rate / when is it/?i. (c) Find the limit of / as $>{it/2]~ .
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Chapter 1: Problem 61 Calculus of A Single Variable 7
Illegal l)nii;s The cost in inillioiis of dollars tor a government.il asency to seize -x'-Zc of an illeiial drua is C = 52S,v 100 - v' (1 < .V < 100. (a) Find the cost of seizing 25'-/c of the drug. (b) Find the cost of seizing >iVA ot the drug. (c) Find the cost of seizing 75% of the drug. (d) Find the limit of C as .v> 100 and interpret its meaning.
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Chapter 1: Problem 62 Calculus of A Single Variable 7
Relativity According to the theory of relativity, the mass /;; of a particle depends on its velocity r. That is. where iii is the mass when the particle is at rest and c is the speed of light. Find the limit of the mass as v approaches r
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Chapter 1: Problem 63 Calculus of A Single Variable 7
Rate of Change A 25-foot ladder is leaning against a house (see figure). If the base of the ladder is pulled away from the house at a rate of 2 feet per second, the top will move down the wall at a rate of : ft/sec 7625 where .v is the distance between the base of the ladder and the house. (a) Find the rate r when .v is 7 feet. (b) Find the rale r when .v is 15 feet. (c) Find the limit of r as .v^25"
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Chapter 1: Problem 64 Calculus of A Single Variable 7
Average Speed On a trip of cl miles to another city, a truck dri\'er's average speed was .v miles per hour. On the return trip the a\'erage speed was y miles per hour. The average speed for the round trip was 50 miles per hour (a) Verifv that v (b) Complete the table. What IS the domain? X 30 40 50 60 y Are the values of y different than you expected'.' Explain. (c) Find the limit of y as v>25 " and inteipret its meaning.
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Chapter 1: Problem 65 Calculus of A Single Variable 7
Numerical and Graphical Analysis Use a graphing utility to complete the table for each function and graph each function to estimate the limit. What is the value of the limit when the power on .V in the denominator is greater than 3? X 1 0.5 0,2 0.1 0.01 0.001 0.0001 fix) (a) lim (c) lim (b) lim (d) lim
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Chapter 1: Problem 66 Calculus of A Single Variable 7
Numerical and Graphical Analysis CdiiskJci the shaded region outside the sector of a circle of radius 10 meters and inside a right triangle (see figure). (a) Write the area A = JiH) of the region as a function of 0. Determine the domain of the function. (b) Use a graphing utilitx to complete the table. 0.3 0.6 0.9 1,2 1.5 no) (c) Use a graphing utility to graph the function o\er the appropriate domain. (d) Find the limit of A as H-^(tt/2)-\
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Chapter 1: Problem 67 Calculus of A Single Variable 7
Numerical and Graphical Reasoning A crossed belt connects a 20-centimeter pulle\ ( lO-cm radius) on an electric motor with a 40-centiiTietcr pulley (20-cm radius) on a saw arbor (see figure). The electric motor runs at 1700 re\cilulions per minute. (a) Determine the number of revolutions per minute of the saw. (b) How does crossing the belt affect the saw in relation to the motor' (c) Let L be the total length of the belt. Write L as a function of 4>. where 4' is measured in radians. What is the domain ol the function' (Hint: Add the lengths of the straight sections of the belt and the length of the belt around each pulley.) (d) LIse a graphing utility to complete the table. ^ 0.3 0.6 0.9 1.2 1.5 L (f) Find Inn L. .for/:)- Use a geometric argument as the basis of a second method of finding this limit. (a) Find lim L. 20 cm (e) Use a graphing utility to graph the function over the appropriate domain.
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Chapter 1: Problem 68 Calculus of A Single Variable 7
True or False? In Exercises 68-71, determine vvhetlier the statement is true or false. If it is false, explain why or give an example that shows it is false. If /i(v) is a polynomial, then the function given by .V - 1 has a vertical asympt
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Chapter 1: Problem 69 Calculus of A Single Variable 7
True or False? In Exercises 68-71, determine vvhetlier the statement is true or false. If it is false, explain why or give an example that shows it is false. A rational t unction has at least one vertical asymptote.
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Chapter 1: Problem 70 Calculus of A Single Variable 7
True or False? In Exercises 68-71, determine vvhetlier the statement is true or false. If it is false, explain why or give an example that shows it is false.Polynomial I unctions ha\e no \erlical asymptotes.
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Chapter 1: Problem 71 Calculus of A Single Variable 7
True or False? In Exercises 68-71, determine vvhetlier the statement is true or false. If it is false, explain why or give an example that shows it is false. II / has a \erlical asymptote at v = 0. then / is undefined al A = 0.
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Chapter 1: Problem 72 Calculus of A Single Variable 7
True or False? In Exercises 68-71, determine vvhetlier the statement is true or false. If it is false, explain why or give an example that shows it is false.Find functions / and .,' such that lull fix) = z/z and lim g{x) = zo but Inn [/(>! - ,t;fv)] ^ 0
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Chapter 1: Problem 73 Calculus of A Single Variable 7
True or False? In Exercises 68-71, determine vvhetlier the statement is true or false. If it is false, explain why or give an example that shows it is false.Prove the leniaiiiiug properties of Theorem 1.15.
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Chapter 1: Problem 74 Calculus of A Single Variable 7
True or False? In Exercises 68-71, determine vvhetlier the statement is true or false. If it is false, explain why or give an example that shows it is false.Pro\e that il lim fix) = cc then lim - = 0.
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Chapter 1: Problem 75 Calculus of A Single Variable 7
True or False? In Exercises 68-71, determine vvhetlier the statement is true or false. If it is false, explain why or give an example that shows it is false.Prove that it Inn - - then liiii /(.v) does not exist
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