Free Printable Applications of the First Derivative

1.

For which value of x is f ' positive and increasing?

a)

a

b)

b

c)

c

d)

d

e)

e

2.

On which of the following intervals is f increasing?

a)

b)

c)

d)

e)

3.

f has a local Max at x = ?

a)

-2.314

b)

-1.332

c)

0.350

d)

0.829

e)

1.234

4.

The function f has a local Max at x = ?

a)

-3

b)

-1

c)

1

d)

3

e)

4

5.

Which of the following could be the graph of f ' , the derivative of f ?

a)

b)

c)

d)

e)

6.

On what intervals is f increasing?

a)

[-2 , 1] only

b)

[-2 , 1] and [1 , 3]

c)

[3 , 5] only

d)

[0 , 1.5] and [3 , 5]

e)

[-2 , -1] , [1 , 2], and [4 , 5]

7.

Let f be the function given by f(x) = x³. What are all values of c that satisfy the conclusion of the Mean Value Theorem on the closed interval [-1, 2]? (No calculator)

a)

0 only

b)

1 only

c)

√3 only

d)

-1 and 1

8.

The following sign chart shows whether $f'\left(x\right)$ is positive, negative, or zero.
There is/are ...

a)

a local maximum at $x=-2$

b)

a local maximum at $x=4$

c)

local maxima at $x=-2$ and $x=4$

d)

no local extrema

9.

If $f\left(x\right)\ =\ \sqrt{1-x^2}$ ,

then $f'\left(1\right)$ is...

a)

$f\left(x\right)$ is not defined at $x=1$

b)

$1$

c)

$\frac{2}{\sqrt{3}}$

d)

$f'\left(1\right)\$ does not exist

10.

A function is decreasing if its first derivative is what?

a)

positive

b)

negative

c)

zero

d)

undefined

11.

If a function switches from increasing to decreasing, what occurs between?

a)

absolute maximum

b)

absolute minimum

c)

relative maximum

d)

relative minimum

12.

Critical numbers occur where the first derivative is what? (Check all that apply).

a)

endpoint

b)

zero

c)

maximum

d)

undefined

13.

A function is increasing if its first derivative is what?

a)

undefined

b)

positive

c)

negative

d)

zero

14.

What is the lowest point on a function called?

a)

relative maximum

b)

relative minimum

c)

absolute maximum

d)

absolute minimum

15.

What is the highest point on a function called?

a)

relative minimum

b)

relative maximum

c)

absolute minimum

d)

absolute maximum

16.

What is the "top of a hill" on a function (even if it is a cusp) called?

a)

relative minimum

b)

relative maximum

c)

absolute minimum

d)

absolute maximum

17.

What is the "bottom of a valley" on a function (even if it is a cusp) called?

a)

relative minumum

b)

relative maximum

c)

absolute minimum

d)

absolute maximum

18.

Absolute extrema can occur at which of the following? (Check all that apply.)

a)

endpoints

b)

relative maximum

c)

relative minimum

d)

critical numbers

19.

A function is decreasing if its first derivative is what?

a)

positive

b)

negative

c)

zero

d)

undefined

20.

What values separate intervals of increasing and decreasing?

a)

possible points of inflection

b)

critical numbers

21.

If a function switches from decreasing to increasing, what occurs between?

a)

absolute maximum

b)

absolute minimum

c)

relative maximum

d)

relative minimum

22.

Identify ALL conditions of Rolle's Theorem.

a)

f must be continuous on [a, b]

b)

$f\left(a\right)=f\left(b\right)$

c)

derivative must equal the slope of secant connecting the endpoints

d)

f must be differentiable on (a, b)

23.

Identify the conclusion of Rolle's Theorem.

a)

$f'\left(c\right)=\frac{f\left(b\right)-f\left(a\right)}{b-a}$

b)

$f'\left(c\right)=0$

24.

Identify the conclusion of the Mean Value Theorem.

a)

$f'\left(c\right)=\frac{f\left(b\right)-f\left(a\right)}{b-a}$

b)

$f'\left(c\right)=0$

25.

Identify ALL conditions of the Mean Value Theorem.

a)

f must be continuous on [a, b]

b)

$f\left(a\right)=f\left(b\right)$

c)

derivative must equal the slope of secant connecting the endpoints

d)

f must be differentiable on (a, b)

26.

Where does the function, f(x) = 2x³ - 9x²+ 12x -3 have relative max/min values?

a)

Rel max x = -1, Rel min x = 2

b)

Rel min x = 1, Rel min x = 2

c)

Rel max x = 1, Rel min x = 2

d)

Rel max x = 2, Rel min x = 1

27.

Given a function g(x), if g'(x)=0 at a certain value of x, then g(x) has _____________ at x.

a)

an inflection point

b)

a critical point

c)

a minimum

d)

a maximum

28.

What is the maximum value of f(x) = x³ - 3x² - 1 on the interval [-3, 2]?

a)

0

b)

-1

c)

2

d)

5

29.

Over what interval(s) is f(x) decreasing?

a)

(-3, 1)

b)

(-∞, -5) ∪ (0, 2)

c)

(-∞, -3) ∪ (1, ∞)

d)

(-5, 0) ∪ (2, ∞)

30.

The derivative f'(x) of the function f(x) is shown. On what interval(s) is the FUNCTION f(x) increasing?

a)

(-∞, -1) only

b)

( 3, ∞) only

c)

(-∞, -1) and ( 3, ∞)

d)

(1, ∞)

Applications of the First Derivative

Total questions: 30

Worksheet time: 36mins

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