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Test your knowledge again:
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!
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) =
x^{2} -
x. Calculate
f'(
x).
f'(x) = 2x - 1
f'(x) = -2x - 1
f'(x) = -2x + 1
Q. If
f'(
x) = 4
x^{2} + 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 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) = \frac{1}{x^{2}} on (0,2)
f(
x) =
x^{2}-
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) =
x^{3} 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.
There is some
c in (-2,1) with
f'(
c) = 3.