A.P. Calculus Holiday Packet

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Kerry Hodges
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1 A.P. Calculus Holiday Packet Since this is a take-home, I cannot stop you from using calculators but you would be wise to use them sparingly. When you are asked questions about graphs of functions, do not automatically reach for your calculator. Unless the problem instructions actually tell you to use a graphing utility, all of these problems can be done without one. These 40 problems should take you three hours, maximum. Have a super holiday and come back refreshed. 1. Find all intervals on which the graph of y = x2 +1 x 2 is concave upward. a. (",) b. (","1) ( 1, ) c. (",0) ( 0, ) d. ( 1," ) 2. If f 3 ( ) = 0, f "( 3) = 6, g( 3) =1, g "( 3) = 1 3, find h " ( 3 ) if h( x) = f ( x ) g( x) a. 18 b. 6 c. -6 d Find all open intervals on which f ( x) = x x 2 + x " 2 is decreasing. a. (",) b. (",0) c. (","2) ( 1, ) d. (","2) ("2,1) ( 1, ) 4. Find all critical values for f ( x) = ( 9 " x 2 ) 3 5 a. 0 b. 3 c. -3, 3 d. -3, 0,3 5. Find all intervals on which the graph of f ( x) = x "1 is concave upward. x + 3 a. (",) b. (","3) c. ( 1," ) d. ("3,) 6. Find dy dx for 5x2 " 2xy + 7y 2 = 0 a. 5x + 7 y 2 b. y " 5x 7y " x c. 10x +12 y d. 5x " y 7y " x

2 7. Given f ( x) =10 " 16, find all c in the interval [-2, 8] that satisfies the Mean Value Theorem. x a. 4 b. 5 c. 8 5 d. ±4 8. Given f x ( ) = "6 x, choose the correct statement. a. The graph of f is concave upward on the interval (",0). b. The graph of f is concave downward on the interval (",0). c. The graph of f is concave upward on the interval (",). d. The graph of f is concave upward on the interval ( 0," ). 9. Find all points of extrema on the interval 0,2" a. "1. "1+ 3 ' & ) b. "," % 2 ( [ ] if y = x + sin x. ( ) c. ("1,0) d. 3" % 2,0 & ( ' 10. Which of the following statements is true of f ( x) = "x 3 +18x 2 "105x +198? a. f is decreasing on the interval ( 6," ). b. f is decreasing on the interval ( 5,7). c. f is increasing on the interval (",5). d. f is increasing on the interval ( 5,7). 11. Find all points of extrema on the interval [ 0,2" ] if y = x cos x. a. &"1. "1+ 3 % 2 ' ) b. "," +1 ( ( ) c. ("1,0) d. % 3" 2, 3" 2 & ( ' 12. Find all points of inflection: f ( x) = 1 12 x 4 " 2x a. ( 2,0) b. ( 2,0), ("2,0) c. ( 0,15) d. 2, 25 " % " ', (2, 25 % ' 3 & 3 &

3 13. Match the graph below with the correct function. a. y = x cos x b. y = 2x sin x c. y = x sin2x d. y = x sin x 14. Find the values of x that give relative extrema for the function f ( x) = 3x 5 " 5x 3. a. Relative maximum: x =1; Relative minimum: x = 5 3 b. Relative maximum: x = "1; Relative minimum: x =1; c. Relative maxima x = ±1; Relative minimum: x = 0 d. Relative maximum: x = 0; Relative minima: x = ±1 15. A particle is moving along the x-axis according to the position function s( t) = 3t 2 " t 3. What is its maximum velocity? a. 1 b. 2 c. 3 d Find the absolute maximum and absolute minimum of f on the interval ( 0,3] : f ( x) = x3 " 4x 2 + 7x a. Maximum: None Minimum: (3, 4) b. Maximum (0, 7) Minimum: (3, 4) c. Maximum: None Minimum: (2, 3) d. Maximum: (0, 7) Minimum: (2, 3) 17. Find the absolute maximum and absolute minimum of f on the interval ("1,2] : f ( x) = "x3 + x 2 + 3x +1 a. Maximum: (1, -2) Minimum: (-1, 2) b. Maximum (1, -2) Minimum: None c. Maximum: None Minimum: None d. Maximum: None Minimum: (-1, 2) x x +1..

4 18. Find the values of x that give relative extrema for the function f x a. Relative maximum: x = -1; Relative minimum: x = 1 b. Relative maxima: x = 1, x = 3; Relative minimum: x = -1 c. Relative maximum: x = 2 d. Relative maximum: x = -1; Relative minimum: x = Find all points of inflection of the function f x 9 ( ),% 2 a. ( 0,0) b. 0,0," ( ) = x 4 " 6x 2. & ( c. 3,"81 ' ( ) d. 0,0 ( ) = ( x +1) 2 x " 2 ( ), 3,"81 ( ). ( ) 20. Find the absolute maximum and absolute minimum of f on the interval ("4,"1] : f x ( ) = x3 + 8x 2 +19x +12 x + 4 a. Maximum: None Minimum: (-2, -1) b. Maximum (-4, 3) Minimum: (-1, 0) c. Maximum: (-4, 3) Minimum: (-2, -1) d. Maximum: None Minimum: (-1, 0) 21. Find dy dx for y = sec3x a. 9sec3x tan3x b. 3sec3x tan3x c. 3sec 2 3x d. 3tan 2 3x 22. Given that f ( x) = "x 2 +12x " 34 has a relative maximum at x = 6, choose the correct statement. a. f " is positive on the interval ( 6," ). b. f " is positive on the interval (",). c. f " is negative on the interval ( 6," ). d. f " is negative on the interval (",6). 23. Find all points of inflection of the function f ( x) = x 4 + x 3. ( ) and %" 1 2," 1 a. 0,0 & ( b. " 1 16' 2," 1 & % ( c. 0, 0 16' ( ) d. ( 0,0) and %" &," ( 256'

( ) = "x 2 +18x " 78 has a relative maximum at x = 9, choose the correct statement. 24. Given that f x a. f " is negative on the interval (",9). b. f " is negative on the interval ( 9," ). c. f " is positive on the interval (",).

5 ( ) = "x 2 +18x " 78 has a relative maximum at x = 9, choose the correct statement. 24. Given that f x a. f " is negative on the interval (",9). b. f " is negative on the interval ( 9," ). c. f " is positive on the interval (",). d. f " is positive on the interval ( 9," ). 25. Given f ( x) = "1 x, find all c in the interval "3," 1 & % ( that satisfies the Mean Value Theorem. 2' a. c = " 3 2 b. c = ± 3 2 c. The Mean Value Theorem doesn t apply because f is not continuous at x = 0. d. The Mean Value Theorem doesn t apply because f " 1 & % ( ) f ("3). 2' 26. Find the value of the derivative at the indicated extremum. a. 2 b. -6 c. 0 d. does not exist 27. Let f " x ( ) = 4x 3 2x and let f ( x) have critical values -1, 0, and 1. Determine which critical values give a relative maximum. a. -1 b. 0 c. 1 d. -1 and 1

28. Which of the following describes the behavior of f ( x) = x 3 + x. a. Relative maximum: x = 0 b. f has no relative extrema c. Relative maximum: x = 1 3 ; relative minimum x = " 1 3 d.

No maximum, minimum at x = b d. No extrema 30. Give the sign of the second derivative of f at the point (1, 1) a. Negative b. Positive c. Zero d. Does not exist 31.

6 28. Which of the following describes the behavior of f ( x) = x 3 + x. a. Relative maximum: x = 0 b. f has no relative extrema c. Relative maximum: x = 1 3 ; relative minimum x = " 1 3 d. Relative minimum: x = Determine from the graph whether f possesses extrema on the interval ( a, b). a. Maximum at x = c, minimum at x = b b. Maximum at x = c, no minimum c. No maximum, minimum at x = b d. No extrema 30. Give the sign of the second derivative of f at the point (1, 1) a. Negative b. Positive c. Zero d. Does not exist 31. Which statement is not true of the graph f x " 1 a. f has a relative maximum at 3, 256 % '. 27 & b. f has a point of infection at (3, 0). c. f has an intercept at (3,0). d. f has a relative maximum at (3, 0). ( ) = ( x +1) ( x " 3) 2?

7 32. f x ( ) = ( x + 2) 3 " 4. The point (-2, -4) is which of the following? a. An absolute maximum. b. An absolute minimum. c. A critical point but not an extremum. d. Not a critical point. 33. The figure in the graph below is the 2nd derivative of a polynomial function, f. Choose a graph of f. a. b. c. d. 34. Let f x ( ) be a polynomial function such that f ("2) = 5, f ("2) = 0 and f ("2) = 3. The point (-2, 5) is which of the following for the graph of f? a. relative maximum b. relative minimum c. intercept d. inflection point 35. Let f x ( ) be a polynomial function such that f ( 4) = "1, f ( 4) = 2 and f ("2) = 0. If x < 4, then f " ( x) < 0 f ( x) > 0. The point (4, -1) is which of the following for the graph of f? and if x > 4, then " " a. relative maximum b. relative minimum c. critical value d. inflection point

8 36. State why Rolle s Theorem does not apply to f ( x) = x 2 3 on the interval [-1, 1]. a. f is not continuous on [-1, 1]. b. f is not defined on the entire interval. c. f is not differentiable at x = 0. d. f ( 1) " f (1). 37. Find the value(s) of c guaranteed by Rolle s Theorem for f ( x) = x 2 + 3x on [ 0,2]. a. c = " 3 2 b. c = 0,3 c. Rolle s Theorem does not apply as f is not continuous on [0, 2] d. Rolle s Theorem does not apply as f ( 0) " f ( 2). 38. Use the graph of f ( x) = x2 applies. x 2 " 4 to determine on which of the following intervals Rolle s Theorem a. [0, 3] b. [-3, 3] c. " 3 2, 3 & % ( d. [-2, 2] 2' 39. Determine the open intervals where the graph of f x ( ) a. concave down ", b. concave down (","1); concave up "1, c. concave down ","1 d. concave up ","1 ( ) and ("1,) ( ) and ("1,) ( ) 1 ( ) = " ( x +1) 2 is concave up or concave down. 40. Consider f ( x) = x 2,a > 0. Determine the effect on the graph of f if a is varied. x 2 + a a. Each y value is multiplied by a. b. As a increases, the vertical tangent lines move further from the origin. c. The graph of the curve is shifted a units to the left. d. As a increases, the curve approaches its asymptote more slowly.

What makes f '(x) undefined? (set the denominator = 0)

What makes f '(x) undefined? (set the denominator = 0) Chapter 3A Review 1. Find all critical numbers for the function ** Critical numbers find the first derivative and then find what makes f '(x) = 0 or undefined Q: What is the domain of this function (especially