June PURDUE UNIVERSITY Stud Guide for the Credit Eams in Single Variable Calculus (MA 65, 66) Eam and Eam cover respectivel the material in Purdue s courses MA 65 (MA 6) and MA 66 (MA 6). These are two separate two hour eaminations. Students who pass Eam will receive credit hours for MA 65, and normall will be placed in MA 7. Those who pass Eam will receive credit hours for MA 66, but onl if the have credit for MA 65, either b credit eamination or other means, and normall will be placed in MA 7. MA 7 and MA 7 are offered in the fall semester onl, but students ma also register in MA 66 or MA 6 instead of MA 7, and in MA 6 instead of MA 7. This stud guide describes briefl the topics to be mastered prior to attempting these eaminations. The material can be studied from man tetbooks, almost all of them entitled Calculus or Calculus with Analtic Geometr. The tetbook currentl used at Purdue is Calculus, Earl Transcendentals, James Stewart, Brooks/Cole Publishing Compan. See also http://www.math.purdue.edu/academic/courses for recent course information and materials. Each eam consists of 5 multiple choice problems with two hours time limit. No books, notes, calculators or an other electronic devices are allowed. The topics covered in the Credit Eams are listed below. Eam : (MA 65) (Two hours). Review of functions and graphs. Trigonometric, eponential and logarithm functions.. Limits and continuit.. Definition of derivative. Rules of differentiation (sum, product, quotient). Chain rule and implicit differentiation. Derivatives of trigonometric, eponential, and logarithmic functions. Related rates and approimations.. Applications of the derivative. Maima and minima. Mean value theorem. Eponential growth and deca. Curve sketching, concavit, asmptotes. Indeterminate forms and L Hôpital s Rule. 5. Definition of definite integral. Fundamental Theorem of Calculus. Indefinite integrals. Integration b substitution. Definition of logarithm. 6. Inverse functions. Eponential and logarithm functions. Inverse trigonometric functions and their derivatives. Eam : (MA 66) (Two hours). Cartesian coordinates and vectors in space. Dot product, cross product.. Techniques of Integration: Integration b parts, trigonometric integrals, trigonometric substitutions, partial fractions. Improper integrals.. Applications of the integral. Volumes b disks and shells. Length of a curve. Area of a surface. Work. Moments and center of gravit.. Sequences and series. Convergence. Integral, comparison, ratio and root tests. Alternating series and absolute convergence. Power series and Talor series. 5. Parametric Equations. Polar coordinates. Conic sections.
Sets of practice problems for each of MA 65 and MA 66 are attached. The correct answers are given on the last page of each set. Naturall, these sets do not cover all points of the courses, but if ou have no difficult with them, ou will probabl do well in the eamination. Additional practice problems are available through the Mathematics Department webpage at www.math.purdue.edu/academic/courses The eamination, like the set of practice problems, is almost entirel manipulative. This does not mean that ou do not need a thorough understanding of the concepts, but rather that ou are not asked to quote an definitions or theorems or offer proofs of an theorems. You are epected to perform the manipulations required with a high degree of understanding and accurac. IMPORTANT. Stud all the material thoroughl.. Solve a large number of eercises.. When ou feel prepared for the eamination(s), solve the practice problems under eam conditions. A strong background in algebra, trigonometr, and basic analtic geometr is indispensable for the stud of calculus. Successful completion of man of the problems in the eamination will depend on our competence in these background areas. SPECIAL NOTE A word of advice concerning the taking of the actual eamination for credit. No one does well on an eamination when he or she is ecessivel fatigued. Therefore, ou are urged to provide ourself an adequate rest period before taking the actual eamination(s). If our trip to the campus necessitates travel into the late hours of the night or an etremel earl departure from our home, ou should allow for a one night rest in the Lafaette area before taking the eamination(s). Man students who are unsuccessful with the eamination(s) tell us that failing to take the above precautions contributed strongl to their inabilit to complete their eamination(s) successfull. Most such students find that their first ear was somewhat less rewarding than it might have been because of the time spent retracing material studied in high school. Please consult our advanced credit schedule for the actual time and place of the eamination(s). The are usuall given both morning and afternoon.
MA 65 PRACTICE PROBLEMS. lim = B. C. D. Does not eist. If =( +)tan, then d d = tan +( + ) sec B. sec C. tan +( +)tan D. tan + sec tan { + a, for <. If h() = determine all values of a so that h is continuous for 8 for all values of. a = B. a = 8 C. a = 9 D. a = There are no values of a.. Evaluate lim + B. C. D. π 5. If f() =, then lim + B. C. 6 6 cos( ). (Hint: cos( ) for all.) Does not eist f() f() D. = Does not eist 6. The equation 5 = has one solution for between and. The solution is in the interval. (, ) B. (, ) C. (, ) D. (, ) (, ) 7. If f() = +,thenf () = B. C. D. 8. If =ln( )+sin,then d d = +cos B. +sin cos C. +sin D. +cos +sin cos
9. Find f () iff() = + ( + ) B. ( + ) C. ( + ) + ( + ) D. ( + ) ( + ) ( + ). Assume that is defined implicitl as a differentiable function of b the equation + +5=. Find d at (, ). d 9 B. 5 C. D. 5. Find the maimum and minimum values of the function f() = +6 on the interval. ma is min is -. B. ma is - min is - C. ma is min is - D. no ma. min is - ma is no min.. For a differentiable function f() itisknownthatf() = 5 and f () =. Use differentials (linear approimation) to get the approimate value of f(.). 6. B. 5. C. 5. D..96.. Water is withdrawn from a conical reservoir, 8 feet in diameter and feet deep (verte down) at the constant rate of 5 ft /min. How fast is the water level falling when the depth of the water in the reservoir is 5 ft? (V = πr h). 5 6π ft/min B. D. 5 /π ft/min π ft/min 5 π ft/min. C. π ft/min. A rectangle is inscribed in the upper half of the circle + = a as shown at right. Calculate the area of the largest such rectangle. a B. a C. a D. a a. a
5. Given that f() is differentiable for all, f() =, and f(7) =, then the Mean Value Theorem states that there is a number C such that <C<7andf (C) = 6 5 C. <C< and f (C) = 6 5 B. <C<7andf (C) = 5 6 D. <C<7andf (C) = <C< and f (C) =. 6. Suppose that the mass of a radioactive substance decas from 8gms to gms in das. How long will it take for gms of this substance to deca to gms? ln ln das B. da C. ln ln das D. das (ln ) das 7. Which of the following is/are true about the function g() =? () g is decreasing for >. () g has a relative etreme value at (, ). () the graph of g isconcaveupforall<. (), () and () B. onl () C. onl () D. () and () () and (). 8. Find where the function f() =/ + is increasing. all B. no C. < D. > =. 9. Let f be a function whose derivative, f,isgivenbf () =( ) ( +)( 5). The function has a relative maimum at = and a relative minimum at =5. B. a relative maimum at = 5 and a relative minimum at =. C. relative maima at =, = and a relative minimum at =5. D. a relative maimum at = 5 and relative minima at =, = a relative maimum at = and relative minima at =, =5.. Find d t +dt at =. d 6 B. C. D. +. 5
. 5 d = B. 7 C. 7 D. 7 +. lim = /7 C. / D. /. +. Calculate the upper sum U f (P ) for the definite integral ( +)d with the partition P = {,,,, } 5+++ B.5+++5 C.5+++5 D.5+++5 5+++. Suppose that a function f has the following properties: 7 f () > for<c, f (c) = and f () < for>c. Which of the following could be the graph of f? B. C. D. c c c c c 5. Find lim cos(). B. 9 C. 9 D. 6. Find lim ( + ) B. C. e 6 D. e e d [ 7. e ln + ] = e ln( + )+ e e B. +e ln + d ( + ) + 8. 9. C. e ln( + )+ e ( + ) D. e + e + d d sin = (cos) sin B. (sin ) sin C. cos D. sin [ sin d d tan e = +(cos)ln] (ln )sin +e B. e +e C. e +e 6 D. e +e 9 e e 6 6
. e d = 9 e B. 9 + 9 e C. D. 9 9 (e ).. + + d = π e +e d = ln +e B. π C. tan D. π 6 B. ln( + e) C. π 6 D. ln e. An equation of the parabola with horizontal ais, verte (, ), and containing the point (, ) is ( ) = ( +) B.( +) = 8( ) C. ( ) = ( +) D. ( +) = 9( ) ( +) = 9 ( ). The ellipse 6( ) + 5( 7) = has one focus at (6, 7) B. (7, 7) C. (, ) D. (, ) (, ) 5. The asmptotes of the hperbola 9 6 8 = have equations: 7=, + = B. 8=, + = C. =, + = D. +7=, + = +8=, + = 6. If f() =,, then the graph of = f () is B. C. D. - - - - - - - - - - - - - - - - - - - - - - - - 6-6 - - 6 - - -6 7. This could be the graph of the function e e,> B. /( + ), > C. /( + ), > D. / ln, > e,> 7
8. Which could be the graph of f() = + B. C. D. 9. If f() = 5 + then (f ) (5) is B. / C. /5 D. /9 /. The horizontal asmptotes to the graph of f() = e +5e are e +e =,= 5 B. = C. = and = 5 D. = This graph does not have an horizontal asmptotes Answers:.D,.A,.D,.A, 5.C, 6.D, 7.B, 8.E, 9.A,.E,.C,.E,.E,.E, 5.A, 6.B, 7.C, 8.C, 9.A,.A,.C,.E,.C,.B, 5.C, 6.C, 7.A, 8.D, 9.C,.E,.B,.A,.C,.A, 5.B, 6.B, 7.A, 8.D, 9.D,.A 8
MA 66 PRACTICE PROBLEMS. Calculate the projection of the vector b = i k onto the vector a = i + j + k. a B. a C. 5 a D. b b. Find the angle between the vectors a = i + j and b = i + j π B. π C. π D. 5π π 6. Find the area of the triangle with vertices at the points (,, ), (,, ) and (,, ). B. C. D.. The area of the region bounded b = and = +isgivenb ( ) ( ) ( ) + B. C. + ( ) ( ) D. E 5. Find the area of the region bounded b = and =. 9/ B. 5/6 C. /6 D. 5/6 /6 6. d = π B. π 6 C. sin D. π 7. π/ cos d = π 8. For the integral B. π + C. D. ( ) / d, (i) choose a trigonometric substitution to simpl the integral and (ii) give the resulting integral (i) = sec u, (ii) tan udu B. (i) = sec u, (ii) tan u sec udu C. (i) = sec u, (ii) tan u sec udu D. (i) =sinu, (ii) cos udu (i) =sinu, (ii) cos udu 9
9. e ln d = 9 e B. 9 e + 9 C. e 9 D. e + e e. Give the form of the partial fraction decomposition of A + + C. A + B + + A + B + + B ( ) B. C + C ( ) A + + D D. ( ) B ( +) + C + A + + B + + ( +)( ). D ( ) C ( ). Indicate convergence or divergence for each of the following improper integrals: (I) d (II) ( ) ( ) d (I) converges; (II) diverges B. (I) converges; (II) converges C. (I) diverges; (II) converges D. (I) diverges; (II) diverges. + d = ln + ln 5 B. ln + ln 5 C. ln + tan 5 tan D. + tan 5 tan ln 5. The volume of the solid generated b revolving about the ais the region in the first quadrant between the graphs of = and = is given b the definite integral C. ( )d B. π[( ) () ]d D. [() ( + ) ]d π( +)d π[( ) () ]d
. Let R be the region between the graphs of = and =. The volume of the solid generated b revolving R about the ais, given b the shell method is D. π ( ) d B. ( π ) d π ( ) d C. ( ) π d 5. The length of the graph of = / for is 5 B. 5 C. 7 9 D. 8 7 ( ( ) / ) π ( ) d 9 ( ) 5 6. A right circular conical tank of height ft. and base radius 5 ft. has its verte at the bottom, and its ais vertical. If the tank is full of water at 6.5 lb./cu. ft., the work required to pump all the water over the top is 6.5π C. 6.5π 6.5π ( ) π ( ) ( ) d B. 6.5π d ( ) ( ( )() d ) d D. 6.5π ( ) ( ) d 7. A force of 9 lb. is required to stretch a spring from its natural length of 6 in. to a length of 8 in. Find the work required to stretch it from its natural length to in. in. lb. B. 8 in. lb. C. in. lb. D. 6 in. lb. 8 in. lb. 8. If R is the semicircular region bounded b the ais and =, <<, the coordinates (, ) of the center of gravit are ( ) ( ) 8 8, B. π π, C. (, ) D. (, ) (, ) 9. The nd Talor polnomial of f() = +sin is + + B. C. + + D. + +. Evaluate the limit lim + ( )n n n. B. C. D. limit does not eist
( ). Evaluate the limit lim n n +. n n! B. C. e D. /e limit does not eist n+. If L = n,thenl = n= B. 6 C. 9 D. /. If L = + ( ) n n,thenl = n n= n= / B. / C. D. / 5/. (n converges when +) p n= p> B. p C. p D. p> ( 5. + n) p converges for n= p p B. p> C. p< D. p> no values of p 6. Which of the following series converge conditionall? (i) ( ) n n n= (ii) n= ( ) n n ln n (iii) n= ( ) n n e n onl (ii) B. onl (i) and (iii) C. onl (i) and (ii) D. all three none of them 7. Which of the following series converge? (i) ( ) n n= n (ii) n= n! 5 (n ) (iii) n= ( ) n onl (ii) B. onl (i) and (iii) C. onl (i) and (ii) D. all three none of them n n 8. The interval of convergence for the power series n ln n is n= < B. < C. D. <<
9. Find the interval of convergence of the power series << B. << C. D. < <<. The fourth term of the Talor series for + about a =is B. C. D.. Use Talor series to approimate.5 n= n n n e d to within. ( e = n= ) n n!.9 B..5 C..55 D..58.6. The first three nonzero terms of the Talor series for f() =( )sin about a = are 5 6 + 5 5 B. + C. 7 6 + 5 5 D. 7 6 + 5 5 7 6 + 5. A point P has polar coordinates (, π ). Which of the following are also polar coordinates of P? I. (, π ) II. (, 5π ), III. (, 7π ) IV. (, 5π ). I and II onl B. I and III onl C. I and IV onl D. II and III onl II and IV onl. Which of the following looks most like the graph of the polar curve r = sin(θ), for θ π? B. C. D.
5. The area inside the curve r =sinθ and outside the curve r = sin θ is given b which of the following definite integrals? C. 5π/6 π/6 π/ π/6 π [sin θ ( sin θ) ]dθ B. [sin θ ( sin θ) ]dθ D. [sin θ ( sin θ)] dθ π 5π/6 π/6 6. The area inside one leaf of the rose curve r =sinθ is π B. π C. π 7. The length of the parametrized curve [sin θ ( sin θ) ]dθ [sin θ ( sin θ)] dθ D. π 8 π = t, =+ t, t is B. 7 C. 7 D. 8 8. Which of the following looks most like the graph of the polar curve r =cosθ? B. C. D. - - 9. The area of the region inside both r =sinθ and r = cosθ is given b π/ π/ ( cosθ) π/ dθ + π/ (sin π/ θ) dθ B. (sin θ)dθ + π/ ( cosθ) dθ C. π π/ [( cosθ) (sin θ) ]dθ D. ( cosθ sin θ) dθ π/ [( cosθ) (sin θ) ]dθ
. Find the length of the curve =sinθ θ cos θ, =cosθ + θ sin θ, θ π π B. π C. π D. π π Answers: A, B, A, A, 5 E, 6 D, 7 E, 8 E, 9 B, C, A, B, D, E, 5 D, 6 A, 7 D, 8 A, 9 E, B, B, B, E, D, 5 E, 6 E, 7 D, 8 A, 9 B, D, E, E, D, E, 5 A, 6 D, 7 C, 8 C, 9 B, A 5