Unit 2

If g(x)=4cosx+2sinx+1, then g′(π/6)=

A: −2+3√

Let g be the function given by g(x)=limh→0sin(x+h)−sinxh. What is the instantaneous rate of change of g with respect to x at x=π3?

A: −3√2

The function f is given by f(x)=1+3cosx. What is the average rate of change of f over the interval [0,π] ?

A: −6/π

The graph of the function f, shown above, consists of three line segments. What is the average rate of change of f over the interval −1≤x≤6 ?

B: 0

Let f be the function given by f(x)=2x3. Selected values of f are given in the table above. If the values in the table are used to approximate f′(0.5), what is the difference between the approximation and the actual value of f′(0.5) ?

B: 0.433

Let f be the function given by f(x)=x3+3x2−4. What is the value of f′(2) ?

B: 24

If f(x)=4x6−3x4+2x3+e2, then f′(x)=

B: 24x5−12x3+6x2

What is the slope of the line tangent to the graph of y=9x2/x+2 at x=1 ?

B: 5

The graph of the trigonometric function f is shown above for a≤x≤b. At which of the following points on the graph of f could the instantaneous rate of change of f equal the average rate of change of f on the interval [a,b] ?

B: B

Let f be the function defined above. Which of the following statements is true?

B: f is continuous but not differentiable at x=2.

The graph of f′, the derivative of a function f, is shown above. The points (2,6) and (4,18) are on the graph of f. Which of the following is an equation of the line tangent to the graph of f at x=2 ?

B: y=5x-4

If f is the function defined by f(x)=x√4, what is f′(x)?

C: 1/4x^−3/4

Selected values of a function f are shown in the table above. What is the average rate of change of f over the interval [1,5] ?

C: 14−2/5−1

Which of the following statements, if true, can be used to conclude that f(2) exists?

C: 2 and 3 only

The derivative of a function f is given by f′(x)=0.1x+e0.25x. At what value of x for x>0 does the line tangent to the graph of f at x have slope 2 ?

C: 2.287

Let g be the function given by g(x)=x4−3x3−x. What are all values of x such that g′(x)=1/2 ?

C: 2.320

The graphs of the functions f and g are shown above. If h(x)=f(x)+4/g(x)+2x, then h′(3)=

C: 3/16

If f(x)=2x2−1/5x+3, then f′(−1)=

C: 3/4

The table above gives the values of the differentiable functions f and g and their derivatives at x=4. What is the value of ⅆⅆx(f(x)g(x)) at x=4 ?

C: 31

If f(x)=x^5, then f′(x)=

C: 5x^4

If f(x)=x√cosx, then f′(x)=

C: cosx−2xsinx/2x√

limh→05ex−5ex+h/3h=

C: −5/3ex

The function f is given by f(x)=(x3+bx+6)g(x), where b is a constant and g is a differentiable function satisfying g(2)=3 and g′(2)=−1. For what value of b is f′(2)=0 ?

D: -22

Below is an attempt to derive the derivative of secx using the product rule, where x is in the domain of secx. In which step, if any, does an error first appear?

D: There is no error.

The graph of the function f, shown above, has a vertical tangent at x=−2 and horizontal tangents at x=−3 and x=−1. Which of the following statements is false?

D: f is not differentiable at x=−3 and x=−1 because the graph of f has horizontal tangents at x=−3 and x=−1.

Let f be the function given by f(x)=1/7x7+1/2x6−x5−15/4x4+4/3x3+6x2. Which of the following statements is true?

D: f′(0.4)<f′(−1.5)<f′(−3.1)

ⅆⅆx(tanx)=

D: sec^2x

The derivative of the function f is given by f′(x)=−3x+4 for all x, and f(−1)=6. Which of the following is an equation of the line tangent to the graph of f at x=−1 ?

D: y=7x+13

If f(x)=1/x7, then f′(x)=

D: −7/x8

ⅆⅆx(cscx)=

D: −cscxcotx