Mathematics 111 (Calculus II) Laboratory Manual

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1 Mathematics (Calculus II) Laboratory Manual Department of Mathematics & Statistics University of Regina nd edition prepared by Patrick Maidorn, Fotini Labropulu, and Robert Petry University of Regina Department of Mathematics and Statistics

2 Contents Module. Inverse Functions. Inverse Functions Exponential and Logarithmic Functions Calculus of Exponential and Logarithmic Functions Inverse Trigonometric Functions L Hopital s Rule Module. Techniques of Integration 7. Integration by Parts Trigonometric Integrals Trigonometric Substitution Partial Fractions Challenge Integration Practice Improper Integrals Module. Integration Applications. Review - Areas Between Curves Volumes by Cross Sections Volumes by Cylindrical Shells Arclength Module Sequences and Series Sequences Series The Integral Test The Comparison Tests The Alternating Series Test Absolute Convergence and the Ratio and Root Tests Strategies for Testing Series Power Series Representations of Functions as Power Series Taylor and Maclaurin Series Module 5. Parameteric Equations and Polar Coordinates 9 5. Curves Defined by Parametric Equations Calculus with Parametric Curves Polar Coordinates Areas and Lengths in Polar Coordinates Answers References i

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4 Introduction One does not learn how to swim by reading a book about swimming, as surely everyone agrees. The same is true of mathematics. One does not learn mathematics by only reading a textbook and listening to lectures. Rather, one learns mathematics by doing mathematics. This Laboratory Manual is a small set of problems that are representative of the types of problems that students of Mathematics (Calculus II) at the University of Regina are expected to be able to solve on quizzes, midterm exams, and final exams. In the weekly lab of your section of Math you will work on selected problems from this manual under the guidance of the laboratory instructor, thereby giving you the opportunity to do mathematics with a coach close at hand. These problems are not homework and your work on these problems will not be graded. However, by working on these problems during the lab periods, and outside the lab periods if you wish, you will gain useful experience in working with the central ideas of elementary calculus. The material in the Lab Manual does not replace the textbook. There are no explanations or short reviews of the topics under study. Thus, you should refer to the relevant sections of your textbook and your class notes when using the Lab Manual. These problems are not sufficient practice to master calculus, and so you should solidify your understanding of the material by working through problems given to you by your professor or that you yourself find in the textbook. To succeed in calculus it is imperative that you attend the lectures and labs, read the relevant sections of the textbook carefully, and work on the problems in the textbook and laboratory manual. Through practice you will learn, and by learning you will succeed in achieving your academic goals. We wish you good luck in your studies of calculus.

6 Module Inverse Functions. Inverse Functions. Determine whether each of the following functions is invertible on its domain. [ (a) f(x) = + x 4 (d) f(x) = cos x on x 0, π (b) f(x) = sin x + cos x (c) f(x) = x x (e) f(x) = { x if x 0 x if x > 0 ] Page. In each case, find a formula for the inverse function f (x). (a) f(x) = x (b) f(x) = x 4x on x (c) f(x) = x + x. In each case, check whether f(x) is a one to one differentiable function, and if it is, find ( f ) (a). (a) f(x) = x 5 + x + 4, a = 8 (c) f(x) = x x, a = (b) f(x) = sin x, a =. Exponential and Logarithmic Functions. Write each expression as a single exponential. (a) ( +x) x (b) e 5x e x Page. Write as a single logarithm. (a) 4 ln(x) ln(x) + (b) 4 log(x + ) log(x) +

7 4 MODULE. INVERSE FUNCTIONS. Solve each equation for x. (a) x = 8 ( ) x+ (b) = (c) 4e x = 6 ( ) x+ 4 (d) = 0 4x (e) ln(ln(x)) = Assume the world s population doubles every 5 years. (a) Find its annual growth rate k in N(t) = N 0 e kt. (b) In 998, Earth s population was 6 billion. Use the model in (a) to predict the population in 00. (c) In what year will Earth s population reach 0 billion, according to this model? 5. Radioactive carbon-4 has a half-life of 570 years. How long will it take for an object to lose 80% of its original C-4 content?. Calculus of Exponential and Logarithmic Functions Page. Find the indicated derivatives. (a) f(x) = e x + e 4x, f (x) (b) f(x) = e sin x, f (x) (c) f(x) = e x x, f (x) (d) f(x) = ln( + x + x), f (0) (e) f(x) = ln(cos x), f (x) ( ) x (f) f(x) = ln x, f (x) +. Integrate the following: (a) e 4x + 4 x dx (b) (c) 0 e x e x dx x dx (d) (e) (f) 4 0 x e x + dx x x + dx e x x dx. Use logarithmic differentiation to find f (x). (a) f(x) = x x (b) f(x) = x sin x (c) f(x) = (x + ) (x 7) 5 (x + ) 6 Find the equation of the tangent line to f(x) = e x at the point x =. 5. Find the area between y = e x and y = e x between x = and x =.

8 . INVERSE TRIGONOMETRIC FUNCTIONS 5.4 Inverse Trigonometric Functions. Find the exact value of each expression. ( ) (a) sin ) (b) cos ( ( ( )) (c) sin sin (d) cos (cos π) Page. Differentiate the following: (a) f(x) = cos x + 5 tan x (b) f(x) = tan x (c) f(x) = sin x 4 (d) f(x) = cos ( sin x ). Find the equation of the tangent line to the graph of f(x) = sec x at the point (x, y) = (, π ). 50 ft θ x As the sun descends, the shadow cast by a 50 ft tall wall lengthens. (a) Express the angle θ as a function of the shadow s length x. (b) Find dθ when the shadow s length is 00 ft. dx 5. Integrate the following: (a) (b) + x dx x x 4 dx (c) tan x x + dx

9 6 MODULE. INVERSE FUNCTIONS.5 L Hopital s Rule Page -0: In each case, find the indicated limit. Note: not all limits allow the use of L Hopital s Rule. sin x. lim x 0 x x + x +. lim x 4x + ln( + x). lim x 0 x x lim x x + x 5 + 4x 5. lim x e x 6. lim x 0 x csc x 7. lim x 0 + xx 8. lim(x + cos x) x x 0 ( 9. lim x 0 + sin x ) x ( 0. lim x x ) e x

10 Module Techniques of Integration. Integration by Parts -0: Integrate the following:. x sin x dx. xe x dx (ln(x)) dx x x + dx Page. x cos x dx 8. cos(x)e x dx 5. e π 0 x ln(x) dx ( x ) (x + 5) cos dx ax e bx dx x 5 sin x dx. Trigonometric Integrals -9: Integrate the following:. sin x cos 5 x dx. sin 5 x cos x dx π. 0 sin x cos x dx cos 4 x dx tan x sec 4 x dx π 0 tan(t) sec (t) dt cos 5 x dx Page 4 5. tan x sec 4 x dx 9. tan 5 x dx 7

11 8 MODULE. TECHNIQUES OF INTEGRATION. Trigonometric Substitution Page 4-9: Integrate the following:. x x dx x 4 x dx 5x dx x dx (4 + x ) x x dx (6 x ) dx x x + 6 x + 4x 5 dx x + (x 6x + ) dx.4 Partial Fractions Page 4. In each case, find the partial fraction decomposition. (a) (b) (x )(x ) x + x x 6 (c) 5x x + x x (d) 7x x + (x )(x x + ). Integrate the following: dx (a) x + x + x + (b) (x + ) (x + ) dx x + 7x (c) x x x dx x + (d) 9x + 6x + 5 dx (e) (f) (g) dx x + x + 0x + x + 6 (x ) (x + 4) dx 5x 4x + x + 5x dx 4x

12 .5. CHALLENGE INTEGRATION PRACTICE 9.5 Challenge Integration Practice -5: Integrate the following: e x. dx e x.. x 5 x + dx sec x dx 5. 0 x + dx e 4x + e x dx Page 5.6 Improper Integrals -0: In each case, determine whether this integral converges or diverges. If it converges, evaluate the integral x dx e x dx dx x dx 9 x f(x) dx, where { f(x) = x 0 < x x < x π 0 ( x ) tan dx xe x dx dx x + dx e x + e x ln x dx Page 5

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14 Module Integration Applications. Review - Areas Between Curves. Find the area of the region R bound by the line y = x and the parabola y = 6 x.. Find the area of the region R enclosed by y = sin x and y = cos x from x = 0 to x = π.. Find the area of the region R enclosed by y = x, y = x 4, x =, and x =. Page 5. Volumes by Cross Sections. The region R is bounded by the curves y = x and y =. R is rotated about the line y =, generating a ring shaped solid. Sketch the region R as well as a typical cross section of the solid. Find the volume of the solid.. Find the volume of the solid S obtained by rotating the region bounded by y = x and y = x about the x-axis. Page 5. Find the volume of the right-circular cone with base radius r and height h. Note: the cone is generated by rotating the triangle with vertices (0, 0), (0, h), and (r, h) about the y-axis. Consider the region R, bound by y = x and y = x. Find the volume of the resulting solid if (a) R is revolved around the x-axis. (b) R is revolved around the y-axis. (c) R is revolved around the vertical line x =. 5. (a) Sketch the curve given by x = y y. (b) Find the volume obtained by rotating the region enclosed by x = 0 and x = y y about the y-axis. 6. (a) R is bounded by y = sin x, y = 0, x = 0, and x = π. Rotate R about the x-axis. Find the volume of the resulting solid. (b) R is bounded by y = sin x, y = cos x, x = 0, and x = π. Rotate R about the x-axis. 4 Find the volume of the resulting solid.

15 MODULE. INTEGRATION APPLICATIONS Page 6 7. A circular man-made lake has[ a 00m diameter and a maximum depth of 0m. Its cross ( x ) ] section is the parabola y = 0. Find the capacity of the lake. 00. Volumes by Cylindrical Shells. Consider the bowl obtained by revolving the region bounded by y = x, y =, and x = 0 about the y-axis. (a) Find its volume using cross sections. (b) Find its volume using cylindrical shells. (c) Compare your answers.. Consider the region bounded by y = x and y = x. Use cylindrical shells to find the volume of the resulting solid if (a) R is revolved about the x-axis. (b) R is revolved about the y-axis. (c) R is revolved about the vertical line x =. Note: Compare your answers with those of question 4 in Section... Find the volume V of the solid generated by revolving the region enclosed by y = x x, y = 0, x = 0, and x = about the y-axis. Determine the volume of the solid obtained by rotating the region bounded by y = x and y = x about the line x = Find the volume of the solid obtained by rotating the region bounded by y = (x ) and y = (x ) about the y-axis. 6. Consider the solid sphere of radius R. A cylinder of radius r < R is bored through the center of the sphere. Find the volume of the remaining solid. 7. Consider f(x) = sin(x ) and g(x) = sin(x ) from x = 0 to x = π. Find the volume of revolution if the region enclosed by f(x) and g(x) is rotated about the y-axis. 8. Let R be the region in the first quadrant bounded by y = (x ) / and let y =. (a) Find the resulting volume if R is rotated about the x-axis. (b) Find the resulting volume if R is rotated about the line y =..4 Arclength. Find the length of the curve f(x) = x between x = 0 and x = Page 6. Find the length of the curve f(x) = e x + 8 e x between x = 0 and x = ln ( ) x (). Find the length of the curve y = + between x = 0 and x =. Hint: consider the curve as a function x(y) instead. Find the length of the curve x = 6 y + y between y = and y =.

16 Module 4 Sequences and Series Sequences -5: Determine whether the given sequence is convergent or divergent.. a n = n4 + 5n + n 4 +. a n = n6 4n + n 5 4n + 7. a n = e n ln n ( ) n a n = ln n + 5 Page 6 5. a n = (n)! (n + )! 6-9: Determine whether the given sequence is increasing or decreasing. Also, determine whether it is bounded. 6. a n = n 5n + 7. a n = n ( 4 n) 8. a n = ln n n a n = + ln n

17 4 MODULE SEQUENCES AND SERIES Series Page 7-6: Determine whether the given series is convergent or divergent. If it is convergent, find its sum. n +. n n 4 n ( ) n + 4 ( n+ ) 5 (n + )(n + ) ( ) 5n ln 7n + 4 [ n(n + ) 5 ] n The Integral Test Page 7-6: Determine whether the given series is convergent or divergent n n 4n + n e n4 ln n n 4 n ( n + 4) 4 4n +

18 THE COMPARISON TESTS 5 4 The Comparison Tests -6: Determine whether the given series is convergent or divergent.. n + n + 5 Page n + 4n n 4n + 5 5n 6 + n 4 + n + n(5 n ) + n n (n + ) n ( + n + 8n ) 5 The Alternating Series Test -5: Determine whether the given series is convergent or divergent ( ) n n n + 4 ( ) n n ( ) n ln n n ( ) n n 4 n ( ) n e n n + Page 7

19 6 MODULE SEQUENCES AND SERIES 6 Absolute Convergence and the Ratio and Root Tests -5: Determine whether the given series is convergent or divergent. Page n n(5 n+ ) ( ) n ne n n n 5 n n! (n)! (n) n (n + 5) n 7 Strategies for Testing Series -5: Determine whether the given series is convergent or divergent. Page n n 5 n(6 n ) + ln(n) e n (n + )! n! n (n 5) 5 n (6n + n + )

20 8. POWER SERIES 7 8 Power Series -5: Find the radius of convergence and the interval of convergence. x n. n n n xn n (x + )n ( ) n (x 5) n 5 n n + 7 (4x 5) n 9 Representations of Functions as Power Series -5: Find a power series representation for f(x) and determine the interval of convergence.. f(x) = x +x. f(x) = 4 x+7. f(x) = x (+x) f(x) = ln(x + ) 5. f(x) = +x Page 8 Page 8 0 Taylor and Maclaurin Series -: Find the Maclaurin series for f(x) and state the radius of convergence.. f(x) = xe x. f(x) = x sin x Page 8. f(x) = x cos 4x 4-5: Find the Taylor series for f(x) at the indicated number a and state the radius of convergence. f(x) = sin x; a = π 4 5. f(x) = e x ; a = 4

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22 Module 5 Parameteric Equations and Polar Coordinates 5. Curves Defined by Parametric Equations -4: In the following problems (a) Sketch the graph of the curve having the indicated parametric equations, and (b) Eliminate the parameter to find a Cartesian equation of the curve.. x = t +, y = t ; t Page 9. x = 4t 4, y = t + 5; t R. x = cos t, y = sin t; 0 t π x = 4 cos t, y = sin t; 0 t π 5. Calculus with Parametric Curves -: Find the equation of the tangent line to the curve at the point corresponding to the given value of the parameter.. x = 4t, y = t + ; t = Page 9. x = e t, y = e t ; t = 0. x = sin t, y = 4 cos t; t = π/4 4-5: Find the points on the curve at which the tangent line horizontal or vertical. x = 4t, y = t 7t 5. x = e t, y = t + e t 9

23 0 MODULE 5. PARAMETERIC EQUATIONS AND POLAR COORDINATES 5. Polar Coordinates Page 9-4: Find a Cartesian equation for the given curve and identify it.. r = 4. r = sin θ. r ( 5 sin θ 4 cos θ ) = θ r cos θ = 4 5-7: Find a polar equation that has the same graph as the given Cartesian equation. 5. x + y = 5 6. x y = x + 5y = : Find the slope of the tangent line to the given curve at the indicated value of θ. 8. r = 4 cos θ; θ = π/6 9. r = + cos θ; θ = π/ 0. r = cos 4θ; θ = π/4 5.4 Areas and Lengths in Polar Coordinates Page 9 -: Sketch the graph of the equation and find the area of the region bounded by the graph.. r = 4 sin θ. r = + cos θ -4: Find the area of the region bounded by one loop of the graph of the give equation.. r = 6 cos 4θ r = cos θ 5-6: Find the area of the region that is outside the graph of the first equation and inside the graph of the second equation. 5. r = 9, r = cos θ 6. r = cos θ

24 5. AREAS AND LENGTHS IN POLAR COORDINATES 7-8: Find the length of the given polar curve. 7. r = e 4θ, 0 θ 8. r = cos ( θ ), 0 θ π

25 Answers. Exercises (page ). (a) No (b) No (c) Yes (d) No (e) Yes. (a) f (x) = (x + ) (b) f (x) = (x + 4) + (c) f (x) = x x +. (a) ( f ) (8) = 4 (b) ( ( ) f ) = (c) Not -. Exercises (page ). (a) 6 x (b) e 4x ( ) ex 4 ( ) 00(x + ) 4. (a) ln (b) log (x) x. (a) x = 5 (b) x = (c) x = ln(4) (a) k = ln() 5 5. (a) 00 years. Exercises (page 4) (b) 8 Billion (c) In 07 (d) x = 4 (log() + ) (e) x = ee (a) f (x) = xe x + e 4x (b) f (x) = ( cos x sin x ) e sin x (c) f (x) = (d) f (0) = (e) f (x) = sec x (f) f (x) = x x x + x e x e x x. (a) 4 e4x + 4 ln x + c (b) (f) e (e ) e (c) ln x + c (d) ex + + c (e) ln x + + c (. (a) f (x) = x x ( + ln(x)) (b) f (x) = x sin x cos(x) ln(x) + sin x x ) (c) f (x) = (x 7)5 (x + ) (x + ) 6 ( 5 x 7 + x + 6(x + ) ) y = e x + e

26 ANSWERS 5. e + e 4.4 Exercises (page 5). (a) π (b) 5π 6 (c) (d) π. (a) f (x) =. y = x + π π 9 ( ) 50 (a) θ = tan x + 5 x + x (b) f (x) = 5. (a) π 6 (b) sin x + c (c).5 Exercises (page 6) (b) Rad ft ( tan x ) + c x( + x) (c) f (x) = x (d) f (x) = x 4 x x e Exercises (page 7). x cos x + sin x + c 4. xe x 9 e x + c. x sin x + 6x cos x 6 sin x + c ] e x ln(x) 9 x = 9 e + 9 ( x ) ( x )] π 5. (x + 0) sin + 48 cos 4 4 = 4π x (ln(x)) x ln(x) + x + c x(x + ) 4 5 (x + ) 5 + c cos(x)ex + sin(x)ex + c a b x e bx a b xebx + a b ebx + c 0. x 5 cos x + 5x 4 sin x + 0x cos x 60x sin x 0x cos x + 0 sin x + c

27 4 ANSWERS. Exercises (page 7). sin x 5 sin5 x + 7 x + c 6. tan x + 7 tan 7 x + c. cos x + 5 cos5 x 7 cos7 x + c x ] π sin 4x = π 0 8 sin x x sin 4x + c 6 tan6 x + 4 tan4 x + c 7. 6 sec t + c 8. sin x sin x + ] π 5 sin5 x 0 9. = tan4 x tan x + ln sec x + c. Exercises (page 8) (. x ) 5 ( x ) + c 5. 4 ( 4 x ) + 4 ( 4 x ) 5 + c 5 ] π. tan(θ) θ ( x 6 tan 5. x + = π 0 ) + 8 x 4 + x + c ( x ) + c ] π 6 tan θ tan = 6 ( x ) + c ] π 8. (tan(θ) θ) = π 0 9. ln ( x 6x + ) + (x ) + c.4 Exercises (page 8). (a) x 9 x (b) 5 x x (c) x x + 4 x (d) 5 x + x + x x +. (a) ln x + ln x + + c (b) ln x ln x + + tan x + c (c) ln x ln x ln x + c (d) 9 ln ( 9x + 6x + 5 ) tan ( x + ) + c (e) ln x + 6 ln x x + + tan ( x ) + c (f) ln x ln x + 4 ( x ) tan (g) x 5 ln 5x + + ln x + c x c

28 ANSWERS 5.5 Exercises (page 9). sin e x + c By substitution and trigonometric substitution.. ( 9 x x + ) 4 ( x + ) 5 + c By substitution and parts 45 ( ( x ) ( x ) ( cos x ) ( x ) ) tan(x) sec(x) ln cos sin + ln + sin ( ) + ln + By parts and trigonometric substitution ( 5. + e x ) 5 ( + e x ) + c By substitution and trigonometric substitution 5.6 Exercises (page 9). Convergent. Convergent. Divergent Convergent π 5. Convergent 5 6. Convergent ln() 7. Convergent 0 8. Convergent π 9. Convergent π 0. Convergent. Exercises (page ) Exercises (page )... 56π 5 π 5 πr h (a) 5π 4 (b) π 5 (c) 7π 0

29 6 ANSWERS 5. (a) y x (b) 6π 5 6. (a) π (b) π π m. Exercises (page ). (a) V = π (b) V = π. (a) 5π 4 (b) V = π 5. 4π 0 7π 5 (c) The same. (c) V = 7π π 0 4π 7. 4π ( R r ) 8. (a) 6π (b) 8π.4 Exercises (page ). Length 9. units. Length = 6 units. Length = 4 units Length = 7 units Exercises (page ). convergent. divergent. convergent convergent 5. convergent 6. increasing, bounded 7. decreasing, bounded 8. increasing,bounded 9. decreasing,bounded

30 ANSWERS 7 Exercises (page 4). divergent. convergent, sum=. convergent, sum= 5 4 convergent, sum= 5 5. divergent 6. divergent Exercises (page 4). convergent. divergent. convergent convergent 5. divergent 6. convergent 4 Exercises (page 5). convergent. divergent. convergent convergent 5. convergent 6. convergent 5 Exercises (page 5). convergent. convergent. convergent convergent 5. divergent 6 Exercises (page 6). convergent. convergent. divergent convergent 5. divergent 7 Exercises (page 6). convergent. convergent. convergent convergent 5. convergent

31 8 ANSWERS 8 Exercises (page 7). R =, x <. R = 4, 4 x 4. R =, x R = 5, 0 < x < 0 5. R = 4, x < 9 Exercises (page 7) n=0 ( ) n xn+ n+, x < ( ) n n 7 n+ xn, x < 7 n=0 n=0 ( ) n n+ n+ nxn, x < ( ) n x n+ n+ + ln, x < n + n=0 n=0 ( ) n xn n, x < 0 Exercises (page 7)... ( ) n n n! xn+, R = n=0 ( ) n xn+ (n + )!, R = n=0 n=0 ( ) n 4n (n)! xn+, R = [ + ( x π 4 )! ( ) x π 4! ( ) x π 4 + 4! ] 5. e [ 8 + (x 4) + 4! (x 4) + 8! (x 4) +, R = ( ) x π ], R =

32 ANSWERS 9 5. Exercises (page 9). line segment: y = x 5, x 6. parabola: x = (y 5) 4. circle: x + y = 4 ellipse: x 6 + y 9 = 5. Exercises (page 9). y = 4 x + 4. y = x + 4. y = x + 4 Horizontal if t = ±, Vertical if t = 0 5. Horizontal if t = ln, No Vertical 5. Exercises (page 0). circle: x + y = 6. circle: x + ( y ) = 9 4. hyperbola: y 4 x 5 = hyperbola: y x = 4 5. r = 5 6. r cos(θ) = 9 7. r ( 4 + sin θ ) = dy dx = dy dx = dy dx = 5.4 Exercises (page 0). A = 8π. A = π. A = 9π 8 A = 5. A = 8 9π 6. A = 8 + 8π ( 7. L = e 8 ) 8. L = 7 4

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34 Appendix A References The problems for this manual were collected from a variety of sources, including instructors s personal class notes and exams, as well as the following resources: Adams, Essex: Calculus: A Complete Course, 8 th Edition, Pearson. Briggs, Chocran: Calculus: Early Transcendentals, Addison Wesley. Dawkins: Paul s Online Math Notes, Edwards, Penny: Calculus, Early Transcendentals, 7 th Edition, Prentice Hall. Tan, Menz, Ashlock: Applied Calculus for the Managerial, Life, and Social Sciences, st Canadian Edition, Nelson. Zill, Wright: Differential Equations with Boundary-Value Problems, 8 th Edition, Brooks/Cole.

Homework Problem Answers

Homework Problem Answers Integration by Parts. (x + ln(x + x. 5x tan 9x 5 ln sec 9x 9 8 (. 55 π π + 6 ln 4. 9 ln 9 (ln 6 8 8 5. (6 + 56 0/ 6. 6 x sin x +6cos x. ( + x e x 8. 4/e 9. 5 x [sin(ln x cos(ln