Die Heuning Pot Literature Guide

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# Derivatives: Rolle With It Quiz

Think you’ve got your head wrapped around Derivatives? Put your knowledge to the test. Good luck — the Stickman is counting on you!
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Q. f'(x) =

$$\lim_{h\to0}\frac{f(a +h)-f(a)}{h}$$
$$\lim_{h\to0}\frac{f(x + h)-f(x)}{h}$$
$$\lim_{h\to0}\frac{f(a +h)}{h}$$
$$\lim_{h\to0}\frac{f(x + h)}{h}$$
Q. Let f(x) = x2 - x. Calculate f'(x).

f'(x) = 2x + 1

f'(x) = 2x - 1
f'(x) = -2x - 1
f'(x) = -2x + 1
Q. If f'(x) = 4x2 + 3, find f'(-2).

11
-19
19
There is insufficient information to answer this question.
Q. If f(x) is a line of the form f(x) = mx + b then

f'(x) = m
f'(x) = mx
f'(x) = b
f'(x) = 0
Q. A graph of the function f(x) is shown below.

PICTURE mult choice 3-3

The function f'(x) is

always positive
always negative
positive when x is less than zero and negative when x is greater than zero.
negative when x is less than zero and positive when x is greater than zero.
Q. A graph of the function f(x) is shown below.

PICTURE mult choice 3-1 without red

Which of the following could be the graph of f'(x)?

PICTURE: mult choice 3-1a
PICTURE: mult choice 3-1b without red
PICTURE: mult choice 3-1c
PICTURE: mult choice 3-1d
Q. A graph of f'(x) is shown below.

PICTURE mult choice 3-2 without red

Which of the following could be the graph of f(x)?

PICTURE: mult choice 3-2a without red
PICTURE: mult choice 3-2b
PICTURE: mult choice 3-2c
PICTURE: mult choice 3-2d
Q. If

•  f is continuous on [a,b
• f is differentiable on (a,b), and
•  f(a) = f(b), then Rolle's Theorem tells us

the number of values c in (a,b) for which f'(c) = 0.

that there is some c in (a,b) with f'(c) = 0.

that there are infinitely many values of c in (a,b) with f'(c) = 0.

the precise value(s) of c in (a,b) for which f'(c) = 0.

Q. For which given function and interval are we allowed to use the Mean Value Theorem?

f(x) = |x| on (-1,1)

f(x) = \frac{1}{x2} on (0,2)
f(x) = x2-x on (-2,-1)

f(x) = \sqrt{x} on (0,1)
Q. What does the Mean Value Theorem tell us about the function f(x) = x3 on the interval (-2,1)?

Nothing. We aren't allowed to use the Mean Value Theorem here.
There is some c in (-2,1) with f'(c) = 0.

f'(0) = 3.

There is some c in (-2,1) with f'(c) = 3.