These CLEP Calculus practice questions are a first-pass diagnostic. They are not official CLEP questions, but they are designed around the core skills students should review: limits, derivatives, applications of derivatives, antiderivatives, definite integrals, and area or accumulated change.
The College Board says the CLEP Calculus exam covers material usually taught in a one-semester college calculus course. The official outline is approximately 60% limits and differential calculus and 40% integral calculus, with 44 questions in about 90 minutes across two sections. Section 1 has no calculator; Section 2 permits an integrated non-CAS graphing calculator for some questions. Always verify current details on the official College Board CLEP Calculus page.
How to Use This Practice Set
Try the 15 problems without notes. Mark each missed answer with a topic label: limit, derivative rule, derivative application, integral rule, definite integral, or setup. The label matters more than the raw score because it tells you what to study next.
15 CLEP Calculus Practice Questions
1. Limit: Evaluate lim as x approaches 4 of (x^2 – 16)/(x – 4).
2. Continuity: For what value of k is f(x) continuous at x = 2 if f(x) = x^2 + k for x < 2 and f(x) = 3x + 1 for x >= 2?
3. Basic derivative: Find d/dx of 5x^4 – 3x^2 + 7.
4. Product rule: Find the derivative of f(x) = x^2 sin x.
5. Chain rule: Find the derivative of g(x) = (3x – 1)^5.
6. Implicit differentiation: If x^2 + y^2 = 25, find dy/dx.
7. Tangent line: Find the equation of the tangent line to y = x^2 + 1 at x = 3.
8. Optimization: A rectangle has perimeter 40. If its width is x, write the area as a function of x and find the width that maximizes area.
9. Motion: A particle has position s(t) = t^3 – 6t^2 + 9t. Find its velocity at t = 4.
10. Antiderivative: Find an antiderivative of 6x^2 – 4x + 3.
11. U-substitution: Evaluate the integral of 2x(x^2 + 1)^4 dx.
12. Definite integral: Evaluate the integral from 0 to 3 of (2x + 1) dx.
13. Average value: Find the average value of f(x) = x^2 on [0, 3].
14. Area: Find the area between y = x and y = x^2 on [0, 1].
15. Accumulated change: Water flows into a tank at rate r(t) = 4t + 2 gallons per minute for 0 <= t <= 5. How much water enters the tank during that time?
Answers and Explanations
1. Factor: (x^2 – 16) = (x – 4)(x + 4). The limit is x + 4 at x = 4, so the answer is 8.
2. Left value at x = 2 is 2^2 + k = 4 + k. Right value is 3(2) + 1 = 7. Set 4 + k = 7, so k = 3.
3. The derivative is 20x^3 – 6x.
4. Use the product rule: f'(x) = 2x sin x + x^2 cos x.
5. Use the chain rule: g'(x) = 5(3x – 1)^4 x 3 = 15(3x – 1)^4.
6. Differentiate both sides: 2x + 2y(dy/dx) = 0. Then dy/dx = -x/y.
7. y’ = 2x, so slope at x = 3 is 6. Point is (3, 10). Tangent line: y – 10 = 6(x – 3), or y = 6x – 8.
8. If width is x, length is 20 – x because 2L + 2x = 40. Area A(x) = x(20 – x) = 20x – x^2. This parabola is maximized at x = 10.
9. Velocity is s'(t) = 3t^2 – 12t + 9. At t = 4, v(4) = 48 – 48 + 9 = 9.
10. An antiderivative is 2x^3 – 2x^2 + 3x + C.
11. Let u = x^2 + 1, so du = 2x dx. Integral becomes u^4 du = u^5/5 + C = (x^2 + 1)^5/5 + C.
12. Antiderivative is x^2 + x. From 0 to 3: (9 + 3) – 0 = 12.
13. Average value = (1/(3 – 0)) times integral from 0 to 3 of x^2 dx = (1/3)(x^3/3 from 0 to 3) = (1/3)(9) = 3.
14. On [0, 1], x is above x^2. Area = integral from 0 to 1 of (x – x^2) dx = [x^2/2 – x^3/3] from 0 to 1 = 1/2 – 1/3 = 1/6.
15. Accumulated water = integral from 0 to 5 of (4t + 2) dt = [2t^2 + 2t] from 0 to 5 = 50 + 10 = 60 gallons.
What Your Score Means
If you missed 0-3 problems, move into timed CLEP-style mixed sets. If you missed 4-7, review the specific topic clusters that caused mistakes. If you missed 8 or more, rebuild foundations in this order: limits, derivative rules, derivative applications, antiderivatives, definite integrals, then mixed applications.
Topics to Study Next
If limits caused the most trouble, practice factoring, rationalizing, one-sided behavior, and continuity before moving back into derivatives. If derivative rules were the problem, drill power, product, quotient, chain, and implicit differentiation until the rule choice is automatic. If integration was weak, start with basic antiderivatives, then add substitution, definite integrals, average value, and area between curves.
For CLEP specifically, do not separate concepts from applications for too long. A student may know the derivative of x^3 but still miss optimization, motion, or tangent-line questions. After each rule review, solve at least three applied problems that use the same rule, then revisit the error log.
ViewMath CLEP Calculus resources can help with extra worked examples and topic-by-topic practice. Browse the CLEP Calculus collection at viewmath.com/books/clep-calculus/.
