AP Calculus BC Unit 5 Progress Check: MCQ Part A

The graph of f′, the derivative of the function f, is shown above for −1<x<5. Which of the following statements is true for −1<x<5 ?

f has two relative minima and one relative maximum.

Let f be the function defined by f(x)=3x^3−36x+6 for −4<x<4. Which of the following statements is true?

f is decreasing on the interval (-2,2) because f'(x)<0 on the interval (-2,2).

Let f be the function defined by f(x)=x33−x22−6x. On which open intervals is f decreasing?

A. -2<x<3 only

Let f be a function with first derivative given by f′(x)=(x+1)(x−2)(x−3). At what values of x does f have a relative maximum?

A. 2 Only

Let f be the function given by f(x)=cos(x^2+x)+2 The derivative of f is given by f'(x)=-(2x+1)sin(x^2+x). What value of c satisfies the conclusion of the Mean Value Theorem applied to f on the interval [1,2]?

B. 1.438 because f'(x)= f(2)-f(1)/(2-1)

Let f be the function defined by f(x)=x^2+1/x+1 with domain [0,∞). The function f has no absolute maximum on its domain. Why does this not contradict the Extreme Value Theorem?

B. The domain of f is not a closed and bounded interval.

Selected values of a continuous function f are given in the table above. Which of the following statements could be false?

By the Mean Value Theorem applied to f on the interval [0,4], there is a value c such that f'(c)=4

Let f be the function given by f(x)=(x^2-9)/sinx on the closed interval [0,5]. Of the following intervals, on which can the Mean Value Theorem be applied to f?

C. I and II only

The derivative of the function f is given by f'(x)= sqrt(x) sin(3sqrt(3sqrt(x)) On which of the following intervals in [0,6pi] is f decreasing?

C. [1.097,4.386], [9.870,17.546]

The concentration of a certain element in the water supply of a town is modeled by the function f, where f(t) is measured in parts per billion and t is measured in years. The first derivative of f is given by f'(t)=1-lnt-sint. At what times t, for 0<t<5 does the concentration attain a local minimum?

C. t=3.353 only

Let f be a differentiable function with f(−3)=7 and f(3)=8. Which of the following must be true for some c in the interval (−3,3) ?

F'(c)=8-7/3-(-3) since the Mean Value Theorem applies.

C. [-1,1]

Let f be the function given by f(x)= sinxcosx/x^2-4 On the closed interval [-2pi, 2pi]. On which of the following closed intervals is the function f guaranteed by the Extreme Value Theorem to have an absolute maximum and an absolute minimum?