Practice Rate of Change in Math

Use these practice problems to test your method after reviewing the concept explanation and worked examples.

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Quick 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.

Showing a random 20 of 50 problems.

Example 1

medium
For f(x)=xf(x) = \sqrt{x}, find the average rate of change on [1,9][1, 9].

Example 2

medium
Find the instantaneous rate of change of f(x)=xf(x) = \sqrt{x} at x=9x = 9.

Example 3

medium
If revenue is R(x)=50xโˆ’x2R(x) = 50x - x^2, find the marginal revenue at x=10x = 10.

Example 4

easy
Find the average rate of change of f(x)=x3f(x) = x^3 on [0,2][0, 2].

Example 5

easy
If f(x)=x2f(x) = x^2, find fโ€ฒ(x)f'(x) using the power rule.

Example 6

easy
Find the average rate of change of f(x)=x2+3xf(x) = x^2 + 3x from x=0x = 0 to x=5x = 5.

Example 7

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 8

easy
Interpret: the velocity of a car is โˆ’3-3 m/s. What does the sign mean?

Example 9

medium
For f(x)=x2โˆ’2xf(x) = x^2 - 2x, compare the average rate on [0,2][0,2] with the instantaneous rate at x=1x=1.

Example 10

easy
Find the instantaneous rate of change of f(x)=x2f(x) = x^2 at x=3x = 3.

Example 11

easy
If f(x)=x2+3xf(x) = x^2 + 3x, find fโ€ฒ(x)f'(x).

Example 12

challenge
For f(x)=lnโกxf(x) = \ln x, show the instantaneous rate at x=ax = a decreases as aa grows, and find it at a=4a = 4.

Example 13

medium
The cost (in dollars) of producing xx items is C(x)=0.5x2+20x+100C(x) = 0.5x^2 + 20x + 100. Find the marginal cost at x=30x = 30.

Example 14

medium
A quantity is Q(t)=e0.5tQ(t) = e^{0.5t}. Find its rate of change at t=0t = 0.

Example 15

medium
The temperature in a room is T(t)=20+5cosโก(ฯ€t12)T(t) = 20 + 5\cos\left(\frac{\pi t}{12}\right) degrees Celsius at time tt hours. Find the instantaneous rate of change of temperature at t=6t = 6.

Example 16

hard
A boat sails north at 1010 km/h and another sails east from the same point at 2424 km/h. How fast is the distance between them changing after 11 hour?

Example 17

medium
Water drains from a tank so that the volume remaining after tt minutes is V(t)=500โˆ’20tโˆ’t2V(t) = 500 - 20t - t^2 litres (0โ‰คtโ‰ค100 \leq t \leq 10). Find the rate at which water is draining at t=3t = 3 minutes.

Example 18

easy
What is the derivative of f(x)=7f(x) = 7 (a constant function)?

Example 19

hard
A car's speed (m/s) at time tt seconds is v(t)=20โˆ’0.5tv(t) = 20 - 0.5t. Find the deceleration.

Example 20

medium
A population grows according to P(t)=200e0.05tP(t) = 200 e^{0.05 t} where tt is years. Find Pโ€ฒ(0)P'(0).
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