Free CLEP Calculus Practice Questions

Sample Practice Questions

Try these 10 sample questions from the CLEP Calculus practice bank. Each includes the answer and explanation. Continue with the interactive practice above to review the full question set.

1. 1As x approaches 3, what is the limit of (x²−9)/(x−3)?

   1. A.0
   2. B.3
   3. C.6Correct answer: C
   4. D.The limit does not exist

   Explanation: OpenStax recommends that we "factor each function and cancel out any common factors." For x ≠ 3 the expression is x + 3, so its limit is 6.

   Reference: Calculus Volume 1: The Limit Laws — §2.3, indeterminate 0/0 problem-solving strategy following Figure 2.24

2. 2What must be true of a function that is differentiable at x = a?

   1. A.It is continuous at a.Correct answer: A
   2. B.It has a local maximum at a.
   3. C.Its derivative is zero at a.
   4. D.It has a vertical tangent at a.

   Explanation: The text states: "If a function is differentiable at a point, then it is continuous at that point." The derivative may be positive, negative, or zero.

   Reference: Calculus Volume 1: Key Concepts, Chapter 3 — §3.2, The Derivative as a Function

3. 3What condition makes F an antiderivative of f on an interval?

   1. A.The derivative of F equals f throughout the interval.Correct answer: A
   2. B.F equals the derivative of f throughout the interval.
   3. C.F equals f at one point of the interval.
   4. D.The derivative of F equals zero throughout the interval.

   Explanation: The definition states "F′(x)=f(x)" for the interval. The antiderivative reverses differentiation.

   Reference: Calculus Volume 1: Antiderivatives — §4.10, The Reverse of Differentiation

4. 4As x approaches zero, find the limit of (√(1+x)−1)/x.

   1. A.0
   2. B.1/2Correct answer: B
   3. C.1
   4. D.3

   Explanation: OpenStax advises "multiplying the numerator and denominator by the conjugate" when a square-root difference occurs. Rationalizing gives 1/(√(1+x)+1), whose limit is 1/2.

   Reference: Calculus Volume 1: The Limit Laws — §2.3, indeterminate 0/0 problem-solving strategy following Figure 2.24

5. 5Let f(x) = |x|. What is true about f at x = 0?

   1. A.It is differentiable and f′(0) = 0.
   2. B.It is differentiable and f′(0) = 1.
   3. C.It is continuous but not differentiable.Correct answer: C
   4. D.It is neither continuous nor differentiable.

   Explanation: A function is not differentiable if "the graph has a sharp corner or cusp." The left and right slopes of |x| at 0 are −1 and 1, although both sides approach f(0) = 0.

   Reference: Calculus Volume 1: Key Concepts, Chapter 3 — §3.2, The Derivative as a Function

6. 6If F and G are antiderivatives of the same function on an interval, how are they related?

   1. A.They differ by a constant.Correct answer: A
   2. B.Their sum is always zero.
   3. C.Their product is constant.
   4. D.They must be identical.

   Explanation: The general antiderivative is "\int{{f\left( x \right)\,dx}} = F\left( x \right) + c" with c arbitrary; the notes also say "they will all differ by a constant". Therefore F−G is constant on the interval.

   Reference: Calculus I: Indefinite Integrals — §5.1, Indefinite Integral Definitions

7. 7Suppose lim(x→2) f(x) = 5 and f(2) = 3. Which statement describes f at x = 2?

   1. A.It is continuous because the limit exists.
   2. B.It is continuous because f(2) is defined.
   3. C.It is discontinuous because its limit differs from f(2).Correct answer: C
   4. D.It is discontinuous because its limit is undefined.

   Explanation: OpenStax requires that "limx→af(x)=f(a)" for continuity. The limit is 5 while the function value is 3, so the equality fails.

   Reference: Calculus Volume 1: Continuity — §2.4, Continuity at a Point, definition

8. 8What is the product rule for differentiating f(x)g(x) when both functions are differentiable?

   1. A.The derivative of f times the derivative of g
   2. B.The derivative of f times g plus f times the derivative of gCorrect answer: B
   3. C.The derivative of f times g minus f times the derivative of g
   4. D.f times g plus the derivative of f times the derivative of g

   Explanation: The product rule combines "the derivative of the first function times the second function plus the derivative of the second function times the first function". Hence (fg)′=f′g+fg′.

   Reference: Calculus Volume 1: Key Concepts, Chapter 3 — §3.3, Differentiation Rules

9. 9Find the most general primitive function whose derivative equals the polynomial 6x².

   1. A.2x³ + CCorrect answer: A
   2. B.3x³ + C
   3. C.12x + C
   4. D.6x³ + C

   Explanation: The power rule uses "bringing the original exponent down in front and multiplying and then subtracting one from the original exponent". Differentiating 2x³ gives 6x², and adding C does not change its derivative.

   Reference: Calculus I: Differentiation Formulas — §3.3, Power Rule

10. 10Let f(x) = 2x + 1 for x < 1 and f(x) = x² + 4 for x ≥ 1. What is lim(x→1⁻) f(x)?

    1. A.1
    2. B.3Correct answer: B
    3. C.5
    4. D.The limit does not exist

    Explanation: The notes say "for the left-handed limit we have \(x \to {a^ - }\)" and "we will only be looking at \(x < a\)". Here values to the left of 1 use 2x+1, which approaches 3.

    Reference: Calculus I: One-Sided Limits — §2.3, Left-handed limit