Free Printable AP Calculus Rates of Change Review

1.

Find the average rate of change of f(x)=2x²+3x-1 from 1 to 2.

a)

18

b)

-9

c)

9

d)

3

2.

Find the equation of the line tangent to the function at the given point

a)

a

b)

b

c)

c

d)

d

3.

Differentiate y = $\frac{2x^2+x-3}{x}$

a)

y = 2x + 1 - 3x^-1

b)

y' = 4x + 1 - 1x^-2

c)

y' = 2 + 3x^-2

d)

y' = 4x + 1 - 3x^-2

4.

If $f\left(x\right)$ has a derivative at x = a, then f is continuous at x = a.

a)

True

b)

False

5.

Is the function is the following graph differentiable at x = 0? Why?

a)

Yes. The limit from the left and right of 0 both approach 0.

b)

No, $\lim_{x\rightarrow0}\left|x\right|\$ does not exist.

c)

No, $\lim_{h\rightarrow0}\frac{\left(\left|x+h\right|-\left|x\right|\right)}{h}$ does not exist.

d)

Yes, the function is differentiable because there is a constant rate of change of -1 from $\left(-\infty,0\right)$ and 1 from $\left(0,\infty\right)$ .

6.

Find the derivative of the funtion. (hint: you will need to multiply first!)

a)

4x⁸+60x⁶+12x³

b)

12x⁷+36x²

c)

32x⁷+36x²

d)

20x⁶

7.

What is the average rate of change of y = 2(3)^x at [2, 4]?

a)

72

b)

3

c)

36.8

d)

630

8.

Graph of a function is given.Where is the function continuous yet NOT differentiable?

a)

x = a, b, c, d

b)

x = b, c, d

c)

x = a, b,

d)

x = b, d

9.

Find all points of the domain of $y\ =\ \left|x-2\right|+3$ where $\frac{dy}{dx}$ does not exist.

a)

$\frac{dy}{dx}$ exists for all real numbers.

b)

$\frac{dy}{dx}$ does not exist at (0,0).

c)

$\frac{dy}{dx}$ does not exist at (2,3).

d)

$\frac{dy}{dx}$ does not exist at (-2,3).

10.

Find the difference quotient when f(x) = x² - x + 1

a)

2x - 1 + h

b)

4x - 5 - h

c)

-2x - 1 + h

d)

x² + 2h

11.

Differentiate h(x) = 3x^5/3 - 6x^1/6 + 3x

a)

h'(x) = 5x^2/3 - x^-5/6 + 3

b)

h'(x) = 3x^2/3 - 6x + 3

c)

h'(x) = 5x^4/3 - 6x + 3

d)

h'(x) = 3x^4/3 - x^-5/6 + 3

12.

Is the above function differentiable at

x = 0? Why?

a)

Yes. The derivative would be equal to zero since there is a vertical tangent.

b)

No. The derivative would not exist at x = 0 since there is a vertical tangent.

c)

No. The derivative does not exist since $\lim_{x\rightarrow0}\ x^{\frac{1}{3}}$ does not exist.

d)

Yes since $\lim_{h\rightarrow0}\ \frac{\left(x+h\right)^{\frac{1}{3}}-x^{\frac{1}{3}}}{h}$ exists.

13.

The following limit is used to find the slope of what?

(select all that apply)

\lim_{h\rightarrow0}\ \frac{f\left(a+h\right)-f\left(a\right)}{h}

a)

Average Rate of Change

b)

Instantaneous Rate of Change

c)

Slope of the Secant Line

d)

Slope of the Tangent Line

14.

Find the derivative of f(x) = 6x³⁰-2x¹⁵ + 4x³ - 2x + 1

a)

f'(x) = 18x²⁹+ 30x¹⁵ + 12x³

b)

f'(x) = 180x²⁹ - 30x¹⁴+ 12x²

c)

f'(x) = 180x²⁹ - 30x¹⁴+ 12x² - 2

d)

f'(x) = 180x²⁹ - 30x¹⁴+ 12x² +1

15.

At which point(s) will the slopes of the tangent line be zero?

a)

at C only

b)

at points A, C and E only

c)

at point B and D only

d)

at points A and E only

16.

which of the following means "second derivative"

a)

f''(x)

b)

y''

c)

d²y/dx²

d)

all of the above

17.

Find the average rate of change on the interval $\left[a,a+h\right]$

a)

$\left(\frac{f\left(a+h\right)-f\left(a\right)}{a}\right)$

b)

$\left(\frac{f\left(a+h\right)-f\left(a\right)}{h}\right)$

c)

$\left(\frac{f\left(a\right)-f\left(a+h\right)}{h}\right)$

d)

$\frac{f\left(a+h\right)}{f\left(a\right)}$

18.

Find the derivative:

f\left(x\right)=x^{\frac{12}{5}}+3x^{\frac{7}{5}}

a)

$f'\left(x\right)=\frac{12}{5}x^{\frac{7}{5}}+\frac{21}{5}x^{\frac{2}{5}}$

b)

$f'\left(x\right)=x^{\frac{7}{5}}+3x^{\frac{2}{5}}$

c)

$f'\left(x\right)=12x^{\frac{7}{5}}+21x^{\frac{2}{5}}$

d)

$f'\left(x\right)=\frac{12}{5}x^{\frac{4}{5}}+\frac{21}{5}x^{\frac{1}{5}}$

19.

Find y'' if y = - 8x³+5x - 12

a)

y'' = -24x²+ 5

b)

y'' = 0

c)

y'' = - 48x

d)

y'' = - 48

20.

Find the derivative of $f\left(x\right)\ =\ \frac{1}{2\sqrt{x}}$

a)

$-\frac{1}{4\sqrt{x^3}}$

b)

$-\frac{1}{\sqrt{x^3}}$

c)

$-x^2$

d)

$-4x^2$

21.

What was the average rate of change in the population from 1900 to 1990?

a)

175 million people/year

b)

1.94 million people/year

c)

2.13 million people/year

d)

-1 million people/year

22.

Find the derivative:

f\left(t\right)=\frac{1}{t}-\frac{3}{t^2}+\frac{5}{t^3}

a)

$f'\left(t\right)=-\frac{1}{t^2}+\frac{6}{t^3}-\frac{15}{t^4}$

b)

$f'\left(t\right)=-t^{-2}+6t^{-3}-15t^{-4}$

c)

$f'\left(t\right)=t^{-1}-3t^{-2}+5t^{-3}$

d)

$f'\left(t\right)=\frac{1}{t^2}+\frac{6}{t^3}-\frac{15}{t^4}$

23.

What is the derivative of xⁿ?

a)

(n-1)xⁿ

b)

nxⁿ⁺¹

c)

(n+1)x^n-1

d)

nx^n-1

24.

Find the slope of f(x) = -3x² – 6x at x = 1.

a)

m = 0

b)

f'(x) = -6x - 6

c)

f'(x) = 6x

d)

m = -12

25.

For what value(s) of x is the function continuous but not differentiable?

a)

x = -1

b)

x = 0

c)

x = 2

d)

x = 3; x = -2

26.

Write the slope of the line tangent to the graph of y=x² – 2 at the point x = -8.

a)

2

b)

-4

c)

-8

d)

-16

AP Calculus Rates of Change Review

Total questions: 26

Worksheet time: 2hrs 42mins

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