Free Printable Applications of Derivatives

1.

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

2.

If $x=0.5$ is a critical point of $f\left(x\right)$ and $f''\left(x\right)\ =\ \cos^2x\cdot\ln x$ then $x=0.5$ is a

a)

local maximum

b)

local minimum

c)

neither

3.

Which of the following is NOT a critical point of $f\left(x\right)\ =\ e^x\cdot x^2$ ?

a)

$x=0$

b)

$x=1$

c)

$x=-2$

4.

Where is the point of inflection for the function $f\left(x\right)\ =\ x^{3\ }+6x^2$ ?

a)

$x=0$

b)

$x=-4$

c)

$x=-2$

d)

$x=4$

5.

On what interval(s) is the function
$f\left(x\right)\ =\ x^{3\ }+6x^2$
concave up?

a)

$\left(-\infty,\infty\right)$

b)

$\left(-\infty,-2\right)$

c)

$\left(-2,\infty\right)$

d)

$\left(0,\infty\right)$

6.

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

7.

How many points of inflection does a parabola have?

a)

0

because the concavity of a parabola never changes

b)

$1$ because there is only one critical point

c)

$2$ because parabolas increase then decrease (or vice versa)

d)

Depends on the parabola

8.

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

9.

The average rate of change of $f\left(x\right)\ =\ \sqrt{x+1}$ on $\left[3,8\right]$ is...

a)

$\frac{1}{5}$

b)

$-\frac{1}{5}$

c)

$\frac{\sqrt{34}}{3}$

d)

$-\frac{\sqrt{34}}{3}$

10.

Which of the following functions does not satisfy the conditions put forth in Rolle's Theorem?

a)

$f\left(x\right)\ =\ \left(x-2\right)^{\frac{2}{3}}\$ on $\left[3,5\right]$

b)

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

c)

$f\left(x\right)=\left(2x-3\right)^4$ on $\left[1,2\right]$

d)

$f\left(x\right)=e^{-x^2}$ on $\left[-4,4\right]$

11.

a)

1/2

b)

-1/2

c)

0

d)

3/2

12.

a)

-4

b)

2

c)

0

d)

-8

13.

The tangent line to f(x)=x³+kx² at x = 1 is parallel to the line containing points (2, 9) and (3, 10). What is the value of k?

a)

-2

b)

-1

c)

1

d)

2

14.

What is the equation of the line tangent to y= x²+2x-1 at x=1?

a)

y=2x

b)

y=4x-2

c)

y=4x

d)

y=2x-2

15.

A function is decreasing if its first derivative is what?

a)

positive

b)

negative

c)

zero

d)

undefined

16.

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

a)

absolute maximum

b)

absolute minimum

c)

relative maximum

d)

relative minimum

17.

A square piece of green origami paper that is 6 inches on a side is being made into a gift box (with no lid) by cutting congruent squares out of each corner, folding up the sides, and taping the edges.

What size squares should you cut out for maximum volume? (do the whole problem)

a)

I should cut out squares that are 1/2 in by 1/2 in

b)

I should cut out squares that are 1 in by 1 in

c)

I should cut out squares that are 3 in by 3 in

d)

I should not cut out any squares

18.

A rectangular solid that has a square base has a surface area of 150 square inches. Find the maximum volume of the solid. (use Calculus)

a)

5

b)

125

c)

150

d)

250

19.

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

a)

endpoint

b)

zero

c)

maximum

d)

undefined

20.

A function is increasing if its first derivative is what?

a)

undefined

b)

positive

c)

negative

d)

zero

Applications of Derivatives

Total questions: 20

Worksheet time: 4hrs 21mins

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