1. Determine the limit (if it exists). + lim A) B) C) D) E) Determine the limit (if it exists).

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1 Please do not write on. Calc AB Semester 1 Exam Review 1. Determine the limit (if it exists) lim x 3 6 x 3 x + 3 A).1 B).8 C) D).7778 E) Determine the limit (if it exists). 1 1cos x lim x x A) B) C) D) does not exist E) 3. Determine the limit (if it exists). 6 sin x lim x 6 x A) B) C) 1 D) does not exist E). f ( x x) f ( x) Find lim x x A) B) 3 C) 1 D) E) Limit does not exist. where f ( x) 3x.

2 5. Find constants a and b such that the function is continuous on the entire real line. 1, x 3 f ( x) ax b, 3 x 7 1, x 7 A) a =, b = B) a =, b = C) a =, b = D) a =, b = E) a =, b = 6. Find the value of c guaranteed by the Intermediate Value Theorem. f ( x) x x + 1, 3, 7, f ( c) 9. A) B) 1 C) D) 6 E) 5 7. Find all values of c such that f is continuous on (, ). x x c f( x) 5, x x c A) c, B) c C) c 3 D) c, E) 5 33 c 8. x 36 Find the vertical asymptotes (if any) of the function f( x). x 9 x+ 18 A) x = 3 B) x = 3 C) x = 6 D) x = 6 E) x = 18

3 9. Find the limit (if it exists). lim xtan x 1 x A) B) C) D) 1 E) Limit does not exist. 1. Find the slope m of the line tangent to the graph of the function g x 6 x at the point A) m 1 B) m C) m 6 D) m 1 E) m 5 1, Find an equation of the tangent line to the graph of the function 3,. A) y 11 x 37 B) y x 7 C) y x D) y 7 E) y 11x f x x 5x at the point 1. The graph of the function f is given below. Select the graph of f.

4 A) D) B) E) C)

5 13. Identify the graph which has the following characteristics. f () f ( x), x Graph 1 Graph Graph 3 Graph A) Graph 3 B) Graph C) Graph D) Graph 1 E) none of the others Suppose the position function for a free-falling object on a certain planet is given by s t 1t v t s. A silver coin is dropped from the top of a building that is 137 feet tall. 5 Determine the velocity function for the coin. A) 5 vt 7t 137 B) v t 7t C) 5 s t 1t 137 D) v t 1t E) 6 v t 5t 15. Suppose the position function for a free-falling object on a certain planet is given by st 16t vt s. A silver coin is dropped from the top of a building that is 138 feet tall. Determine the average velocity of the coin over the time interval 1,. A) 16 ft/sec B) 8 ft/sec C) 8 ft/sec D) 16 ft/sec E) 5 ft/sec

6 16. Suppose the position function for a free-falling object on a certain planet is given by s t 1t v t s. A silver coin is dropped from the top of a building that is 1366 feet tall. Find the instantaneous velocity of the coin when t 3. A) ft/sec B) 16 ft/sec C) 5 ft/sec D) 8 ft/sec E) 39 ft/sec 17. Suppose the position function for a free-falling object on a certain planet is given by s t 13t v t s. A silver coin is dropped from the top of a building that is 1368 feet tall. Find the time required for the coin to reach ground level. Round your answer to the three decimal places. A).85 sec B) sec C).6 sec D) sec E) 1.58 sec 18. Suppose the position function for a free-falling object on a certain planet is given by s t 1t v t s. A silver coin is dropped from the top of a building that is 1376 feet tall. Find velocity of the coin at impact. Round your answer to the three decimal places. A) ft/sec B) ft/sec C) ft/sec D) 67.9 ft/sec E) ft/sec 19. The volume of a cube with sides of length s is given by respect to s when s 11 centimeters. A) 363 cm B) 11 cm C) 1331 cm D) cm E) 3993 cm V 3 s. Find the rate of change of volume with. Determine all values of x, (if any), at which the graph of the function has a horizontal tangent. 3 y( x) x 9x A) x B) x and x 6 C) x and x 6 D) x 6 E) The graph has no horizontal tangents.

7 1. The length of a rectangle is 3t 5and its height is t, where t is time in seconds and the dimensions are in inches. Find the rate of change of area, A, with respect to time. A) da t 15 t square inches/second dt B) da 3 t 15 t square inches/second dt C) da 3 t 1 t square inches/second dt D) da t t square inches/second dt E) da t 3 15 t 5 square inches/second dt. A population of 53 bacteria is introduced into a culture and grows in number according to the 8t equation Pt 531 where t is measured in hours. Find the rate at which the population t is growing when t =. Round your answer to two decimal places. A) 91.3 bacteria per hour B) bacteria per hour C) 1.95 bacteria per hour D) 69.7 bacteria per hour E) 78.8 bacteria per hour 3. Find the derivative of the function. 5 y cos x +3 A) 5 y sin x +3 B) y x sin x 5 +3 C) 5 y sin x +3 D) y x 5 cos x 5 +3 E) 5 y sin x +3. x y Use implicit differentiation to find an equation of the tangent line to the ellipse A) y 7x 9 B) y 11x 11 C) y 7x 1 D) y 1x 11 E) y x at 1,7.

8 5. Evaluate the derivative of the function at the given point. 3 f() t t 1, A) 3 f '(3) 3 3, B) 3 f '(3) C) 3 f '(3) D) 3 f '(3) E) 3 f '(3) 8 6. Find / tan 6x y 6 x. dy dx by implicit differentiation given that A) dy 6sec 6x y 6 dx B) dy 36sec 6x y dx C) dy sec6x y tan 6x y 6 dx D) dy 6cos 6x y 6 dx E) dy 1 cos 6 x y 6 x dx 6 7. Find dy/dx by implicit differentiation. x 5x 1xy y 36 A) dy x 5 1y dx y 1x B) dy x 5 1y dx x 1y C) dy x 5 1y dx y 1x D) dy x 5 1y dx y 1x E) dy x 5 1y dx y 1x

9 8. Find d y/dx in terms of x and y. x y 6 A) d y 6 dx y B) d y x dx y C) x y d y y dx y D) x y d y y dx y E) d y y dx x 9. A spherical balloon is inflated with gas at the rate of cubic centimeters per minute. How fast is the radius of the balloon increasing at the instant the radius is 5 centimeters? A) dr 1 cm/min dt 5 B) dr 5 cm/min dt C) dr 1 cm/min dt 75 D) dr 1 cm/min dt 5 E) dr 75 cm/min dt 3. Locate the absolute extrema of the function A) no absolute max; absolute min: (, 31) B) absolute max: ( 1, 5) ; absolute min: (, 31) C) absolute max: (, 31) ; no absolute min D) absolute max: (, 31) ; absolute min: ( 1, 5) E) no absolute max or min f x x x ( ) + 8 1,. on the closed interval

10 31. Locate the absolute extrema of the function A) absolute maximum:, 5 absolute minimum:, B) absolute maximum:, absolute minimum:, 5, absolute minimum:, C) absolute maximum: D) absolute maximum:, absolute minimum:, 5, E) absolute maximum: absolute minimum:, 5 g( x) x 5,. on the closed interval 3. x 1 x1 Sketch the graph of the function f( x) x 1 x function on the interval,. A) left endpoint:, 1 absolute minimum right endpoint:,7 absolute maximum B) left endpoint:, absolute minimum right endpoint:,16 absolute maximum C) left endpoint:, 1 absolute minimum right endpoint:,16 absolute maximum D) left endpoint:, 1 absolute minimum right endpoint: 1,1 absolute maximum E) left endpoint: 1,1 absolute minimum right endpoint:,16 absolute maximum and locate the absolute extrema of the

11 33. Determine whether Rolle's Theorem can be applied to the function f ( x) x 1 x + 3 on the closed interval [,8]. If Rolle's Theorem can be applied, find all values of c in the open interval (,8) such that f( c). A) Rolle's Theorem applies; c = 6 B) Rolle's Theorem applies; c = 5 C) Rolle's Theorem does not apply D) Rolle's Theorem applies; c = 6 E) Rolle's Theorem applies; c = 7 3. Determine whether the Mean Value Theorem can be applied to the function f ( x) x on the closed interval [ 3,5]. If the Mean Value Theorem can be applied, find all numbers c in the open interval f(5) f( 3) ( 3,5) such that f() c. 5 3 A) MVT applies; c = B) MVT applies; c = C) MVT applies; c = 1 D) MVT applies; c = 3 E) MVT applies; c = Determine the open intervals on which the graph of concave upward., A) concave downward on ; concave upward on, B) concave upward on, C) concave downward on, D) concave upward on, ; concave downward on, E) concave downward on,1 ; concave upward on 1, f ( x) 3 x + 6 x + 7 is concave downward or 36. Find the limit. 7 x lim x x 3 A) 3 B) 7 C) 1 D) E) does not exist

12 37. Find the limit. 6 x 1 lim x x + 8 A) B) 1 C) D) 3 E) Find the limit. lim 5 x + 7 x 16 x 8 x A) 5 B) 5 16 C) 5 D) 5 E) 39. Find the length and width of a rectangle that has perimeter meters and a maximum area. A) 5, 15 B) 1, 19 C) 1, 1 D) 11, 9 E) 1, 7. Find the indefinite integral 8z 3 A) z 3z C B) 8z 3z C C) z 3z D) 8 E) none of the above dz.

13 1. Find the indefinite integral 5sin x 7cos x dx. A) 5cos x 7sin x C B) 5cos x 7sin x C C) 5cos x 7sin x C D) 5cos x 7sin x+c E) 7cos x 5sin x C. The diagram below shows upper and lower sums for the function using subintervals. Use upper and lower sums to approximate the area of the region using the subintervals. A) lower:.175 ; upper: 1.15 B) lower:.5 ; upper:.9 C) lower: 1.7 ; upper: 3.6 D) lower:.85 ; upper: A rectangular page is to contain 1 square inches of print. The margins on each side are 1 inch. Find the dimensions of the page such that the least amount of paper is used. A) 1, 1 B) 13, 13 C) 1, 1 D) 11, 11 E) 1, 1

14 . The graph of the function g( y) y is given below. Which of the following definite integrals yields the area of the shaded region? A) y dy B) y dy C) y dy D) y dy E) y dx 5. Determine the area of the given region. y x(1 x) A) 5 6 B) 1 6 C) 6 7 D) 1 3 E) 1 18

15 6. 13 Use your calculator to graph the function 3 3 x x and determine all absolute extrema on the interval,3. A) absolute maximum: 3,13 absolute minimum:, B) absolute maximum: 6, 3,, 3, absolute minimum: C) absolute maximum: 3,13 absolute minimum:,3 D) 6 absolute maximum:, 3 absolute minimum:,3 E) 13 absolute maximum: 1, 3,, 3, absolute minimum: 7. The graph of f is shown below. For which values of x is f ' x zero? A) x ; x B) x ; x C) x6; x D) x ; x 1 E) x ; x 1

16 8. The graph of f is shown below. On what interval is f ' an increasing function? A), B), C), D) 1, E), 9. The graph of f is shown below. For which value of x is f ' x minimum? A) x B) x C) x D) x E) x 6 5. Write an equation of the line that passes through the point,3 and is perpendicular to the line x y 5. A) x y7 B) x y1 C) x y 7 D) x y1 E) x y 5