Name Date Period. AP Calculus AB/BC Practice TEST: Curve Sketch, Optimization, & Related Rates. 1. If f is the function whose graph is given at right

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1 Name Date Period AP Calculus AB/BC Practice TEST: Curve Sketch, Optimization, & Related Rates. If f is the function whose graph is given at right Which of the following properties does f NOT have? (A) lim f ( ) on, (B) f ( ) 0 lim f ( ) lim f ( ) (D) differentiable at (E) local maimum at. Find the critical values (numbers), 0, of the function g( ) sin 0,. n n n (A) 0, n 0,,, (B) 0, n 0,,, 0, n 0,,, n n (D) 0, n 0,,, (E) 0, n 0,,, in the interval. Determine the absolute maimum value of (A) abs ma (B) abs ma 8 abs ma 0 f ( ) on the interval,. 8 (D) abs ma 7 (E) none of these. ( parts) Let f be the function defined by on, f ( ). (i) Find the derivative of f. (A) (D) f ( ) (B) f ( ) (E) f ( ) f ( ) (ii) Find all the critical points of f in,. (A) (B) (D) (iii) Determine the absolute maimum value of f 7 (A) abs. ma value (B) abs. ma value (F) (E),. f ( ) f ( ) (F) abs. ma value

2 (D) abs. ma value (E) abs. ma value (F) abs. ma value. An advertisement is run to stimulate the sale of cars. After t days, t 8, the number of cars sold is given by N( t) 000 t t. On what day does the maimum rate of growth sales occur? (A) on day 7 (B) on day on day (D) on day 6 (E) on day 6. Determine of the function f ( ) 6 satisfies the hypothesis of the MVT on the interval0,6, and if it does, find all numbers c satisfying the conclusion of that theorem. (A) c, (B) c, c (D) c (E) c (F) hypothesis not satisfied / 7. Determine if the function / (A) f ( ) satisfies the hypothesis of the MVT on the interval0,. If it does, find all possible values of c that satisfy the conclusion of the MVT. c (B) c c (D) c (E) c (F) hypothesis not satisfied 8. As a pre-graduation present, Natalie received a sports car which she drives very fast but very smoothly. She always covers the miles from her home to her favorite stores in Austin in less than 8 minutes. To slow her down, the local DPS officer, Fred Meaney, decides to change the posted speed limit. Which of the speed limits below is the highest he can post, but still catch her speeding at some point on her trip? (A) Speed Limit mph (B) Speed Limit 70 mph Speed Limit 6 mph (D) Speed Limit 0 mph (E) Speed Limit 60 mph / / 9. Determine the increasing and decreasing properties of the function f ( ) domain. (A) inc:,, dec:, inc:,,, dec:, (E) inc:, over its (B) inc:,,, dec:,,,, dec:,,, (D) inc:, dec: 0. Find the values of at which the graph of y cos changes concavity on,. (A) (B) there are no values of (D), 6 (E) (F), (G) 6 6 6

3 . The derivative of a function f is given for all by f ( ) 6 g( ) where g is some unspecified function. At which point(s) will f have a local maimum? (A) local maimum at (B) local maimum at local maimum at (D) local maimum at (E) local maimum at,. If f is a continuous function on, whose graph is at right, which of the following properties are satisfied? I. f ( ) 0, on II. f has eactly local etrema III. f has eactly critical points. (A) all of them (B) B only none of them (D) A and C only (E) C only (F) B and C only (G) A only. Suppose g is the inverse function of a differentiable function f and G( ) g( ). If f () 7 and f (), find G (7). 9 (A) G(7) (B) G(7) G(7) 6 (D) G(7) (E) G(7). On, the function f ( ) 6 tan has an inverse g. Find the value of (6) g. (Hint: find the value of f (0).) 6 (A) g(6) (B) g(6) g(6) (D) g(6) (E) g(6) 6. Determine the derivative of f ( ) sin / (A) (D) f ( ) f ( ) (B) (E) f ( ) f ( ) 0 (F) f ( ) f ( ) 0

4 6. Find the derivative of f ( ) cosarctan 6 (A) f ( ) (D) f ( ) 6 6 / 6 / (B) (E) f ( ) f ( ) 6 / 6 6 / (F) f ( ) f ( ) 6 6 / 6 6 / 7. Find the derivative of f when (A) (D) ( ) tan ln f f ( ) (B) f ( ) (E). f ( ) f ( ) (F) f ( ) f ( ) 8. A canvas wind shelter like the one at right is to be built for use along parts of the Guadalupe River. It is to have a back, two square sides, and a top. If 7 square feet of canvas is to be used in the construction, find the depth of the shelter for which the space inside is maimized assuming all the canvas is used. 7 7 (A) depth feet (B) depth feet depth feet (D) depth 7 feet (E) none of these 9. A right circular cylinder is inscribed in a sphere with diameter cm as shown. If the cylinder is open at both ends, find the largest possible surface area of the cylinder. (A) A 8 cm (D) A cm (G) A cm (B) (E) (H) A 6 cm A 8 cm A cm A 6 cm (F) A cm (I) A cm 0. Find an equation for the tangent line to the graph of f ( ) having maimum slope. (A) y (B) y y (D) y 0 (E) none of these

5 . If the radius and the height of a right circular cone both increase at a constant rate of centimeters per second, at what rate, in cubic centimeters per second, is the volume increasing when the height is 9 centimeters and the radius is 6 centimeters? (A) (B) 0 (D) (E) 08. An equation of the tangent line to y at its point of inflection is (A) y 6 6 (B) y y 0 (D) y (E) y. The area of a circular region is increasing at a rate of 96 square meters per second. When the area of the region is 6 square meters, how fast, in meters per second, is the radius of the region increasing? (A) 6 (B) 8 6 (D) (E). The sides of the rectangle at right increase in such a way that dz dt and d dt. At the instant when and y, dt dt what is the value of d dt? k. For what value of k will have relative maimum at? (A) (B) (D) (E) None of these 6. The point on the curve (A) (B) 0 y 0 that is nearest the point (D) 0, occurs where y is (E) None of these 7. A point moves on the -ais in such a way that its velocity at time t t 0 is given by what value of t does v attain its maimum? (A) (B) e e (D) e (E) There is no maimum value for v. ln t v. At t

6 8. At 0, which of the following is true of the function f defined by f ( ) e? (A) f is increasing (B) f is decreasing f is discontinuous (D) f has a relative minimum (E) f has a relative maimum 9. If a function f is continuous for all and if f has a relative maimum at, and a relative minimum at,, which of the following statements must be true? (A) The graph of f has a point of inflection somewhere between and f 0 (B) The graph of f has a horizontal asymptote (D) The graph of f has a horizontal tangent line at (E) The graph of f intersects both aes y arctan? 0. What are the coordinates of the inflection point on the graph of (A),0 (B) 0,0 0, (D), (E),. The Mean Value Theorem guarantees the eistence of a special point on the graph of y between 0,0 and,. What are the coordinates of this point? (A), (B),, (D), (E) None of these. If y is a function of such that y 0 for all and y 0 for all, which of the following could be part of the graph of y f ( )?

7 . The derivative of f ( ) attains its maimum value at (A) B) 0 (D). Given the function defined by f ( ) 0, find all values of for which the graph of f is concave up. (A) 0 (B) 0 or 0 (D) (E). If f ( ), then the set of values for which f increases is,,,, (A) (B) (D) 0, (E),0 0, 6. Let g be a continuous function on the closed interval 0,. Let g(0) and g() 0. Which of the following is NOT necessarily true? h 0, g( h) g( ) 0, (A) (B) a, b 0,, if a b g a 0, g( h) h 0. g( h) h 0,, lim g ( ) g ( h ) h h (D) (E), then ( ) g( b). 7. If f ( ) and the domain is the set of all such that 0 9, then the absolute maimum value of the function f occurs when is (A) 0 (B) (D) 6 (E) 9 8. If f is a continuous function defined for all real numbers and if the maimum value of f ( ) is and the minimum value of f ( ) is 7, then which of the following must be true? I. The maimum value of f is. II. The maimum value of f ( ) is 7. III. The minimum value of f is 0. (A) I only (B) II only I and II only (D) II and III only (E) I, II, and III

8 9. Let f be the function given by f ( ). What are the values of c that satisfy the conclusion of the Mean Value Theorem of differential calculus on the closed interval 0,? (A) 0 only (B) only only (D) 0 and (E) and 0. The graph of y f ( ),7 is shown below. How many points of inflection does this graph have on this interval? on the closed interval (A) one (B) two three (D) four (E) five. The graph of y f ( ) is shown at right. On dy which of the following intervals are 0 d and d y 0? d I. a b II. b c III. c d (A) I only (B) II only III only (D) I and II (E) II and III. A street light is on top of a 0 foot pole. Neil, who is 6 feet tall, walks away from the pole at a rate of feet per second. At what speed is the tip of Neil s shadow moving from the base of the pole when he is 0 feet from the pole? (A) 9 ft/sec (B) 0 ft/sec ft/sec (D) ft/sec (E) 8 ft/sec. A baseball diamond is a square with side 90 feet. If a batter hits the ball and runs towards first base with a speed of ft/sec, at what speed is his distance from second base decreasing when he is two thirds of the way to first base? (A) 0 ft/sec (B) 0 ft/sec ft/sec (D) 0 ft/sec (E) ft/sec

9 . A point is moving on the graph of 6y y. When the point is at,, its y-coordinate is increasing at a speed of units per second. What is the speed of the -coordinate at that time and in which direction is the -coordinate moving? (A) 8 units/sec, increasing (D) units/sec, decreasing (B) 7 units/sec, decreasing 7 units/sec, increasing (E) 8 units/sec, decreasing (F) units/sec, decreasing. In the right-triangle shown at right, the angle is increasing at a constant rate of radians per hour. At what rate is the side length of increasing when feet? (A) 8 ft/hour (B) ft/hour 0 ft/hour (D) 6 ft/hour (E) ft/hour

10 Part II: Free Response. 000 AB The figure above shows the graph of f, the derivative of the function f, for 7 7. The graph of f has horizontal tangent lines at,, and, and a vertical tangent line at. (a) Find all values of, for 7 7, at which f attains a relative minimum. Justify your answer. (b) Find all values of, for 7 7, at which f attains a relative maimum. Justify your answer. (c) Find all values of, for 7 7, at which f ( ) 0. (d) At what value of, for 7 7, does f attain its absolute maimum? Justify your answer.. Consider the curve given by y y 6 dy y y (a) Show that d y (b) Find all points (, y) on the curve whose -coordinate is, and write an equation for the tangent line at each of these points. (c) Find the -coordinate of each point on the curve where the tangent line is vertical.. 00 AB Let h be a function defined for all 0 such that h() and the derivative of h is given by h( ) for all 0. (a) Find all values of for which the graph of h has a horizontal tangent, and determine whether h has a local maimum, a local minimum, or neither at each of these values. Justify your answers. (b) On what intervals, if any, is the graph of h concave up? Justify your answer. (c) Write an equation for the line tangent to the graph of h at. (d) Does the line tangent to the graph of h at lie above or below the graph of h for? Why?

11 . 00 AB6 form B (related rates) Ship A is traveling due west toward Lighthouse Rock at a speed of kilometers per hour (km/hr). Ship B is traveling due north away from Lighthouse Rock at a speed of 0 km/hr. Let be the distance between Ship A and Lighthouse Rock at time t, and let y be the distance between Ship B and Lighthouse Rock at time t, as shown in the figure at right. (a) Find the distance, in kilometers, between Ship A and Ship B when km and y km. (b) Find the rate of change, in km/hr, of the distance between the two ships when km and y km. (c) Let be the angle shown in the figure. Find the rate of change of, in radians per hour, when km and y km AB AB6 Related Rate

12 AB (Calculator) AB