Rate of change is defined as the rate at which one quantity changes concerning another. Rate of Change helps us to understand how a function generally behaves. Is it gaining height overall? Going down? Some functions, like sin(x) and cos(x), that are oscillating functions, could even have zero net change.

In this article, we will learn What is Rate of Change, Rate of Change Formula, Practice Questions on the same and others in detail.

What is Rate of Change?

The rate of change quantifies the amount of change in one variable (dependent variable) relative to a change in another variable (independent variable).

Rate of change

To calculate the rate of change for a function f(x), select two points a and b on the x-axis and evaluate f(a) and f(b) and take their difference, {f(b) – f(a)}, then take the difference (b – a) Finally, divide the values to determine the rate of change for the function.

Rate of Change Formula

The rate change formula is studied under three cases that include:

Rate of Change = (Change in Quantity 1) / (Change in Quantity 2)

Rate of Change in Algebra

Δy/Δx = (y₂ – y₁)/(x₂ – x₁)

Rate of Change of Functions

Rate of Change = {f(b) – f(a)}/(b – a)

Rate of Change Practice Problems with Solutions

Problem 1: Consider the function f(x) = x², and compute the rates of change from x = -2 to x = 0.

Solution:

Given, f(x) = x²

Now we solve using Formula for Rate of Change = {f(b) – f(a)}/{b – a}

Rate of Change = {f(0)-f(-2)}/{0-(-2)}

= {0²-(-2)²}/2

Rate of Change = (0 – 4)/2 = -2

Problem 2: Consider the function f(x) = x², and compute the rates of change from x = 1 to x = 3.

Solution:

Given f(x) = x²

Now we solve using Formula for Rate of Change = {f(b) – f(a)}/{b – a}

Rate of Change = {f(3) – f(1)}/{3 – 1}

Rate of Change = (3²-1²)/2

Rate of Change = (9-1)/2 = 4

Problem 3: Calculate the average rate of change of a function, f(x) = 3x + 12 as x changes from 5 to 8 .

Solution:

Given,

• f(x) = 3x + 12
• a = 5
• b = 8

f(5) = 3(5) + 12

f(5) = 15 + 12 = 27

f(8) = 3(8) + 12

f(8) = 24 + 12 = 36

Average rate of change is,

A(x) = Rate of Change = {f(b) – f(a)}/(b – a)

A(x) = {f(8) – f(5)}/(8 – 5)

A(x) = 36-27/3

A(x) = 9/3 = 3

Problem 4: Consider the function f(x) = x², and compute the rates of change from x = -5 to x = 5.

Solution:

Given f(x) = x²

Now we solve using Formula for Rate of Change = {f(b) – f(a)}/{b – a}

Rate of Change = f(5)-f(-5)/5-(-5)

Rate of Change = {5²-(-5)²}/{5+5}

Rate of Change = (25-25)/10 = 0

Problem 5: Consider the function f(x) = x³, and compute the rates of change from x = -3 to x = 3.

Solution:

Given f(x) = x³

Now we solve using Formula for Rate of Change = {f(b) – f(a)}/{b – a}

Rate of Change = {f(3)-f(-3)}/{3-(-3)}

Rate of Change = {3³-(-3)³}/{3+3}

Rate of Change = {81-(-81)}/6 = 162/6 = 27

Problem 6: Calculate the average rate of change of the function f(x) = x2 – 9x in the interval 2 ≤ x ≤ 7.

Solution:

Given,

• f(x) = x² – 8x
• a = 2
• b = 7

f(a) = f(2) = (2)² – 8(2) = 4 – 16 = -12

f(b) = f(7) = (7)² – 8(7) = 49 – 56 = -7

Rate of Change = {f(b)-f(a)}/{b-a}

= {-7 – (-12)}/{-7 + (-12)}

= {-7 + 12}/{-7 – 12}

= -5/19

Problem 7: Consider the function f(x) = x, and compute the rates of change from x = -12 to x = 12.

Solution:

Given f(x) = x

Now we solve using Formula for Rate of Change = {f(b) – f(a)}/{b – a}

Rate of Change = {f(12)-f(-12)}/{12-(-12)}

Rate of Change = {12-(-12)}/{12+12}

= (12+12)/24 = 24/24 = 1

Problem 8: Consider the function f(x) = x⁴, and compute the rates of change from x = -2 to x = 2.

Solution:

Given f(x) = x⁴

Now we solve using Formula for Rate of Change = {f(b) – f(a)}/{b – a}

Rate of Change = {f(2) – f(-2)}/{2 – (-2)}

Rate of Change = {2⁴-(-2)⁴}/{2+2}

= {16-16}/4 =0

Problem 7: Find the average rate of change of the volume of water in the tank from t = 2 minutes to t = 5 minutes. V(t) = 2t³+3t²+5t

Solution:

Average rate of change of V(t) from t = a to t = b is given by: {V(b) – V(a)}/{b – a}

Here,

• a = 2 and b = 5

Calculate: V(5) and V(2)

V(5) = 2(5)³ + 3(5)² + 5(5) = 2.125 + 3⋅25 + 25

V(5) = 2⋅125 + 3⋅25 + 25

V(5) = 250 + 75 + 25 = 350

V(2) = 2(2)³+3(2)²+5(2)

V(2) = 2⋅8+3⋅4+10

V(2) = 16+12+10 = 38

Now, find average rate of change:

{V(5) – V(2)}/{5 – 2}= {350 – 38}/{5 – 2}

= 312/3 = 104

Rate of Change practice problems

Problem 1: Consider the function f(x) = x³, and compute the rates of change from x = 1 to x = 3.

Problem 2: Consider the function f(x) = x³, and compute the rates of change from x = 2 to x = 3.

Problem 3: Consider the function f(x) = x-1, and compute the rates of change from x = 4 to x = 6.

Problem 4: Consider the function f(x) = x², and compute the rates of change from x = 3 to x = 4.

Problem 5: Consider the function f(x) = x⁴ and compute the rates of change from x = 7 to x = 4.

Problem 6: Consider the function f(x) = x²+1, and compute the rates of change from x = 2 to x = 1.

Problem 7: Consider the function f(x) = x, and compute the rates of change from x = 4 to x = 2.

Also Read,

• Average and Instantaneous Rate of Change
• Derivatives as Rate of Change

Frequently Asked Questions

What is the Rate of Change?

Rate of change is a measure of how a quantity changes over time or with respect to another variable. In mathematics, it often refers to the derivative in calculus, which provides the instantaneous rate of change of a function.

What is the Significance of Sign of Rate of Change?

• A positive rate of change indicates that the function is increasing over the interval.
• A negative rate of change indicates that the function is decreasing over the interval.
• A zero rate of change indicates that the function is constant over the interval or has a local maximum or minimum.

How do you Interpret Rate of Change in Real-World Contexts?

In real-world scenarios, the rate of change can represent various phenomena:

• Velocity is the rate of change of position.
• Acceleration is the rate of change of velocity.
• Growth rate in populations or finance is the rate of change of size or value.

How do you Find Rate of Change from a Graph?

• For average rate of change, use the slope of the secant line connecting two points on the curve.
• For instantaneous rate of change, use the slope of the tangent line at a specific point on the curve. This can be estimated using a derivative or visually by drawing the tangent.

How can Technology (like graphing calculators or software) help in Finding Rate of Change?

Technology, such as graphing calculators, software (like GeoGebra, Desmos, or MATLAB), and computer algebra systems (CAS), can:

• Compute derivatives and average rates of change.
• Visualize functions and their derivatives.
• Analyze the behavior of functions over intervals.