Test Information Guide: College-Level Examination Program

Transcription

1 Test Information Guide: College-Level Examination Program Calculus X/ 01 The College Board. All righte reserved. Ccilege Board, College-Level Examination Program, CLEF', and the acorn logo are registered hadern arka of the College Boeud.

2 Calculus Description of the Examination The Calculus examination covers skills and concepts that are usually taught in a one-semester college course in calculus. The content of each examination is approximately 60% limits and differential calculus and 40% integral calculus. Algebraic, trigonometric, exponential. logarithmic and general functions are included. The exam is primarily concerned with an intuitive understanding of calculus and experience with its methods and applications. Knowledge of preparatory mathematics, including algebra. geometry. trigonometry and analytic geometry is assumed. The examination contains 44 questions, in two sections, to be answered in approximately 90 minutes. Any time candidates spend on tutorials and providing personal information is in addition to the actual testing time. Section I: 7 questions, approximately 50 minutes. No calculator is allowed for this section. Section : 17 questions, approximately 40 minutes. The use of an online graphing calculator (non-cas) is allowed for this section. Only some of the questions will require the use of the calculator. Graphing Calculator A graphing calculator is integrated into the exam software, and it is available to students during Section of the exam. Since only some of the questions in Section actually require the calculator. students are expected to know how and when to make appropriate use of it. The graphing calculator, together with a brief tutorial, is available to students as a free download for a 0-day trial period. Students are expected to download the calculator and become familiar with its functionality prior to taking the exam. In order to answer some of the questions in Section of the exam. students may be required to use the online graphing calculator in the following ways: Perform calculations (e.g., exponents, roots, trigonometric values, logarithms). Graph functions and analyze the graphs. Find zeros of functions. Find points of intersection of graphs of functions. Find minima/maxima of functions. Find numerical solutions to equations. Generate a table of values for a function. Knowledge and Skills Required Questions on the exam require candidates to demonstrate the following abilities: Solving routine problems involving the techniques of calculus (approximately 50% of the exam) Solving nonroutine problems involving an understanding of the concepts and applications of calculus (approximately 50% of the exam) The subject matter of the Calculus exam is drawn from the following topics. The percentages next to the main topics indicate the approximate percentage of exam questions on that topic. For more information about downloading the practice version of the graphing calculator, please visit the Calculus exam description on the CLEP website,

3 CALCULUS 10% Limits Statement of properties. e.g., limit of a constant, sum, product or quotient Limit calculations. including limits xiim sin x involving infinity, e.g., lim is nonexistent and x oo Continuity 50% Differential Calculus The Derivative Definitions of the derivative e.g.. f'(a) = lim f (x) x- a X a sin x ltm = 0 x-4 00 X (a) and f, f (. x + h) f (x) (x) = It m h o0 Derivatives of elementary functions Derivatives of sums, products and quotients (including tan x and cot.v ) Derivative of a composite function (chain rule ), e.g., sin (ax + b), ae 1". ln(kx) Implicit differentiation Derivative of the inverse of a function (including arcsin x and arc tan.v ) Higher order derivatives Corresponding characteristics of graphs of f, f', and f" Statement of the Mean Value Theorem; applications and graphical illustrations Relation between differentiability and continuity Use of L'Hospital's Rule (quotient and indeterminate forms) Applications of the Derivative Slope of a curve at a point Tangent lines and linear approximation Curve sketching: increasing and decreasing functions; relative and absolute maximum and minimum points; concavity; points of inflection Extreme value problems Velocity and acceleration of a particle moving along a line Average and instantaneous rates of change Related rates of change 40% Integral Calculus Antiderivatives and Techniques of Integration Concept of antiderivatives Basic integration formulas Integration by substitution (use of identities. change of variable) Applications of Antiderivatives Distance and velocity from acceleration with initial conditions Solutions of y' = k y and applications to growth and decay The Definite Integral Definition of the definite integral as the limit of a sequence of Riemann sums and approximations of the definite integral using areas of rectangles Properties of the definite integral The Fundamental Theorem:,d ix f (t) dt = f (x) and CLX a Flx) dr = F(b) - F(a) Applications of the Definite Integral Average value of a function on an interval Area. including area between curves Other (e.g., accumulated change from a rate of change)

4 CALCULUS Notes and Reference Information ( I) Figures that accompany questions are intended to provide information useful in answering the questions. All figures lie in a plane unless otherwise indicated. The figures are drawn as accurately as possible EXCEPT when it is stated in a specific question that the figure is not drawn to scale. Straight lines and smooth curves may appear slightly jagged. () Unless otherwise specified, all angles are measured in radians, and all numbers used are real numbers. () Unless otherwise specified. the domain of any function f is assumed to be the set of all real numbers x for which f is a real number The range of f is assumed to be the set of all real numbers f( x) where x is in the domain of f. (4) In this test, In x denotes the natural logarithm of x (that is, the logarithm to the base e (5) The inverse of a trigonometric function f may be indicated using the inverse function notation, f -I or with the prefix "arc" (e.g.. sin -i x = arcsin x Sample Test Questions The following sample questions do not appear on an actual CLEP Examination. They are intended to give potential test-takers an indication of the format and difficulty level of the examination, and to provide content for practice and review. Knowing the correct answers to all of the sample questions is not a guarantee of satisfactory performance on the exam. Section I Directions: A calculator will not be available for questions in this section. Some questions will require you to select from among five choices. For these questions. select the BEST of the choices given. Some questions will require you to enter a numerical answer in the box provided. If y = (x +1)(x -1)(x +5), then dy = dr (A) x -1 (B) x +10x x +10x -1 (D) +5x - x (E) +0x +50x. At which of the five points on the graph in the dy figure above are -di and r both negative? di d (A) A (B) B C (D) D (E) E. Which of the following is an equation of the line tangent to the graph of f (x) = x - x at the point where x =? (A) y - 6 = 4(x - ) (B) y - 6 = 5(x - ) y - 6 = 6(x - ) (D) y - 6 = 11(x - ) (E) y - 6 = I(x - ) 4. (el + dr = (A) ex + C (B) ex +e+c +ex+c e x+i (D) x+1 +ex+c x +I + e (E) +C x + I

5 X What is Bin x Ko + x 4x (A) (B) (D) 1 (E) The limit does not exist 7. 1(x 1).1i dx = 5 (A) + C 5 (B) 1 X +x x+c 1 X X + C (D) x + x +C (E) x x + C 8. Let f and g be the functions defined by f(x) = sin x and g(x) = cos x. For which of the following values of a is the line tangent to the graph of f at x = a parallel to the line tangent to the graph of g at x = a? ft (A) 0 (B) (D) (E) The graph of f", the second derivative of the function f. is shown in the figute above. On what intervals is the graph of f concave up? (A) ( oe, oo) (B) (-00, 1) and (,00) (-00, 1) (D) ( I, ) (E) (I, 00) 9. The acceleration, at time t, of a particle moving along the x-axis is given by ao) = At time t = 0, the velocity of the particle is 0 and the position of the particle is 7. What is the position of the particle at time t? (A) 10t + 7 (B) 60t + 7t + 6t +7 (D) (E) ts + t + 7t

6 10. If f(x) = -- sill: then f(x) = (A) cos x xcos x sin x (B) x xcos x sin x (D) (E) 4x x xcos sin x x sin x xcos x 4x cos( tr + h ) cos x 1.What is lim? h- h (A) 00 (B) 1 0 (D) I (E) 00 1.If x + y = xy, then dy = dx (A) (B) x + y xy xy xy + xy x y xy x y xy (D) y xy xy x (E) 6xy x y 11. The piecewise linear graphs of the functions f and g are shown in the figure above. If h(x) = f (g(x)), what is the value of h'()? (A) 4 (B) -- 4 I (D) 1 (E ) 4 14.For which of the following functions does dy _ dy 9 dx L y = ex II. y = y = sin x (A) I only (B) II only III only (D) I and II (E) II and III

7 CALCULUS 15. The vertical height. in feet, of a ball thrown upward from a cliff is given by s(t) = 16/ + 64t + 00, where t is measured in seconds. What is the height of the ball, in feet, when its velocity is zero? 16. Which of the following statements about the curve y = x4 - x is true? (A) The curve has no relative extremum. (B) The curve has one point of inflection and two relative extreina. The curve has two points of inflection and one relative extremum. (D) The curve has two points of inflection and two relative exttema. (E) The curve has two points of inflection and three relative extreina. - (sin 17. (cos x)) = (A) cos(cos x) (B) sin( sin (sin( sin x))cos x (D) (cos(cos x))sin x (E) (sin(cos x))sin x 18. Let f be the function defined by f (x) = x` 5 x- 5 for x * 5 0 for x = 5 Which of the following statements about f are true? lim f (x) exists. x--+5 f (5) exists. III. f(x) is continuous at x = 5. (A) None (B) I only II only (D) I and II only (E) I, H, and III 19. What is the average rate of change of the function f defined by f(x) = 100 x on the interval [0, 4]? (A) 100 (B) (D) 1500 (E) If the functions f and g are defined for all real numbers and f is an antiderivative of g, which of the following statements is NOT necessarily true? (A) If g(x) > 0 for all.v, then if is increasing. (B) If g(a) = 0, then the graph of f has a horizontal tangent at x = a. If f (x) = 0 for all x, then g(x) = 0 for all x. (D) If g(x)= 0 for all x, then f (x) = 0 for all x. (E) if is continuous for all x. 10

8 C IL $ 1. If f (x) = arctan(trx), then 110) = (A) (B) -1 0 (D) 1 (E) tr 5. The function f is given by f (x) = sin (14 Which of the following is the local linear approximation for f at x = 0? (A) y = 1x (B) y = -1x y = 1 + 1x (D) y = 1-1x (E) y = -1 +1x 6. What is the area of the region in the first quadrant that is bounded by the line y = 6x and the parabola y = x?. A rectangle with one side on the axis and one side on the line x = has its upper left veitex on the graph of y = x, a.s indicated in the figure above. For what value of x does the area of the rectangle attain its maximum value? 4 (A) (B) - 1 (D) - (E) - 4. Let f (x) = x + x, If h is the inverse function of f, then ii() = 1 1 (A) (B) - 1 (D) 4 (E) Let F be the number of trees in a forest at time t, in years. If F is decreasing at a rate given by the equation df = -F and if dt F(0) = 5000, then F(t) = (A) 50001' (B) 5000e t (D) t- (E) e-1 7. Let f be a differentiable function defired on the closed interval Ea, b] and let c be a point in the open interval (a, b) such that AO = 0, r(x) > 0 when a 5. x < c, and f' (x) < 0 when c < x 5 b. Which of the following statements must be tnie? (A) f (c) = 0 (B) r (c) = 0 f (c) is an absolute maximum value of f on [a, b]. (D) f (c) is an absolute minimum value of f on [a, (E) The graph of f has a point of inflection at x = c.

9 8. The function f is continuous on the open, interval ( it, 4 If f (x ) = cos x I for xsin x x 0, what is the value of f( 0)? (A) 1 (B) 1 0 (D) (E) 1 g(0) = 0 et) > 0 for all values of t 9. The function g is differentiable and satisfies the conditions above. Let F be the function given by F(x) = f: g(t) di. Which of the following must be true? (A) F has a local minimum at x = 0. (B) F has a local maximum at x = 0. The graph of F has a point of inflection at x = 0. (D) F has no local minima or local maxima on the interval 0 5 x < 00. (E) F10) does not exist 0. The Riemann sum V( i " 1 on the closed A I 50) 50 interval [0,1] is an approximation for which of the following definite integrals? (A) f l x dx (B) (D) sox dx 50 "( x dx T) ) dr (E) sro i(7- dx F X 6x + 8 Idx = (A) (B) (D) 1 (E) 8. A particle moves along the x axis, and its velocity at time t is given by v(t) = 1 t + 1t + 8. What is the maximum acceleration of the particle on the interval 0 5 5? (A) 9 (B) 1 14 (D) 1 (E) 44 x f (x) = x k for x 0 and x -4 forx=0.let f be the function defined above, where k is a constant. For what value of k is f continuous at x =0? k= 4. If f is a continuous, even function such that f (x) dr = 4, then (51(x) + 1) cix = (A) 9 (B) 4 19 (D) 14 (E) 6 1

10 CALCULUS Section II Directions: A graphing calculator will be available for the questions in this section. Some questions will require you to select from among five choices. For these questions. select the BEST of the choices given. If the exact numerical value of your answer is not one of the choices, select the choice that best approximates this value. Some questions will require you to enter a numerical answer in the box provided. 5. The graph of the function f, shown in the figure above, has a vertical tangent at x =1 and a horizontal tangent at x =. x 0 1 For each of the following values of x indicate whether f is not continuous, continuous but not differentiable, or differentiable. f is not continuous at.v. f is continuous but not differentiable at x. f is differentiable at.v. 6. f In(10x) dr J5 X (A) 1.8 (B) (D) (E) Click on your choices. 7. The graph of the function f is shown in the figure above. What is lim f (x)? x---) (A) 1 (B) 0 1 (D) (E) The limit does not exist. 1

11 8. If the function f is continuous for all real numbers and lim f(a h) f(a) = 7, then /7-40 which of the following statements must be true? (A) f (a) = 7 (B) f is differentiable at x = a. f is differentiable for all real numbers. (D) f is increasing for x > 0. (E) f is increasing for all real numbers. 9. Let f be a function with second derivative given by flx) = sin(x) cos(4x). How many points of inflection does the graph of f have on the interval [0,101? (A) Six (B) Seven Eight (D) Ten (E) Thirteen 41. The function f is given by f(x) = x + 1. What is the average value of f over the closed interval [1, al? 4. Starting at t = 0, a particle moves along the x-axis so that its position at time t is given by x(t) = t 4-5t + t. What are all values of t for which the particle is moving to the kft? (A) 0 < t < 0.91 (B) 0.0 < t < < t < 0.91 (D) < t <.000 (E) There are no values of t for which the particle is moving to the left. 40. The area of the region in the first quadrant between the graph of y = xvri-7-7 and the x-axis is (A) (B) rr, 8 (c) 1 (D) 1- '6 (E) 4. The function f has a ielative maximum value of at x = 1, as shown in the figure above. If h(x) = x f(x), then ii(1) = (A) 6 (B) 0 (D) (E) 6 14

12 CALCULUS 44. icos x sin x dx = (A) (B) (D) (E) c o X+ C COS X sin 6 sin x C cos X + C + C cos x sin X + C 6 f i(x) = Nr; sin x 45. The first derivative of the function f is given above. If f(0) = 0, at what value of x does the function f attain its minimum value on the closed interval [0, 10]? (A) 0 (B) (D) 6.8 (E) 9.4 ex) = tan( I +x 47. Let g be the function with first derivative given above and g(1) = 5. If f is the function defined by f(x) = ln(g(x)), what is the value of CO)? (A) 0.11 (B) (D) (E) Let r(t) be a differentiable function that is positive and increasing. The rate of increase of r is equal to 1 times the rate of increase of r when r(t) = (A)a (B) Vi-7 (D) (E) The function f is differentiable on [a, 6] and a < c < b. Which of the following is NCR necessarily true? (A) fa f(x) dx = f ( x ) + f ( x ) eir (B) There exists a point d in the open interval f (a, b) such that r(d) f (bb a (a) f(x) dx 0 a (D) lim f (x) = f (c) x-+c (E) If k is a real number. then sa k f(x) dx = kl f(x) dx. 49. The function f is shown in the figure above. At which of the following points could the derivative of f be equal to the average rate of change of f over the closed interval [-, 4]? (A) A (B) B C (D) D (E) E 15

13 50. d dr S x t dt = i (A) x (B) x 1 ( D ) x y (E) x + C 51. A college is planning to construct a new parking lot. The parking lot must be rectangular and enclose square meters of land. A fence will surround the parking lot, and another fence parallel to one of the sides will divide the parking lot into two sections. What are the dimensions. in meters. of the rectangular lot that will use the least amount of fencing? (A) 1,000 by 1,500 (B) 0J by /10- by 041"0 (D) by 04 (E) 04i by 4041 x I 4 5 f ( ) The function f is continuous on the closed interval [1, 51 and has values that are given in the table above. If two subintervals of equal length are used, what is the midpoint Riemann sum approximation of J,5 f(x) dr? (A) (B) 9 14 (D) (E) 5 5. The graph of the continuous function f consists of three line segments and a semicircle centered at point (5, 1), as shown above. If F(x) is an antiderivative of f(x) such that F(0) =, what is the value of F(9)? (A) +?-r-- (B) 9+ n ( C ) 5-- (D) 7 - (E) 1-7r 54. A spherical balloon is being inflated at a constant rate of 5 cm'/sec. At what rate, in cm/sec. is the radius of the balloon changing when the radius is cm? (The volume of a sphere with radius r is V = 4. ) (A) 5 167r (B) 5 87r (D) (E) r 7r 5 7r 16

14 CALCULUS 55. R is the region below the curve y = x and above the x-axis from x = 0 to x = b, where b is a positive constant. S is the region below the curve y = cos x and above the.v axis from x = 0 to x = b. For what value of h is the area of R equal to the area of S? (A) 0.79 (B) (D) (E) The population P of bacteria in an experiment d P grows according to the equation = kp, d where k is a constant and t is measured in hours. If the population of bacteria doubles every 4 hours, what is the value of k? (A) 0.09 (8) (D).485 (E) Let f be the function defined by f(x) = ex. and let g be the function defined by g(x) = x. At what value of x do the graphs of f and g have parallel tangent lines? (A) (B) (D) (E) d (sin (5x)) = dr (A) cos-i (5x) (B) 5cos-I (5x) 1 (D) (E) 5, - 5x x 59. The graph of the function f is shown in the figure above. What is the value of lim (4 1(x))? r--4 (A) 4 (8) 0 (D) (E) 8 17

15 CALCULUS 60. During a snowstorm. the rate. in inches per hour, at which snow falls on a certain town is modeled by the function R(t)= cos(t)-0.9t +.0, whew t is measured in hours and 0 t 4. Based on the model. what is the total amount of snow. in inches. that fell on the town from t = 0 to t=4? (A) 1. (B) (D) 5.6 (E)

16 Study Resources To prepare for the Calculus exam. you should study the contents of at lea.st one introductory college level calculus textbook. which you can find in most college bookstores. You would do well to consult several textbooks. because the approaches to certain topics may vary. When selecting a textbook. check the table of contents against the knowledge and skills required for this exam. Visit for additional calculus resources. You can also find suggestions for exam preparation in Chapter IV of the Official Snkly Guide. In addition. many college faculty post their course materials on their schools' websites. Section 1 1. C g. D 4. C 5. B 6. D 7. A 8. D 9. D O. B 1. B. B. C 4. D C 7. D 8. D 9. B 0. D 1. E. B. B 4. B 5. A C 8. B /9. A 0. A 1. C. D B 5. See below Answer Key Section 6. B 7. E 8. B 9. B 40. B B 4. E 44. A 45. D 46. C 47. A 48. B 49. B 50. C 51. C 5. D 5. D 54. A 55. D 56. C 57. D 58. A 59. E 60. D 5..V f is not continuous at.7c. 0 f is continuous but not differentiable at.v. I f is differentiable at x. 19