Free Practice Quiz Question List

3 Rates of Change and Derivatives Online Quiz Questions

Use this free practice quiz with 20 questions to review 3 Rates of Change and Derivatives, test your knowledge, and prepare for your next test or exam.

20 questions

True or false

1 point

True or false: If position is measured in meters and time is measured in seconds, the units of its derivative with respect to time are meters per second.

A
True
B
False

True or false

1 point

True or false: If $f(x)=12$ for every $x$ , then $f'(x)=0$ .

A
True
B
False

Choose one

1 point

What geometric quantity represents the average rate of change of $f$ from $x=a$ to $x=b$ ?

A
The slope of the tangent line at one endpoint
B
The slope of the secant line through the two endpoint points
C
The height of the graph at the midpoint
D
The derivative evaluated at every point in the interval

Fill in the blank

1 point

If $s(t)$ gives an object's position, its instantaneous velocity is $v(t)=$ .

Written response

1 point

Let $f(x)=3x^4-5x^2+7$ . What is the instantaneous rate of change $f'(1)$ ?

True or false

1 point

True or false: In $f'(a)=\lim_{h\to0}\frac{f(a+h)-f(a)}{h}$ , the value $h=0$ is substituted into the quotient before taking the limit.

A
True
B
False

Choose one

1 point

Suppose $f(10)=100$ and $f'(10)=1.8$ . Using a linear approximation, estimate $f(10.5)$ .

A
99.1
B
100.5
C
100.9
D
101.8

Choose all

1 point

Select all situations in which an object is speeding up.

A
If $v(t)>0$ and $a(t)>0$
B
If $v(t)>0$ and $a(t)<0$
C
If $v(t)<0$ and $a(t)<0$
D
If $v(t)<0$ and $a(t)>0$

Fill in the blank

1 point

The average rate of change of $f$ from $x=a$ to $x=b$ is .

Written response

1 point

An object's position is $s(t)=t^3-4t$ , measured in meters. What is its acceleration at $t=2$ seconds? Enter the numerical value in $\text{m/s}^2$ .

Choose all

1 point

Select all features that can prevent a function from being differentiable at a point.

A
A sharp corner
B
A cusp
C
A smooth point on a differentiable curve
D
A discontinuity
E
A certain vertical tangent

Choose one

1 point

For $f(x)=x^4-3x^2+5$ , what is the value of $f'(2)$ ?

A
$8$
B
$20$
C
$26$
D
$32$

Choose one

1 point

At hour 10, a temperature function satisfies $T'(10)=-1.5$ degrees Celsius per hour. What is the best interpretation?

A
The temperature is increasing by approximately $1.5$ degrees Celsius per hour
B
The temperature is decreasing by approximately $1.5$ degrees Celsius per hour
C
The temperature is exactly $-1.5$ degrees Celsius at hour 10
D
The temperature has remained constant for $1.5$ hours

Open ended

1 point

Explain the difference between average and instantaneous rates of change. Include the relevant formulas, the geometric meaning of each rate, and one correct example involving motion.

Choose one

1 point

If $f(x)=5$ , what is $f'(x)$ ?

A
$5$
B
$0$
C
$x$
D
$5x$

Choose one

1 point

An object's position is $s(t)=t^2+3t$ meters. What is its average velocity from $t=1$ second to $t=3$ seconds?

A
$5\text{ m/s}$
B
$6\text{ m/s}$
C
$7\text{ m/s}$
D
$9\text{ m/s}$

Choose one

1 point

A function gives temperature in degrees Celsius as a function of distance in kilometers. What are the units of its derivative?

A
$\text{km}/^{\circ}\text{C}$
B
$^{\circ}\text{C}/\text{km}$
C
$^{\circ}\text{C}\cdot\text{km}$
D
$\text{km}\cdot{}^{\circ}\text{C}$

Choose one

1 point

At a particular time, an object has velocity $v(t)=-4\text{ m/s}$ and acceleration $a(t)=-2\text{ m/s}^2$ . Which interpretation is correct?

A
The object is moving in the positive direction and slowing down.
B
The object is moving in the negative direction and slowing down.
C
The object is moving in the negative direction and speeding up.
D
The object is stationary.

Written response

1 point

A machine has produced $f(100)=250$ units after 100 minutes, and its production rate is $f'(100)=4$ units per minute. Using a linear approximation, estimate the production after 103 minutes. Enter the value in units.

Written response

1 point

A cost model satisfies $C'(200)=15$ dollars per item. What is the estimated increase in cost, in dollars, for producing one additional item near a production level of 200 items?