Rate of Change Examples in Math

Start with the recap, study the fully worked examples, then use the practice problems to check your understanding of Rate of Change.

This page combines explanation, solved examples, and follow-up practice so you can move from recognition to confident problem-solving in Math.

Concept Recap

A measure of how quickly one quantity changes with respect to another; the ratio of the change in output to the change in input.

How much does the output change for each unit increase in input? That ratio is the rate of change.

Read the full concept explanation →

How to Use These Examples

Read the first worked example with the solution open so the structure is clear.
Try the practice problems before revealing each solution.
Use the related concepts and background knowledge badges if you feel stuck.

What to Focus On

Core idea: Rate of change is the ratio $\frac{\Delta y}{\Delta x}$ — average over an interval, or instantaneous at a point as $\frac{dy}{dx}$ .

Common stuck point: The procedure for rate of change is the easy part; the trap is using two points when the question asks for the rate at a single instant. Asking "Am I measuring output change divided by input change, and is it over an interval (average) or at one point (instantaneous)?" first is what keeps a correct-looking calculation from being attached to the wrong concept.

Sense of Study hint: Ask: Am I measuring output change divided by input change, and is it over an interval (average) or at one point (instantaneous)?

Worked Examples

Example 1

easy

The position of a particle at time

t

seconds is

s(t) = 3t^2 - 2t + 1

metres. Find the average rate of change of position from

t = 1

t = 4

, and the instantaneous rate of change at

t = 2

Answer

Average rate of change:

13

m/s; instantaneous rate at

t=2

10

m/s

First step

Average rate of change:

\frac{s(4) - s(1)}{4 - 1}

Full solution

2
$s(4) = 3(16) - 2(4) + 1 = 48 - 8 + 1 = 41$ . $s(1) = 3 - 2 + 1 = 2$ .
3
Average rate: $\frac{41 - 2}{3} = \frac{39}{3} = 13$ m/s.
4
Instantaneous rate: $s'(t) = 6t - 2$ . At $t = 2$ : $s'(2) = 12 - 2 = 10$ m/s.

The average rate of change is the slope of the secant line between two points. The instantaneous rate of change is the derivative evaluated at the specific time. They are equal only when the function is linear.

Example 2

medium

Water drains from a tank so that the volume remaining after

t

minutes is

V(t) = 500 - 20t - t^2

litres (

0 \leq t \leq 10

). Find the rate at which water is draining at

t = 3

minutes.

Example 3

medium

Find the average rate of change of

f(x) = x^3 - 2x

[1, 3]

Example 4

medium

A particle's position is

s(t) = 4t^2 - 3t

metres. Find the average velocity from

t = 1

t = 4

and the instantaneous velocity at

t = 4

Example 5

medium

The position of an object is

s(t) = t^3 - 6t^2 + 9t

. When is the object at rest?

Example 6

hard

Find the average rate of change of

f(x) = x^2

[a, a + h]

. What does this approach as

h \to 0

Practice Problems

Try these problems on your own first, then open the solution to compare your method.

Example 1

easy

Find the average rate of change of

f(x) = x^2 + 3x

from

x = 0

x = 5

Example 2

medium

The temperature in a room is

T(t) = 20 + 5\cos\left(\frac{\pi t}{12}\right)

degrees Celsius at time

t

hours. Find the instantaneous rate of change of temperature at

t = 6

Example 3

easy

Find the average rate of change of

f(x) = x^2

[1, 3]

Example 4

easy

Find the instantaneous rate of change of

f(x) = x^2

x = 3

Example 5

easy

A car's position is

s(t) = 4t

(meters). What is its velocity?

Example 6

easy

f(x) = 3x + 2

, what is its rate of change?

Example 7

easy

Find the average rate of change of

f(x) = x^3

[0, 2]

Example 8

easy

Population grows by

P(t) = 100 + 5t

. What is the growth rate?

Example 9

easy

A balloon's volume is

V(t) = 2t^2

. Find the rate of change of volume at

t = 3

Example 10

easy

Interpret: a tank's water level changes at rate

-2

cm/min. What is happening?

Example 11

medium

For

f(x) = x^2 - 2x

, compare the average rate on

[0,2]

with the instantaneous rate at

x=1

Example 12

medium

A ball's height is

h(t) = 20t - 5t^2

meters. Find its velocity at

t = 1

Example 13

medium

Find when the ball

h(t) = 20t - 5t^2

reaches its highest point.

Example 14

medium

A quantity is

Q(t) = e^{0.5t}

. Find its rate of change at

t = 0

Example 15

medium

The area of a circle is

A = \pi r^2

. Find

\frac{dA}{dr}

r = 3

Example 16

medium

If revenue is

R(x) = 50x - x^2

, find the marginal revenue at

x = 10

Example 17

medium

Water flows so that volume is

V(t) = t^3 - 6t^2 + 9t

. When is the flow rate zero?

Example 18

challenge

A spherical balloon's volume is

V = \frac{4}{3}\pi r^3

. If

r

increases at

2

cm/s, find

\frac{dV}{dt}

when

r = 5

Example 19

challenge

A 13 m ladder slides down a wall; its base moves out at 2 m/s. How fast does the top descend when the base is 5 m out?

Example 20

challenge

For

f(x) = \ln x

, show the instantaneous rate at

x = a

decreases as

a

grows, and find it at

a = 4

Example 21

medium

For

f(x) = \sqrt{x}

, find the average rate of change on

[1, 9]

Example 22

medium

The temperature is

T(t) = 20 + 4t - t^2

(degrees). Find the rate of change at

t = 1

Example 23

easy

Find the average rate of change of

f(x) = 2x + 7

[1, 5]

Example 24

easy

Find the average rate of change of

f(x) = x^2 + 1

from

x = 2

x = 4

Example 25

easy

f(x) = x^2

, find

f'(x)

using the power rule.

Example 26

easy

Find

f'(x)

for

f(x) = 5x^3

Example 27

easy

f(x) = x^2 + 3x

, find

f'(x)

Example 28

medium

Find the average rate of change of

f(x) = \frac{1}{x}

[1, 4]

Example 29

medium

f(x) = x^3 - 2x^2 + 5

, find

f'(2)

Example 30

medium

The cost (in dollars) of producing

x

items is

C(x) = 0.5x^2 + 20x + 100

. Find the marginal cost at

x = 30

Example 31

medium

Find the instantaneous rate of change of

f(x) = \sqrt{x}

x = 9

Example 32

medium

For

f(x) = \sin x

, find the average rate of change on

[0, \pi/2]

to 3 decimal places.

Example 33

medium

A population grows according to

P(t) = 200 e^{0.05 t}

where

t

is years. Find

P'(0)

Example 34

medium

f(x) = \ln x

, find

f'(1)

Example 35

hard

A spherical balloon has radius increasing at

0.3

cm/s. Find the rate of change of its surface area

A = 4\pi r^2

when

r = 5

cm.

Example 36

hard

Find the slope of the tangent line to

y = x^3

x = 2

Example 37

hard

A cone-shaped tank fills with water; volume

V = \frac{1}{3}\pi r^2 h

where

r = h/2

. If water flows in at

2

^3

/min, find the rate at which the water level rises when

h = 4

Example 38

hard

Find all

x

where the instantaneous rate of change of

f(x) = x^3 - 3x

equals zero.

Example 39

hard

A car's speed (m/s) at time

t

seconds is

v(t) = 20 - 0.5t

. Find the deceleration.

Example 40

hard

h(t) = -16t^2 + 80t

models a ball's height in feet, find the times when the ball's instantaneous velocity equals its average velocity on

[0, 5]

Example 41

hard

A boat sails north at

10

km/h and another sails east from the same point at

24

km/h. How fast is the distance between them changing after

1

hour?

Example 42

challenge

Use the limit definition of the derivative to find

f'(x)

for

f(x) = \frac{1}{x}

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Related Concepts

Slope Derivative Related Rates

Background Knowledge

These ideas may be useful before you work through the harder examples.

slope