10 free sample questions with answers and explanations. See how you'd score on the real CLEP exam.
A tank is being filled with water at a rate of 10 cubic meters per hour. How much water will be in the tank after 3 hours?
Explanation
30 cubic meters is correct because the rate of change is 10 cubic meters per hour, so by multiplying this rate by the time, 3 hours, we get 10 * 3 = 30 cubic meters.
The area of a circle is increasing at a rate of 4π square centimeters per minute. If the radius is increasing at a rate of 1 centimeter per minute, what is the radius of the circle?
Explanation
2 centimeters is correct because we are given the rate of change of the area, which is 4π square centimeters per minute, and the rate of change of the radius, which is 1 centimeter per minute. So, by using the formula for the area of a circle, A = πr², we get dA/dt = 2πr * dr/dt. Then, 4π = 2πr * 1, so r = 2 centimeters.
A car is traveling at a velocity of 60 miles per hour. How far will it travel in 2 hours?
Explanation
120 miles is correct because the rate of change is 60 miles per hour, so by multiplying this rate by the time, 2 hours, we get 60 * 2 = 120 miles.
A water tank is being filled at a rate of 2 cubic meters per minute. How much water will be in the tank after 5 minutes?
Explanation
10 cubic meters is correct because the rate of change is 2 cubic meters per minute, so by multiplying this rate by the time, 5 minutes, we get 2 * 5 = 10 cubic meters.
Calculate the area between the curves y = x² and y = x from x = 0 to x = 1.
Explanation
1/6 is correct because the area between curves can be calculated by integrating the difference between the functions from 0 to 1, ∫[0,1] (x - x²) dx = [0.5x² - (1/3)x³] from 0 to 1 = 0.5 - 1/3 = 1/6.
Calculate the area between the curves y = 2x and y = x² from x = 0 to x = 2.
Explanation
4/3 is correct because to find the area between curves, we calculate the integral of the difference between the functions from 0 to 2, ∫[0,2] (2x - x²) dx, which equals [x² - (1/3)x³] from 0 to 2 = 4 - 8/3 = 4/3.
Find the average value of f(x) = x + 1 on [0, 2].
Explanation
2 is correct because the average value is (1/(b-a)) * integral of f(x) from a to b, which equals (1/(2-0)) * integral of (x + 1) from 0 to 2, and this equals (1/2) * [x^2/2 + x] from 0 to 2, which equals (1/2) * ((2^2/2) + 2) = (1/2) * (2 + 2) = 2.
Find the average value of f(x) = 3 on [1, 3].
Explanation
3 is correct because the average value is (1/(b-a)) * integral of f(x) from a to b, which equals (1/(3-1)) * integral of 3 from 1 to 3, and this equals (1/2) * [3x] from 1 to 3, which equals (1/2) * 3 * (3 - 1) = (1/2) * 3 * 2 = 3.
Find the average value of f(x) = x on [2, 4].
Explanation
3 is correct because the average value is (1/(b-a)) * integral of f(x) from a to b, which equals (1/(4-2)) * integral of x from 2 to 4, and this equals (1/2) * [x^2/2] from 2 to 4, which equals (1/2) * (1/2) * (4^2 - 2^2) = (1/4) * (16 - 4) = (1/4) * 12 = 3.
If ∫[a,x] f(t) dt = 5, what is the value of ∫[x,a] f(t) dt?
Explanation
-5 is correct because by the Fundamental Theorem of Calculus, the integral from a to x is the negative of the integral from x to a, since the limits are reversed.