AP Packet Questions

The function f given by f(x)= 3x^5 -4x^3 -3x has a relative maximum at x= a) -1 b) - root 5/5 c) 0 d) root 5/5 e) 1

A. -1

At what values of x does f(x)= 3x^5 - 5x^3 +15 have a relative maximum? a) -1 only b) 0 only c) 1 only d) -1 and 1 only e) -1, 0 and 1

A. -1 only

The absolute maximum value of f(x)= x^3 -3x^2 +12 on the closed interval [-2,4] occurs at x= a) 4 b) 2 c) 1 d) 0 e) -2

A. 4

The graph of f', the derivative of f, is shown in the figure above. Which of the following describes all relative extrema of f on the open interval (a, b)? (there is a graph in this question) a) one relative maximum and two relative minima b) two relative maxima and one relative minimum c) three relative maxima and one relative minimum d) one relative maximum and there relative minima e) three relative maxima and two relative minima

A. One relative maximum and two relative minima

The graphs of the derivatives of the functions f, g, and h are shown above. Which of the following functions f, g or h have a relative maximum on the open interval a<x<b ? (graphs are involved) a) f only b) g only c) h only d) f and g only e) f, g and h

A. f only

(CALC) The first derivative of the function f is given by f'(x)= cosx^2/ x -1/5. How many critical values does have on the open interval, (0,10) ? a) one b) three c) four d) five e) seven

B. three

A polynomial p(x) has a relative maximum at (-2, 4), a relative minimum at (1,1), and a relative maximum at (5,7) and no other critical points. How many real zeros does p(x) have? a) one b) two c) three d) four e) five

B. two

An equation to the line tangent to y= x^3 + 3x^2 +2 at its point of inflection is a) y= -6x-6 b) y= -3x +1 c) y= 2x +10 d) y= 3x-1 e) y= 4x+1

B. y= -3x+1

if f''(x)= x(x+1)(x-2)^2, then the graph of f has inflection points when x= a) -1 only b) 2 only c) -1 and 0 only d) -1 and 2 only e) -1, 0 and 2 only

C. -1 and 0 only

If the derivative of f is given by f'(x)= e ^x -3x^2, at which of the following values of x does f have a relative maximum value? a) -0.46 b) 0.20 c) 0.91 d) 0.95 e) 3.73

C. 0.91

What are all values of x for which the function f is defined by f(x)= x^3 +3x^2 -9x +7 is increasing? a) -3<x<1 b) -1 <x<1 c) x <-3 or x>1 d) x <-1 or x > 3 e) all real numbers

C. x <-3 or x>1

(CALC) The graph of the function y= x^3 +6x^2 +7x- 2cos x changes concavity at x= a) -1.58 b) -1.63 c) -1.67 d) -1.89 e) -2.33

D. -1.89

What is the x coordinate of the point of inflection on the graph y= 1/3 x^3 + 5x^2 +24? a) 5 b) 0 c) -10/3 d) -5 e) -10

D. -5

If c is the number that satisfies the conclusion of the Mean Value Theorem for f(x)= x^3 - 2x^2 on the interval 0≤x≤2, then c= a) 0 b) 1/2 c) 1 d) 4/3

D. 4/3

Which of the following pairs of graphs could represent the graph of a function and the graph of its derivative? (graphs) a) I only b) II only c) III only d) I and II e) II and III

D. I and III (two quadratic functions and two exponential)

The graph of a twice-differentiable function f is shown in the figure above. Which of the following is true? (there is a graph in this question) a) f(1) <f'(1) <f"(1) b) f(1) <f"(1) < f'(1) c) f'(1) < f(1) < f"(1) d) f''(1)< f(1) < f'(1) e) f"(1) < f'(1) < f(1)

D. f''(1)< f(1) < f'(1)

The figure above shows the graph of f', the derivative of a function of f. The domain of the function f is the set of all x such that -3≤x≤3. a) For what values of x, -3<x<3, does f have a relative maximum? A relative minimum? Justify your answer. b) For what values of x is the graph of f concave up? Justify your answer. c) sketch a possible graph:

a) F(x) has a relative max @ x= 2 since f' changes from + to F(x) has a relative min @ x= 0 since f' changes from to + b) F is concave up whenever f''>0, which means when f' is increasing. -1<x<1 and 2<x<3

The graph of f is shown in the figure above. Which of the following could be the graph of the derivative of f? (graph) - slope is zero at 1

A. (graph increasing then hitting a maximum and coming back down gradually hitting a minimum and coming back up)

The graph of the derivative of f is shown in the figure above. Which of the following could be the graph of f? (graphs with this one) picture is a regular quadratic- there is a slope of 0 at negative 1

B. (graph with -1 slope in the middle and not so steep)

If the graph of y= x^3 +ax^2 +bx -4 has a point of inflection at (1,-6) what is the value of b? a) -3 b) 0 c) 1 d) 3 e) cannot be determined

B. 0

Let f be the function given by f(x)= cos (2x) +ln (3x). What is the least value of x at which the graph of f changes concavity? a) 0.56 b) 0.93 c) 1.18 d) 2.38 e) 2.44

B. 0.93

The graph of y= f(x) is shown in the figure above. On which of the following intervals are dy/dx >0 and d2y/dx2 <0? (there is a graph) a) I only b) II only c) III only d) I and II e) II and III

B. II only (between b and c)

If g is a differentiable function such that g(x) <0, for all real numbers x and if f'(x)= (x^2 -4) (g(x)) which of the following is true? a) f has a relative maximum at x =-2 and a relative minimum at x =2 b) f has a relative minimum at x= -2 and a relative maximum at x=2 c) f has relative minima at x=-2 and at x=2 d) f has relative maxima at x= -2 and at x=2 e) it cannot be determined if f has any relative extrema

B. f has a relative minimum at x= -2 and a relative maximum at x=2

If f is the function defined by f(x)= 3x^5 -5x^4, what are all the x-coordinates of points of inflection for the graph of f? a) -1 b) 0 c) 1 d) 0 and 1 e) -1, 0 and 1

C. 1

The graph of y= 3x^4 -16x^3 + 24x^2 +48 is concave down for a) x<0 b) x>0 c) x<-2 or x >- 2/3 d) x < 2/3 or x >2 e) 2/3 <x <2

E. 2/3 <x<2

The graph of the derivative of f is shown in the figure above. Which of the following could be f? (the picture is a negative quadratic)

E. graph decreases then hits a minimum and goes through the center at +1 slope and hits a max and goes back down

The graph of y= h(x) is shown above. Which of the following could be the graph of y= h'(x)? - positive cubed function

E. simple quadratic