WORKBOOK. MATH 32. CALCULUS AND ANALYTIC GEOMETRY II.

WORKBOOK. MATH 32. CALCULUS AND ANALYTIC GEOMETRY II. DEPARTMENT OF MATHEMATICS AND COMPUTER SCIENCE Contributors: U. N. Iyer, P. Laul, I. Petrovic. (Many problems have been irectly taken from Single Variable Calculus, 7E by J. Stewart, an Calculus: One Variable, 8E by S. Sallas, E. Hille, an G. Etgen. ) Department of Mathematics an Computer Science, CP 315, Bronx Community College, University Avenue an West 181 Street, Bronx, NY 10453. PL, 2015 (Version 1) This version has been reformatte by Kerry Ojakian (August 2015) 1

MTH 32 2 Contents 1. Recall from MTH 31 3 2. Area between curves 6 3. Volumes 8 4. Volumes by Cylinrical Shells 9 5. Review Chapter 5 10 6. Inverse Functions 11 7. Exponential Functions an their erivaties 14 8. Logarithmic functions 20 9. Derivatives of Logarithmic Functions 22 10. Inverse Trigonometric Function 24 11. Hyperbolic Functions 27 12. Intermeiate forms an L Hospital s Rule 31 13. Review Chapter 6 32 14. Integration by Parts 34 15. Trigonometric Integrals 35 16. Trigonometric Substitutions 37 17. Integration of Rational Functions by Partial Fractions 38 18. Strategy for Integration 39 19. Improper Integrals 40 20. Review Chapter 7 41 21. Arc Length 42 22. Area of surface of revolution 43 23. Curves efine by parametric equations 44 24. Calculus with parametric curves 45 25. Polar Coorinates 46 26. Areas an Lengths in Polar Coorinates 47 27. Conic Sections 48 28. Conic Sections in Polar Coorinates 49 29. Review Chapter 10 50 30. Practice Problems 51

MTH 32 3 1. Recall from MTH 31 (1) State the efiniton of the area A of the region uner the graph of a continuous functon using limit Riemann sums. Draw an illustration to explain this proceure. (2) Draw an illustration of four rectangles to estimate the area uner the parabola y = x 2 from x = 1 to x = 3 using (a) left enpoints; (b) right enpoints; (c) mipoints; () Guess the actual area. (3) What is the Definite Integral of a function f from a to b? (4) The symbol was introuce by an is calle an. It is an elongate S an is chosen because an integral is a. (5) In the notation b f(x)x, a f(x) is calle, a an b are calle, a is the, an b is the. (6) The symbol x simply inicates that. (7) The sum n i=1 f(x i ) x is calle, name after the German mathematician. (8) Theorem: If f is continuous on [a, b], or if f has only a finite number of jump iscontinuites on [a, b], then f is ; that is, exists. (9) Theorem: If f is integrable on [a, b], then f(x)x = where x = an x i =. (10) Properties of Definite Integrals: (a) a f(x)x =. b (b) a f(x)x =. a (c) b cf(x)x =. a () b [f(x) + g(x)]x =. a (e) b [f(x) g(x)]x =. a (f) b cx =. a (11) Fill in the blanks with the appropriate inequality. (a) If f(x) 0 for a x b, then b f(x)x 0. a (b) If f(x) g(x) for a x b, then b f(x)x b a g(x)x. a b (c) If m f(x) M for a x b, then m(b a) f(x)x M(b a). a (12) State the Funamental Theorem of Calculus. (13) Fin the erivative of: (a) g(x) = x 0 4 + t2 t = (b) h(x) = x 0 sin ( πt 2 2 ) t =

(c) k(x) = x 6 cos(t 2 + 5)t = 0 (Hint: Do not forget the Chain rule). (14) Evaluate the integral: (a) 3 0 r4 + r 2 + 5 r (b) ( ) 8 1 1 u + 2 3u2 u (c) 3 r 4 + r 2 + 5 r 1 r 9 () 8 ( 3 u + 3u 2 ) u 0 (e) π/4 sec(θ) tan(θ) θ π/4 (f) ( ) 8 3 5 u 1 u (g) π cos(t) for π/2 t π/2 f(t)t where f(t) = π/2 sin(t) for π/2 t π. (15) Complete the table: cf(x)x [f(x) + g(x)]x [f(x) g(x)]x kx x n x sin(x)x cos(x)x sec(x) tan(x)x sec 2 (x)x csc(x) cot(x)x csc 2 (x)x (16) Fin the general inefinite integral: (a) ( x 5 + 3 x 7 )x (b) ( x 5 + 3 x 7 ) 2 x (c) sec(θ)(sec(θ) + 3 tan(θ))θ () ( u 4 + 3 + 1 ) u u 4 (e) ( ) sin(2t) t cos(t) (f) ( x 5 + x 7 x 3 ) x MTH 32 4

(17) Evaluate the following efinite integrals: (a) 3 2 (x5 + x 3 ) 2 x (b) 7 ( ) x5 + x 7 x 1 MTH 32 5 (c) π 3 sec(θ)(sec(θ) + 3 tan(θ))θ π 3 () 7 0 (u2 + 3) 2 u (e) π/2 π/2 (t2 sin(t))t (f) π (sin(t))t π/2 (g) π sin(t) t π/2 (h) ( ) 8 1 1 x + 5 x7 x (18) The acceleration function in km/sec 2 is a(t) = 3t + 7 where time t is in secons, an 0 t 10. Let v(t) (in km/sec) an s(t) (in km) be the velocity an position functions respectively with the initial velocity, v(0) = 45 km/sec an the initial position s(0) = 4 km. Fin the velocity function an the position functions. Then, fin the total istance covere. (19) State the Substitution Rule for integration. (20) Fin 4x 3 x 4 + 10x (21) Fin x 3 cos(x 4 + 10)x (22) Fin 3x 5x (23) Explain the Substitution Rule for Definite Integrals. (24) Fin 3 (25) Fin 7 2 1 3x + 5x 1 (3x + 5) x 5 (26) Fin 5 x5 1 + x 0 2 x (27) Suppose thatf is a continuous function on [ a, a]. (a) If f is even on [ a, a] then a f(x)x = a (b) Fin 3 3 (x4 + 2x 2 1)x (c) If f is o on [ a, a] then a f(x)x = a () Fin 3 3 (x5 + 2x 3 x)x (28) Fin the integral: (a) x 7 x 8 + 5 x (b) 1 (3 7t) t 8 (c) csc 2 (1/x) x x 2 () x 6 cos(x 7 + 5)x (e) x 3 sec(x 4 + 5) tan(x 4 + 5) x (f) (5t 7) 3.55 t (g) 5 1 0 (1 + x) x 5 (h) π/2 π/2 x8 sin(x)t (i) a x a 0 2 x 2 x (j) a a2 x a 2 x

MTH 32 6 2. Area between curves (1) Theorem: The area A of the region boune by the curves y = f(x), y = g(x), an the lines x = a, x = b where f an g are continuous an f(x) g(x) for all x [a, b] is A =. Explain with an illustration. (2) Fin the area of the shae region: Y y = (x 1) 2 y = x + 1 (3, 4) (0, 1) X (3) Theorem: The area between the curves y = f(x) an y = g(x) an between x = a an y = b is A =. Explain with an illustration. (4) Fin the area of the region boune by the curves y = sin x, y = cos x, x = π/2, an x = 3π/2. You will nee to graph these two functions on the given omain. (5) Sometimes we encounter regions boune by curves obtaine when x is a function of y. In this case, if x R enotes the right han sie curve, an x L enotes the left han sie curve, then the area A = c (x R x L )y where c, are limits for y.

MTH 32 7 (6) Fin the area of the shae region: Y x = y 2 + 1 x = y 2 1 X (7) Try as many problems (# 1-30) as time permits you from the textbook.

MTH 32 8 3. Volumes (1) What is the formula for the volume of a right cycliner? (Note, the base may be an irregular shape). Draw an illustration. (2) What is the formula for the volume of a circular cyliner? Draw an illustration. (3) What is the formula for the volume of a rectangular box (also calle, a rectangular parallelepipe). Draw an illustration. (4) Let S be a soli that lies between x = a an x = b. If the cross-sectional area of S in the plane P x, through x an perpenicular to the x-axis, is A(x), where A is a continuous function, then the volume of S is V = lim n n A(x i ) x =. i=1 Draw an illustration. (5) Fin the volume of a sphere of raius r. Draw an illustration. (6) The above is an example of the volume of solis of revolution. (7) Fin the volume of the soli obtaine by rotating the region boune by y = 36 x 2, y = 0, x = 2, x = 4, about the x-axis. (8) Fin the volume of the soli obtaine by rotating the region boune by y = 1 + sec x, y = 3, about the line y = 1. Draw an illustration. (9) Set up an integration to fin the volume an raw an illustration each of the soli obtaine by rotating the region boune by y = 0, y = cos 2 x, π/2 x π/2 (a) about the x-axis; (b) about the line y = 1. (10) Fin the volume of a soli S where the base of S is the region enclose by the parabola y = 1 x 2 an the x-axis. The cross-sections perpenicular to the y-axis are squares. (11) Try as many problems as time permits you to from the textbook on this section (# 1-34, 54-60).

MTH 32 9 4. Volumes by Cylinrical Shells (1) Sometimes the methos of the previous section may not be feasible. So, we use the metho of cylinrical shells. (2) What is the formula for the volume of a cyliner? Draw an illustration. (3) What is the formula for the volume of a cylinrical shell? Draw an illustration. Write in the form of V = (circumference) (height) r. (4) Theorem: The volume of the soli obtaine by rotating about the y-axis the region uner the curve y = f(x) from x = a to x = b is V =... Draw an illustration. (5) Fin the volume an raw an illusration of the soli S obtaine by rotating the region boune by π (a) y = cos(x 2 ), y = 0, x = 0, x = about the y-axis. 2 (b) y = x 2, y = 4x x 2 about the y-axis. (c) x = y 2 + 1, x = 2, about the line y = 2. () x 2 y 2 = 5, x = 3, about the line y = 5. (6) Try as many problems from the textbook as time permits you to (# 1-26).

MTH 32 10 5. Review Chapter 5 (1) Fin the area of the region boune by the curves an raw an illustration (a) y = x 2 an y = 6x x 2. (b) x + y = 0 an x = y 2 + 2y. (2) Fin the volume of the soli obtaine by rotating the region boune by the given curves about the specific axis an raw an illustration (a) y = x 2 + 1 an y = 4 x 2 about the line y = 2 (b) x 2 y 2 = 25, x = 8 about the y-axis. (3) Fin the volume of the soli S whose base is a square with vertices (1, 0), (0, 1), ( 1, 0), (0, 1) an the cross-section perpenicular to the x-axis is a semicircle. (4) Fin the volume of the soli obtaine by rotating the region boune by the curves y = x, y = x 2 about the line y = 2. (5) Do as many problems from the textbook as time permits you to (Review section of chapter 5: # 1,2, 7-10,14,15, 22-26).

MTH 32 11 6. Inverse Functions (1) Define a function. (2) What is the vertical line test for a graph? Draw an illustration. Y X (3) Define a one-to-one function. (4) What is the horizontal line test for a graph of a function? Explain with an illustration. Y X (5) Present an example of a function which is one-to-one. Explain why it is one-to-one. (6) Present an example of a function which is not one-to-one. Explain. (7) Define the inverse of a one-to-one function. How are the omains an ranges of a one-to-one function an those of its inverse relate? What happens if a function is not one-to-one? Why can its inverse be not efine?

MTH 32 12 (8) What are the cancellation equations for a function an its inverse? Explain with an example each. (9) Present steps to fin the inverse of a one-to-one function. (10) Fin the inverse function of f(x) = 3 x 5. Check that your caniate inverse is inee the inverse. Draw the graph y = f(x) an the that of the inverse on the same coorinate plane. Y X (11) Fin the inverse function of f(x) = x 2 + 3. Check that your caniate inverse is inee the inverse. Draw the graph y = f(x) an the that of the inverse on the same coorinate plane. (12) Theorem: If f is a one-to-one function efine on an interval, then its inverse function f 1 is also continuous. (13) Theorem: If f is a one-to-one ifferentiable function with the inverse function f 1 an f (f 1 (a)) 0, then the inverse function is ifferentiable at a an (f 1 ) (a) = 1 f (f 1 (a)). Copy own the proof to this theorem from the textbook. (14) Determine whether the following functions are one-to-one (raw graphs): (a) y = 1 + sin x for 0 x π. (b) y = 1 + cos x for 0 x π. (c) y = x for 2 x 2 () y = x for 0 x 2 (e) f(t) is your height at age t. (15) Fin the inverse of (a) f(x) = 3x 5 4x + 1 (b) f(x) = x 2 10x for x 5.

MTH 32 13 (16) Show that f is one-to-one. Fin (f 1 ) (a). Calculate f 1. Fin the omain an range for both f an f 1. Calculate (f 1 ) an check that it agrees with your previous calculation. Sketch the graph of f an f on the same coorinate plane. (a) f(x) = 1 x 2, a = 4 (b) f(x) = x 3, a = 4. Y X (17) Fin (f 1 ) (a): (a) f(x) = 5x 3 4x 2 + 11x + 9, a = 9. (b) f(x) = 3x 3 + 5x 2 + 3x + 5, a = 4. (18) Do as many problems from the textbook as time permits you to (section 6.1: # 1-16, 22-28, 34-44).

MTH 32 14 7. Exponential Functions an their erivaties (1) Recall the efinition of an exponential function. First, the basics. Let a be a positive real number. Then (a) for n a positive integer, a n = ; (b) for n = 0, a n = ; (c) for n a negative integer, say n = k, a n = ; () for x = p q a rational number, ax = ; (e) for x an irrational number, a n =. Explain in etail. (2) Properties of exponents: For any real numbers x, y an a, b positive real numbers, (a) a x+y = ; (b) a x y = ; (c) (a x ) y = ; () ((ab) x = ; a ) x (e) = ; b (f) 0 0 =. (3) Let f(x) = b x. Here b is calle, an x is calle the. (a) What happens when b < 0? (b) What happens when b = 0? (c) What happens with b = 1? (4) Draw the graph of the function f(x) = 2 x. Y X (5) Draw the graph of the function f(x) = ( 3 x. 1 (6) Draw the graph of the function f(x) = ( ) 2 x 1 (7) Graph the function f(x) =. 3 ) x

MTH 32 15 (8) For b > 1, the function f(x) = b x is an function. (9) For 0 < b < 1, the function f(x) = b x is a function. (10) The horizontal asymptote for the graph of f(x) = b x, b > 0, b 1 is. (11) The y-intercept for the graph of f(x) = b x, b > 0, b 1 is. (12) If a > 1, then lim x a x = an lim x a x =. (13) If 0 < a < 1, then lim x a x = an lim x a x =. (14) Fin lim x (3 2x + 2). (15) For the exponential function f(x) = a x, a > 0, a 1, use limits to escribe (a) f (0) (b) f (x) Therefore, f (x) = f(x) f (0). Geometrically, f (0) is. (16) Define the Euler number e. e 2.718. (17) Let f(x) = e x. Then (a) f (0) =. (b) f (x) =. (c) That is, x (ex ) =. (18) Let u = u(x). Then by the Chain Rule, x (eu ) =. (19) Fin the erivatives of: (a) y = e x2 (b) y = e cos x (c) y = sin(x)e x3 () y = e x5 e x2 (20) Since x (ex ) = e x, we have e x x =. (21) Fin cos(x)e sin(x) x.

MTH 32 16 (22) Graph f(x) = e x. Y X (23) Let f(x) = e x. Then, (a) it is an function. (b) lim x f(x) = an lim x f(x) =. e 3x (24) Fin lim x 1 + e an lim e 3x 3x x 1 + e. 3x (25) Use transformations to graph: f(x) = e x 1 Y X

MTH 32 17 f(x) = e x 1 Y X f(x) = e x Y X

MTH 32 18 f(x) = e x Y X f(x) = e x Y X (26) Start with the graph of f(x) = 3 x. Write the equation of the graph that results from (a) shifting 4 units ownwar; (b) shifting 4 units upwar; (c) shifting 4 units left; () shifting 4 units right; (e) reflecting about the x-axis; (f) reflecting about the y-axis.

(27) Fin the following limits: (a) lim x (0.99) x 1 (b) lim x 3 e3 x 1 (c) lim x 3 + e3 x e 4x e 4x () lim x e 4x + e 4x (28) Fin the erivative of: (a) f(x) = e 4x2 +8x (b) f(x) = e x + x e ex MTH 32 19 (c) f(x) = 1 + e x () f(x) = e 2x sin(x) (e) f(x) = x 2 + 4x + e 3x (29) Fin the integrals (a) x 3 e x4 x (b) (1 + e x ) 2 x e 4x (c) 1 1 + e x x 0 e x (30) Fin the volume of the soli obtaine by rotating the region boune by the curves y = e x, y = 0, x = 0, x = 1 about (a) the x-axis; (b) the y-axis. (31) Do as many problems from the textbook as time permits you to (section 6.2: # 1,2, 6-12, 22-50, 78-90).

MTH 32 20 8. Logarithmic functions (1) The exponential function f(x) = a x, a > 0, a 1 is a one-to-one function. The inverse function of this exponential function is calle, the logarithmic function, enote log a (x). That is, log a (x) = y. (2) The omain of f 1 (x) = log a (x) is. (3) The range of f 1 (x) = log a (x) is. (4) f f 1 (x) = x =. (5) f 1 f(x) = x =. (6) Prove the following: (a) log a (xy) = log a (x) + log a (y) (b) log a ( x y ) = log a(x) log a (y) (c) log a (x r ) = r log a (x) (7) Fix a > 1. Graph f(x) = a x an f 1 (x) = log a (x) on the same coorinate plane. Y X (8) For a > 1, we have lim x log a (x) = an lim x 0 + log a (x) =. (9) Fin the values: (a) log 4 (64) (b) log 64 (4) (c) log 3 (405) log 3 (5) () log 6 (2) + log 6 (3) (10) The logarithm function with base e is calle the an is enote by. (11) Therefore, ln(x) = y. (12) We have ln(e x ) = for x. (13) We have e ln(x) = for x.

MTH 32 21 (14) ln(e) =. (15) Solve for x: (a) ln(x) = 4 (b) ln(3x + 4) = 7 (c) e 4x+5 = 7 (16) Express as a single logarithm: (a) ln(x) + 3 ln(y) 1 ln(z) 2 (b) 4( 1 ln(a) ln(b) + 3 ln(x)) 3 (17) Use the formula log a (t) = ln(t) ln(a) to fin an approximate value of log 3(5). (18) We have lim x ln(x) = an lim x 0 + ln(x) = (19) Fin the limits: (a) lim x [ln(5 + x) ln(4 + x)] (b) lim x 0 ln(sin(x) + 1). (c) lim x 5 + ln(3x 15) (20) Solve for x: (a) ln(x) + ln(x 3) = ln 40 (b) e 2x 4e x 60 = 0 (21) Express as a single logarithm: 1 ln(y + 4 7)3 4 ln(y 2 + 2y 3) (22) Expan the expression: ln(a 5 b 3 4 c 7 ). (23) Do as many problems from the textbook as time permits you to (section 6.3: # 1-18, 26-36, 46-52).

MTH 32 22 9. Derivatives of Logarithmic Functions (1) Complete an prove the formula: (ln x) =. x (2) Fin the following erivatives: (a) x ln(cos(x)) (b) x ln(4x2 + 5x + 7) ( ) x + 4 (c) x ln x 7 () x ln( x ) (3) Since x ln( x ) =, we have the integration,. (4) Recall the integration formula for n 1 (why?) x n x =. (5) Fin the following integrals: (a) x 2 x 3 + 1 x (b) ln(x) x x (c) tan(x)x (6) Complete an prove the following formulae: (a) x log a x =. (b) For a 1, x ax =. What happens when a = 1? (c) For a 1, a x x = (7) Fin the erivatives of (a) y = ln(x x 2 + 5x) (b) y = ln(sin(ln x)) (8) State the steps in logarithmic ifferentiation.

(9) Use logarithmic ifferentiation to fin y x : (a) y = (x2 + 1) 3 4 4x + 5 (3x 7) 1 3 (b) y = x x MTH 32 23 (c) y = (x3 + 4x 2 ) sin 2 (x) e 2x (x + 1) () y = x sin(x) (e) y = (cos(x)) ln(x) (f) y = (ln(x)) x (10) Fin the integrals of (a) x 3 3 x4 x (b) cos(ln x) x x (c) x 3x + 8 () e x e x + 2 x (e) sin(x) 4 + cos(x) x (11) Do as many problems from the textbook as time permits you to (section 6.4: # 1-30, 42-54, 70-82).

MTH 32 24 10. Inverse Trigonometric Function (1) Define the Inverse Sine function (or the arcsine function). Draw the relevant graphs an explain the range an omain for each. (2) sin 1 x = y for x an y. (3) sin 1 (sin(x)) = for x. (4) sin(sin 1 (x)) = for x. (5) State an prove the formula for x (sin 1 (x)) for x. (6) Let f(x) = sin 1 (x 2 9). Fin (a) Domain of f(x) (b) The erivative f (x) (c) Domain of f (x) (7) Define the Inverse Cosine function (or the arccosine function). Draw the relevant graphs an explain the range an omain for each. (8) cos 1 x = y for x an y. (9) cos 1 (cos(x)) = for x. (10) cos(cos 1 (x)) = for x. (11) State an prove the formula for x (cos 1 (x)) for x. (12) Let f(x) = cos 1 (2x 3). Fin (a) The omain of f(x) (b) The erivative f (x) (c) The omain of f (x) (13) Define the Inverse Tangent function (or the arctan function). Draw the relevant graphs an explain the range an omain for each. (14) tan 1 x = y for x an y. (15) tan 1 (tan(x)) = for x. (16) tan(tan 1 (x)) = for x. (17) State an prove the formula for x (tan 1 (x)) for x. (18) State an prove the formula for x (csc 1 (x)) for x. (19) State an prove the formula for x (sec 1 (x)) for x. (20) State an prove the formula for x (cot 1 (x)) for x.

MTH 32 25 (21) Fill in the table: Function f(x) Derivative f (x) Function f(x) Integral f(x)x x n 1 x cos(x) sin(x) sin(x) tan(x) sec 2 (x) csc(x) sec(x) cot(x) Function f(x) Derivative f (x) Function f(x) Integral f(x)x e x a x ln(x) log a (x) sin 1 (x) tan 1 (x) sec 1 (x)

MTH 32 26 (22) Fin the exact values: (a) tan 1 ( 1 3 ) (b) sec 1 ( 2) (c) cot 1 ( 3) () arcsin( 1 2 ) (e) tan(sin 1 ( 3 )) (Use right triangle) 4 (f) tan 1 (cos 1 ( 1 4 )) (g) sin(arccos( 2 3 )) (h) sin(tan 1 (4) + tan 1 (5)) (23) Fin the erivative of: (a) f(x) = sin 1 (tan 1 (x)) (b) f(x) = cos 1 ( sin(x)) (c) f(x) = tan 1 (sin 1 (cos 1 (x))) () f(x) = cos 1 (4x 3 7x 2 + 8x) (24) Fin the limits (a) lim x tan 1 (4x 3 + 5x + 2) (b) lim x 3 + tan 1 (ln(x 3)) (25) Fin the integrals: (a) 1 9 x 2 x (b) 1 x 2 + 16 x (c) 1 x x 2 9 x () π 2 0 (e) cos(x) 1 + sin 2 (x) x e 3x x 1 e 6x (f) x 2 1 + x 6 x (26) Do as many problems from the textbook as time permits you to (section 6.6: # 4-14, 22-36, 42-46, 58-70).

MTH 32 27 11. Hyperbolic Functions (1) Define the Hyperbolic functions (a) sinh(x) (b) cosh(x) (c) tanh(x) () csch(x) (e) sech(x) (f) coth(x) (2) Graph the functions y = sinh(x), y = cosh(x), y = tanh(x) on three separate coorinate systems. (3) Prove the following Hyperbolic ientities: (a) sinh( x) = sinh(x) (b) cosh( x) = cosh(x) (c) cosh 2 (x) sinh 2 (x) = 1 () 1 tanh 2 (x) = sech 2 (x) (e) sinh(x + y) = sinh(x) cosh(y) + cosh(x) sinh(y) (f) cosh(x + y) = cosh(x) cosh(y) + sinh(x) sinh(y) (4) Complete an prove the formulae: (a) x (sinh(x)) (b) x (cosh(x)) (c) x (tanh(x)) () x (csch(x)) (e) x (sech(x)) (f) x (coth(x)) (5) Complete an prove the following formulae: (a) sinh 1 (x) = x (b) cosh 1 (x) = x (c) tanh 1 (x) = x (6) Complete an prove the following formulae: (a) x (sinh 1 (x)) (b) x (cosh 1 (x)) (c) x (tanh 1 (x)) () x (csch 1 (x)) (e) x (sech 1 (x)) (f) x (coth 1 (x))

(7) Prove the ientities: (a) cosh(x) sinh(x) = e x (b) 1 + tanh(x) 1 tanh(x) = e2x (8) Fin the limits: (a) lim x sinh(x) (b) lim x sinh(x) (c) lim x cosh(x) () lim x cosh(x) (e) lim x tanh(x) (f) lim x tanh(x) (g) lim x csch(x) (h) lim x csch(x) (i) lim x 0 + csch(x) (j) lim x 0 csch(x) (k) lim x sech(x) (l) lim x sech(x) (m) lim x coth(x) (n) lim x coth(x) (o) lim x 0 + coth(x) (p) lim x 0 coth(x) MTH 32 28

MTH 32 29 (9) Fill in the table: Function Derivative Function Integral x n, n 1 cos(x) sin(x) tan(x) csc(x) sec(x) cot(x) e x a x ln(x) log a (x) sin 1 (x) tan 1 (x) sec 1 (x) sinh(x) cosh(x) tanh(x) csch(x) sech(x) coth(x) sinh 1 (x) cosh 1 (x) tanh 1 (x) csch 1 (x) sech 1 (x) coth 1 (x) 1 x sin(x) sec 2 (x)

(10) Fin the erivatives: (a) (cosh(ln x)) x (b) x (sinh(1 + x2 )) (c) (cosh(x) sinh(x)) x () x (cosh 1 (3x + 5)) (e) x (sinh 1 (4x 2 5)) (f) x (tanh 1 (e 2x3 )) (11) Fin the integrals: (a) cosh(3x + 4)x (b) tanh(5x 7)x (c) cosh(x) 3 + sinh(x) x () 1 x 9 + x 2 MTH 32 30 (12) Do as many problems as time permits you to from the textbook (section 6.7: # 6-24, 30-46, 58-68).

MTH 32 31 12. Intermeiate forms an L Hospital s Rule (1) State L Hospital s Rule. (2) Use L Hospital s rule when appropriate. When not appropriate, say so. x 2 + 2x 15 (a) lim x 3 x 3 1 cos(x) (b) lim x 0 sin(x) e 3x (c) lim x x 3 sinh(x) x () lim x 0 (e) lim x 0 x 2 x tan 1 (4x) (f) lim x xe x 2 (g) lim x x sin( 1 x ) (h) lim x 0 (cot(x) 1 x ) (i) lim x 0 (csc(x) cot(x)) (j) lim x 0 +(tan(2x)) x (k) lim x (e x + 1) 1 x (3) Prove that lim x e x x n = for any natural number n. (4) Do as many problems as time permits you to from the textbook (section 6.7: # 1-66, 70-74).

MTH 32 32 13. Review Chapter 6 (1) Graph (a) y = e x+3 (b) y = ln(x + 1) (2) Fin the exact value: (a) log 5 + log 6 log 3 (b) cot(sin 1 ( 3/2)) (3) Solve for x: (a) ln(2 + e x ) = 4 (b) cos(x) = 2/2 (4) Differentiate: (a) y = (arcsin(4x 2 + 5x)) 3 (b) y = log 3 (1 + x 4 ) (c) y = (sin(x)) x () y = (x 3 + 3x 2 ) 4 (3x 2 + 5) 3 (2x + 5) 2 (e) y = x 2 tanh 1 (x) (5) Fin the limits: (a) lim x arctan(x ( 3 x) (b) lim x 3 + 5 ) x x (c) lim x 0 + x 2 ln(x) (6) Evaluate the integrals: (a) 1 0 ye 3y2 y (b) tan(x) ln(cos(x))x (c) 2 tan x sec 2 (x)x

MTH 32 33 (7) Fill in the table: Function Derivative Function Integral x n 1 x cos(x) sin(x) tan(x) csc(x) sec(x) cot(x) e x a x ln(x) log a (x) sin 1 (x) tan 1 (x) sec 1 (x) sinh(x) cosh(x) tanh(x) csch(x) sech(x) coth(x) sinh 1 (x) cosh 1 (x) tanh 1 (x) csch 1 (x) sech 1 (x) coth 1 (x) (8) Do as many problems as time permits you to from the textbook (Review section from Chapter 6: # 4-48, 62-78, 92-106).

MTH 32 34 14. Integration by Parts (1) State an prove the formula for integration by parts. (2) Fin the integrals: (a) x cos(x)x (b) x 2 cos(x)x (c) ln(x)x () e x cos(x)x (e) sin 1 (x)x (f) p 5 ln(p)p (g) t sinh(5t)t (h) t sinh(mt)t (i) t 3 e t2 t (3) Prove: x n e x x = x n e x n x n 1 e x x (4) Prove: (ln(x)) n x = x(ln(x)) n n (ln(x)) n 1 x (5) Prove: sec n (x)x = tan(x) secn 2 (x) + n 2 sec n 2 (x)x for n 1. n 1 n 1 (6) Prove: tan n (x)x = tann 1 (x) tan n 2 (x)x (for n 1). n 1 (7) Do as many problems as time permits you to from the textbook (Section 7.1: # 1-54).

MTH 32 35 15. Trigonometric Integrals (1) Fin the integrals (note, one of the exponents is o what is the strategy here?) (a) sin 3 (x)x (b) cos 3 (x) sin 2 (x)x (2) When the exponents of sin an cos are both even, use: sin 2 (x) = 1 (1 cos(2x)); 2 cos 2 (x) = 1 (1 + cos(2x)); 2 Sometimes we will nee, sin(x) cos(x) = 1 2 sin(2x). Prove these formulae. (3) Fin: (a) sin 2 (x) cos 5 (x)x (b) cos 2 (x) sin 5 (x)x (c) sin 2 (x) cos 2 (x)x () cos 4 (x)x (e) cos 5 (x)x (f) sin 3 (x) cos 3 (x)x (4) Recall the formulae: (a) tan(x)x = (b) sec(x)x = (c) tan n (x)x = () sec n (x)x = (e) ln( x ) = (5) Fin the integrals: (a) tan 2 (x)x (b) sec 2 (x)x (c) tan 3 (x)x () sec 3 (x)x (e) sec 4 (x)x (f) sec 5 (x)x (g) tan(x) sec(x)x (h) tan 2 (x) sec(x)x (i) tan(x) sec 2 (x)x (j) tan 3 (x) sec 5 (x)x (power of tan an sec are both o, an power of tan is greater than 1. In this case, let u = sec(x)). (k) tan 4 (x) sec 6 (x)x (power of sec is even, an power of tan is greater than 0. In this case, let u = tan(x)). (l) tan 5 (x) sec 6 (x)x

MTH 32 36 (6) Complete an prove the formulae: (a) sin(a) cos(b) = (b) sin(a) sin(b) = (c) cos(a) cos(b) = (7) Fin sin(5x) cos(4x)x (8) Fin sin(5x) sin(4x)x (9) Fin cos(5x) cos(4x)x (10) Do as many problems as time permits you to from the textbook (Section 7.2: # 1-32).

MTH 32 37 16. Trigonometric Substitutions (1) Here are some trigonometric substitutions: Expression Substitution Interval Ientity a2 x 2 a2 + x 2 x2 a 2 (2) Fin the integrals: (a) x 2 36 x 2 x (b) x 2 x 36 + x 2 (c) x x2 4 x () (e) π/2 0 x x2 10x + 26 cos x 1 + sin 2 x x (3) Do as many problems as time permits you to from the textbook (Section 7.3: # 1-30).

MTH 32 38 17. Integration of Rational Functions by Partial Fractions (1) Evaluate the integrals: (a) 2t 3 t + 5 t (b) x 2 + x 5 x 2 + 2x 35 x (c) x x 2 (x 2) 2 () 3x 2 + x + 4 x 4 + 3x 2 + 2 x (e) 3 x 1/3 x 2 + x x (f) sin(x) cos 2 (x) 3 cos(x) x (g) cosh(t) sinh 2 (t) + sinh 4 (t) t. (2) Do as many problems as time permits you to from the textbook (Section 7.4: # 1-30, 39-50).

MTH 32 39 18. Strategy for Integration (1) Write own a complete strategy for integration. (2) Evaluate the integral: (a) 1 x 1 0 x 2 4x 5 x (b) 2 x 2 2 x 0 1 x 2 (c) 2 1 ex 1 x () 2x 1 2x + 3 x (e) 1 x 2 1 x 0 2 x (f) (x + sin(x)) 2 x (g) x x + x x x (h) x 2 4x 2 1 (3) Do as many problems as time permits you to from the textbook (Section 7.5: # 1-60.)

MTH 32 40 19. Improper Integrals (1) Describe Improper Integrals of Type 1. (2) Determine whether each integral is convergent or ivergent. (a) 1 0 4 x 1 + x (b) 1 x 0 (1 + x) 5 4 (c) 0 2r r () cos(πt)t (e) 1 x for p an integer. 1 xp (3) Describe Improper Integrals of Type 2. (4) Determine whether each integral is convergent or ivergent. (a) 8 6 (b) 8 6 (c) 3 1 4 (x 6) x 3 4 3 x x 6 1 x 3 x (5) Describe the Comparison Test for Improper Integrals. (6) Determine whether each integral is convergent or ivergent. (a) 2 + e x x 1 x (b) arctan(x) x 0 2 + e x (7) Do as many problems as time permits you to from the textbook (Section 7.8: # 1,2 5-32, 49-54).

Evaluate the integral: (1) 2 1 x5 ln(x)x (2) 1 0 arctan(x) 1 + x 2 x (3) sec 2 (θ) tan 2 (θ) θ (4) e x cos(x)x (5) x e x 1 e 2x (6) 1 tan(θ) 1 + tan(θ) θ (7) 1 (8) 1 0 (9) 1 ln(x) x x 4 1 2 3x x tan 1 (x) x x 2 MTH 32 41 20. Review Chapter 7 (10) Do as many problems as time permits you to from the textbook (Review section from Chapter 7: # 1-26, 41-50).

MTH 32 42 21. Arc Length (1) Describe the formula for calcuating the arc length of a curve. Explain how you arrive at the formula. (2) Fin the length of the curve: (a) y = xe x, 0 x 2 (Just set up the integral). (b) y = x3 3 + 1 4x, 1 y 3 (c) x = ln(cos(y)), 0 y π/3 () y = x x 2 + sin 1 ( x) (e) x = 1 e y, 0 y 2 (3) Do as many problems as time permits you to from the textbook (Section 8.1: # 1-18).

MTH 32 43 22. Area of surface of revolution (1) Give a formula for the surface area of a right circular cyliner. Explain with an illustration. (2) Give a formula for the surface area of a circular cone. Explain with an illustration. (3) Give a formula for the surface area of a ban (frustum of a cone). Explain with an illustration. (4) Give a formula for the surface area of the surface obtaine by rotating the curve y = f(x), a x b, about the x-axis. Explain with an illustration. (5) Set up an integral to calculate the area of the surface obtaine by rotating the curve: (a) y = x 2, 1 x 2 about the x-axis (b) y = x 2, 1 x 2 about the y-axis (6) Fin the area of the surface obtaine by rotating the curve about the x-axis: (a) y = 1 + e x, 0 x 1 (b) y = x2 6 + 1 2x, 1 2 x 1. (7) Fin the area of the surface obtaine by rotating the curve about the y-axis: (a) y = 1 x 2, 0 x 1 (b) y = x2 4 ln(x) 2, 1 x 2 (8) Do as many problems as time permits you to from the textbook (Section 8.2: # 1-16).

MTH 32 44 23. Curves efine by parametric equations (1) Explain the term parametric curve. Explain using the parametric equation x = t 2, y = t 3 4t. Ientify the initial point an the terminal point. (2) Explain the parametric curve x = cos(t), y = sin(t).

MTH 32 45 24. Calculus with parametric curves (1) Explain when a parametric curve has a (a) horizontal tangent (b) vertical tangent (2) Fin area uner the curve: x = cos(2t), y = sin(2t), 0 t π/4. (3) Develop the formula for the length of a curve C parametrize by x = f(t), y = g(t), α t β where f, g are coninuous on [α, β]an C is traverse exactly once as t increases from α to β. (4) Fin the length of the parametric curve x = e t + e t, y = 5 2t for 0 t 3. (5) Develop the formula for the formula for the surface area of the surface obtaine by rotating about the x-axis, the parametric curve x = f(t), y = g(t), for α t β, g(t) 0, an f, g continuous on [α, β]. (6) Fin the area of the surface area obtaine by rotating x = t 3, y = t 2 for 0 t 1.

(1) Complete the table: Rectangular Coorinates Given x r = y θ = MTH 32 46 25. Polar Coorinates Polar coorinates Given x = r y = θ (2) Explain the corresponence above with illustrations (one each for r > 0 an r < 0). (3) Fin the rectangular coorinates of (a) (3, 3π/4) (b) ( 3, 3π/4) (c) (2, 7π/6) () ( 2, 7π/6) (e) ( 2, 8π/3) (4) Fin polar coorinates with 0 θ < 2π an (a) (a) r > 0 an (b) r < 0. (b) ( 3, 3) (c) (1, 3) () (3, 3) (e) ( 3, 1) (5) Sketch the region: (6) 2 r 4, π/3 < θ < 5π/3 (7) r 2, π θ 2π (8) Ientify the curve by fining a Cartesian equation. (a) r = 4 sec(θ) (b) θ = π/3 (c) r = tan(θ) sec(θ) (9) Fin a polar equation: (a) y = x (b) 4y 2 = x (c) xy = 4 (10) Sketch the curve with the given polar equation by first sketching the graph of r as a function of θ in cartesian coorinates: (a) r = 1 + 2 cos(θ) (b) r = 3 + sin(θ) (c) r = 2 cos(4θ) () r = ln(θ), θ 1 (11) Do as many problems as time permits you to from the textbook (Section 10.3: # 1-12, 15-26, 29-46).

MTH 32 47 26. Areas an Lengths in Polar Coorinates (1) Let R be the region boune by the polar curve r = f(θ) an by the rays θ = a an θ = b where f is a positive continuous function an where 0 < b a 2π. Draw a possible such region. (2) Draw a small sector in the region above, an escribe its area. (3) Fin an approximate area of the area of the region R. (4) Using Riemann sums to present a formula for the area of the region R.. (5) Fin the area of the region boune by the curve (a) r = tan(θ), π/6 θ π/3 (b) r = 1 sin(θ) for 0 θ π (c) r = 1 sin(θ) for 0 θ 2π () r 2 = 9 sin(2θ) for one loop. (6) Fin the area of the region that lies insie the first curve an outsie the secon curve: r = 2 + sin(θ), r = 3 sin(θ) (7) Fin the area of the region that lies insie both curves: r = 3 + 2 cos(θ), r = 3 + 2 sin(θ). (8) Describe the formula for the length of a curve with polar equation, r = f(θ), a θ b. (9) Fin the length of the polar curve r = 5 θ for 0 θ 2π. (10) Fin the length of the polar curve r = 2(1 + cos(θ)) for 0 θ 2π. (11) Do as many problems as time permits you to from the textbook (Section 10.4: # 1-32, 45-48).

MTH 32 48 27. Conic Sections (1) Define a parabola. Explain with an illustration. Ientify the focus, irectrix, axis an vertex. (2) Write an equation an raw illustrations of the parabola with focus (0, p) an irectrix y = p (3) Write an equation an raw illustrations of the parabola with focus (p, 0) an irectrix x = p (4) Fin the focus an irectrix of the parabola y 2 + 12x = 0 an sketch. (5) Fin the focus an irectrix of the parabola x 2 + 5x = 0 an sketch. (6) Define an ellipse. Explain with an illustration. Ientify the foci, vertices, the major axis, an the minor axis. (7) Write an equation of the ellipse with a b > 0 an foci (±c, 0) for c 2 = a 2 b 2. Explain with an illustration. (8) Write an equation of the ellipse with a b > 0 an foci (0, ±c) for c 2 = a 2 b 2. Explain with an illustration. (9) Sketch the graph of 4x 2 + 9y 2 = 121. Ientify the foci. (10) Sketch the graph of 9x 2 + 4y 2 = 36. Ientify the foci. (11) Sketch the graph of 2x 2 + 3y 2 = 6. Ientify the foci. (12) Fin an equation of the ellipse with foci (0, ±3) an vertices (0, ±4) (13) Define a hyperbola. Explain with an illustration. Ientify the foci an the asymptotes. (14) Write an equation an raw an illustration of a hyperbola with foci (±c, 0) where c 2 = a 2 +b 2 an vertices (±a, 0). (15) Sketch the graph of 4x 2 9y 2 = 121. Ientify the foci an the asymptotes. (16) Sketch the graph of 3x 2 2y 2 = 6. Ientify the foci an the asymptotes. (17) Write an equation an raw an illustration of a hyperbola with foci (0, ±c) where c 2 = a 2 +b 2 an vertices (0, ±a). (18) Fin the foci an equation of the hyperbola with vertices (±1, 0) an asymptotes y = ±3x. (19) Fin the foci an equation of the hyperbola with vertices (0, ±1) an asymptotes y = ±3x. (20) Fin an equation of the ellipse with foci (2, 2), (4, 2) an vertices (1, 2), (5, 2). (21) Fin the conic 9x 2 4y 2 36x + 8y 4 = 0. Ientify all the important characteristics of this conic. (22) Do as many problems as time permits you to from the textbook (Section 10.5: # 1-48).

MTH 32 49 28. Conic Sections in Polar Coorinates (1) State an explain the Theorem 1 from your text. (2) Present polar equation for a conic. Explain when you obtain an ellipse, a parabola, or a hyperbola. 6 (3) Sketch the conic r = 3 6 sin(θ). (4) If the conic in your previous question is rotate through an angle π/3 about the origin, then fin a polar equation an graph the resulting conic. (5) State Kepler s laws of planetary motion. (6) Write the polar equation of an ellipse. Explain the parameters being use (7) Explain the terms perihelion, aphelion, perihelion istance an aphelion istance. (8) Do as many problems as time permits you to from the textbook (Section 10.6: # 1-16).

MTH 32 50 29. Review Chapter 10 (1) Sketch the polar curve: (a) r = 3 + 3 cos(3θ) 3 (b) r = 2 2 cos(θ) (2) Fin the area enclose by the inner loop of the curve r = 1 2 sin(θ). (3) Fin the area of the region that lies insie the curve r = 2 + cos(2θ) but outsie the curve r = 2 + sin(θ). (4) Fin the length of the curve: (a) x = 2 + 3t, y = cosh(3t), 0 t 1 (b) r = sin 3 (θ/3), 0 θ π (5) Fin an equation of the parabola with focus (2, 1) an irectrix x = 4. (6) Fin an equation of the ellipse with foci (3, ±2) an major axis with length 8. (7) Do as many problems as time permits you to from the textbook (Review section of Chapter 10: # 9-16, 31-40,45-56).

MTH 32 51 30. Practice Problems Below is a collection of problems from all chapters in no particular orer. (1) Assume that f is a continuous function an that x 0 tf(t)t = (a) Determine f(0). (b) Fin the zeros of f, if any. (2) Apply the Mean Value Theorem to the function F (x) = x a f(t)t 2x 4 + x 2. on [a, b], to obtain the Mean Value Theorem for Integrals: If f is continuous on [a, b], then there is at least one number c in (a, b) such that b a f(x)x = f(c)(b a). (3) Let f be continuous an let the functions F an G be efine by F (x) = x c f(t)t, an G(x) = 0 x f(t)t, where c, [a, b]. Show that F an G iffer by a constant. x 2 + x, 0 x 1 (4) (a) Sketch the graph of the function f(x) = 2x, 1 < x 3. (b) Fin the function F (x) = x f(t)t, 0 x 3, an sketch its graph. 0 (c) What can you say about f an F at x = 1? (5) A rectangle has one sie on the x-axis an the upper two vertices on the graph of y = 1/(1 + x 2 ). Where shoul the vertices be place so as to maximize the area of the rectangle? (6) Fin the vertex, focus, axis an irectrix an then sketch the parabola. (a) y 2 = 2(x 1) (b) y 3 = 2(x 1) 2 (c) y = x 2 + x 1 (7) Compare: [ x ] x (a) f(t)t to x a a t [f(t)]t. [ ] (b) f(x)x to x x [f(x)]x. (8) Let f(x) = x 3 x. Evaluate 2 f(x)x without computing the integral. 2 (9) As a particle moves about the plane, its x-coorinate changes at the rate of t 2 units per secon an its y coorinate changes at a rate of t units per secon. If the particle is at the point (3, 1) when t = 4 secons, where is the particle 5 secons later? (10) Let f be continuous an efine F by x [ t ] F (x) = t f(u)u t. Fin (a) F (x). (b) F (1). 1

MTH 32 52 (11) Let f be a continuous function. Show that (a) b+c f(x c)x = b f(x)x. a+c a (b) 1 bc f ( ) x b c ac c x = f(x)x, if c 0. a (12) Prove that the hyperbolic cosine function is even an the hyperbolic sine function is o. (13) Use integration to fin the area of the trapezoi with vertices ( 2, 2), (1, 1), (5, 1), (7, 2). (14) Sketch the region boune by the curves an fin the volume of the soli generate by revolving the region about the x- or y- axis, as specifie below. (a) y = 1 x, y = 0, revolve aroun the x-axis (b) x = 9 y 2, x = 0, revolve aroun the y-axis. (15) Sketch the region Ω boune by the curves an use the shell metho to fin the volume of the soli generate by revolving Ω about the y-axis. (a) x = y 2, x = 2 y. (b) x = y, x = 2 y 2. (c) f(x) = cos 1 πx, y = 0, x = 0, x = 1. 2 (16) Fin the rectangular coorinates of the given point. (a) [3, 1π] 2 (b) [2, 0] (c) [ 1, 1π] 4 (17) A ball of raius r is cut into two pieces by a horizontal plane a units above the center of the ball. Determine the volume of the upper piece by using the shell metho. (18) Determine the exact value of the given expressions: (a) tan 1 0 (b) sin[arccos( 1)] 2 (c) arctan(sec 0) (19) Write the equation in rectangular coorinates an ientify the curve. (a) r sin θ = 4 (b) θ 2 = 1 9 π2 (c) r = 4 1 cos θ (20) The region boune between the graphs y = x an the x-axis for 0 x 4 is revolve aroun the line y = 2. Fin the volume of the soli that is generate. (21) Show that for a 0, x a 2 + (x + b) 2 = 1 a arctan 1 ( ) x + b + C a (22) Assume that f(x) has a continuous secon erivative. Use integration by parts to erive the ientity f(b) f(a) = f (a)(b a) b a f (x)(x b)x (23) Fin the area uner the graph of y = x 2 9 x from x = 3 to x = 5. (24) Fin values of a an b such that cos ax b lim = 4. x 0 2x 2

MTH 32 53 (25) Fin the center, the vertices, the foci, the asymptotes, an the length of the transverse axis of the given hyperbola. Then sketch the figure. (a) (x 1)2 (y 3)2 = 1 16 16 (b) y 2 /9 x 2 /4 = 1 (c) 4x 2 8x y 2 + 6y 1 = 0. (26) An object starts at the origin an moves along the x-axis with velocity v(t) = 10t t 2, 0 t 10. (a) What is the position of the object at any time t, 0 t 10? (b) When is the object s velocity a maximum, an what is its position at that time? (27) (a) Calculate the area A of the region boune by the graph of f(x) = 1 an the x-axis x2 for x [1, b]. (b) The results in part (a) epens on b. Calculate the limit of A(b) as b. (28) Fin the inicate limit ln(sec x) (a) lim x 0 (b) lim x 0 + x 2 x x+sin x x (c) lim 1/2 x 1/4 x 1 x 1 () lim x x 2 + 2x x) ( (e) lim 1+2 x ) 1/x x 0 2 (29) Plot the given points in polar coorinates. (a) [ 2, 0] (b) [ 1, 1π] 3 (c) [ 1, 2, π] 3 3 (30) Sketch the region boune by the curves an fin its area (a) 4x = 4y y 2, 4x y = 0. (b) y = e x, y = e, y = x, x = 0. (c) 4y = x 2 an y = 8. x 2 +4 () y = sin 2 x, y = tan 2 x, x [ π, ] π 4 4 (e) The region in the first quarant boune by the x-axis, the parabola y = x2, an the 3 circle x 2 + y 2 = 4. (31) The base of a soli is the region boune by x = y 2 an x = 3 2y 2. Fin the volume of the soli given that the cross sections perpenicular to the x-axis are (a) rectangles of height h. (b) equilateral triangles. (32) Fin an equation for the ellipse that satisfies the given conitions. (a) foci at (3, 1), (9, 1); minor axis 10. (b) center at (2, 1); vertices at (2, 6), (1, 1). (33) Fin the arc length of the following graphs an compare it to the straight line istance between enpoints of the graph. (a) f(x) = 1 4 x2 1 ln x, x [1, 5]. 2 (b) f(x) = ln(sin x), x [π/6, π/2]

MTH 32 54 (34) Fin a formula for the istance between [r 1, θ 1 ] an [r 2, θ 2 ]. (Hint: Recall the Cosine Law of triangles). (35) Let f(x) be a continuous function on [0, ). The Laplace transform of f is the function F efine by F (s) = 0 e sx f(x)x. The omain of F is the set of all real numbers s such that the improper integral converges. Fin the Laplace transform of each of the following functions an give the omain of F. (a) f(x) = cos 2x. (b) f(x) = e ax. (36) At time t a particle has position x(t) = 1 + arctan t, y(t) = 1 ln( 1 + t 2 ). Fin the total istance travele from t = 0 to t = 1. (37) Write the equation in polar coorinates. (a) y = x (b) (x 2 + y 2 ) 2 = 2xy (38) Fin the area of the given region. (a) r = 2 tan θ an the rays θ = 0 an θ = 1π. 8 (b) r = a(4 cos θ sec θ) an the rays θ = 0 an θ = 1π. 4 (39) Calculate ( the erivative: ) (a) 3 sin tt x x 2 t ( ) (b) 2x t 1 + t x tan x 2 t. (c) H (3) given that H(x) = 1 x [2t x 3 3H (t)]t. (40) Show that the set of all points (a cos t, b sin t) with real t lie on an ellipse.

MTH 32 55 (41) Evaluate the integrals: (a) e kx x (b) π/2 π/6 (c) 3 0 cos x x 1+sin x r r r 2 +16 () x+1x x 2 (e) sin(e 2x ) x e 2x (f) sinh 2x e cosh 2x x (g) ln(x+a) x x+a (h) e ln x x (i) ( ) 2 0 a y2 1 y3 a y 2 (j) sin x x x (k) 1 x sin 2 x (l) tan x ln(sec x)x (m) 1+tanh x cosh 2 x (n) x (o) 8 5 (p) 1 x ln[1+(ln x) 2 ] x x x 2 16 0 cos2 πx sin πxx 2 2 (q) 1 x(1+ x) x (r) ln 3 ln 2 e x 1 e 2x x (a) x x 2 + 6xx (b) e px x, p > 0 0 (c) cos xx () tan 2 (2x)x (e) x10 x2 x (f) tanh x (g) 1 0 (h) log 5 x x 5p x+1 x+1 x x (i) ln(π/4) e x sec e x x 0 (j) sin(ln x)x (k) sinh ax x cosh 2 ax (l) ln(x+1) x+1 x (m) x (x 2 4x+4) 3/2 (n) x 3 x x 3 +x 2 (o) tan 2 x sec 2 xx (p) π/4 sin 5x cos 2xx 0 (q) x (r) x 4 16 e x 1+e 2x x (42) Evaluate the erivative: (a) y = e2x 1 e 2x +1 (b) y = ln(cos(e 2x )) (c) g(x) = tan 1 (2x) () f(x) = x 2 sec ( ) 1 1 x (e) θ(r) = sin 1 ( 1 r 2 ) (f) y = (tan x) sec x (g) f(x) = x sin 1 (2x) (h) f(x) = 2 5x 3 ln x (i) f (e) where f(x) = log 3 (log 2 x) (j) y = x c 2 x 2 + c 2 sin ( ) 1 x c, c > 0 (k) y = ln(cosh x) (l) f(x) = arctan(sinh x) (m) y = sec h(3x 2 + 1) (n) y = cosh x 1+sec hx (o) f(x) = (sinh x) x