20 CLEP Calculus practice questions, with answers and video walkthroughs

Twenty practice questions for CLEP Calculus, mixed the way the real exam is: about 60 percent limits and differential calculus, the rest integral calculus and its applications. Score 16 of 20 here and you are in passing range. Miss more than four, and the pattern of your misses points straight at the family to review next. Every question below is explained on video, including why the wrong answers trap most test-takers.

What students preparing for this exam tell us is that the surprise is not the difficulty of any single problem. It is the breadth. Miss an entire family, like related rates or the Fundamental Theorem, and you bleed points fast. Use these the same way you would the real thing: drill in the exam's format until the move each question wants feels automatic. For the full plan around them, see the CLEP Calculus pillar guide.

Watch the full video walkthrough above, then test yourself on the twenty questions below. Each one is explained on the video, including why the wrong answers trap most test-takers.

Limits and continuity (questions 1 to 5)

1. Evaluate lim_x → 3(x²-9)/(x-3). 6. Factor: ((x-3)(x+3))/(x-3)=x+3, so the limit is 3+3=6.

2. Evaluate lim_x → 0(sin 3x)/(x). 3. Write (sin 3x)/(x)=3 · (sin 3x)/(3x); since (sin u)/(u) → 1, the limit is 3.

3. Find a so that f(x)= x², x less than 2; ax+1, x ≥ 2 is continuous at x=2. (3)/(2). Match pieces at x=2: 2²=a(2)+1 ⇒ 4=2a+1 ⇒ a=(3)/(2).

4. Evaluate lim_x → 0(1-cos x)/(x²). (1)/(2). Multiply by (1+cos x)/(1+cos x): (1-cos²x)/(x²(1+cos x))=(sin²x)/(x²) · (1)/(1+cos x) → 1 · (1)/(2)=(1)/(2).

5. Evaluate lim_x → 0frac√(x+1)-1x. (1)/(2). Multiply by the conjugate: fracxx(√(x+1)+1)=frac1√(x+1)+1 → (1)/(2).

Differentiation (questions 6 to 10)

6. Differentiate f(x)=(2x+1)⁵. 10(2x+1)⁴. Chain rule: 5(2x+1)⁴ · 2=10(2x+1)⁴.

7. Find the slope of the tangent line to y=x² at x=3. 6. y'=2x, so at x=3 the slope is 2(3)=6.

8. Differentiate f(x)=x³ln x. 3x²ln x + x². Product rule: 3x²ln x + x³ · (1)/(x)=3x²ln x + x².

9. Differentiate f(x)=e²xsin x. e²x(2sin x+cos x). Product rule: 2e²xsin x + e²xcos x=e²x(2sin x+cos x).

10. A ladder leans against a wall with x²+y²=25. When x=3, y=4, and the base slides out at (dx)/(dt)=2 ft/s, find (dy)/(dt). -(3)/(2) ft/s. Differentiate: 2x(dx)/(dt)+2y(dy)/(dt)=0 ⇒ (dy)/(dt)=-(x)/(y)(dx)/(dt)=-(3)/(4)(2)=-(3)/(2).

Derivatives and integrals (questions 11 to 15)

11. Find all x where f(x)=x³-3x has a horizontal tangent. x= ± 1. f'(x)=3x²-3=0 ⇒ x²=1 ⇒ x= ± 1.

12. Differentiate f(x)=tan x and give f' ((π)/(4) ). sec² x; 2. (d)/(dx)tan x=sec² x; at (π)/(4), sec(π)/(4)=sqrt 2, so sec²(π)/(4)=2.

13. Evaluate int_0² 3x² dx. 8. ∫ 3x² dx=x³; evaluate x³big|_0²=8-0=8.

14. Evaluate ∫ e³x dx. (1)/(3)e³x+C. ∫ e³x dx=(1)/(3)e³x+C (divide by the inside coefficient 3).

15. Evaluate int_1⁴ (2x+3) dx. 24. ∫(2x+3) dx=x²+3x; (16+12)-(1+3)=28-4=24.

Integration and applications (questions 16 to 20)

16. Compute (d)/(dx)int_0x sin(t²) dt. sin(x²). By the Fundamental Theorem of Calculus, (d)/(dx)int_0x f(t) dt=f(x)=sin(x²).

17. Evaluate ∫ x ex dx. ex(x-1)+C. Integration by parts with u=x, dv=ex dx: xex-∫ ex dx=xex-ex=ex(x-1)+C.

18. A particle moves with s(t)=t³-6t²+9t. At what times is it at rest (v=0)? t=1 and t=3. v(t)=3t²-12t+9=3(t-1)(t-3)=0 ⇒ t=1,3.

19. A rectangle has perimeter 20. What dimensions maximize its area? 5 × 5. With w=10-ell, A=ell(10-ell); A'=10-2ell=0 ⇒ ell=5, so 5 × 5.

20. A particle has velocity v(t)=2t-4. Find the total distance traveled on [0,3]. 5. Total distance =int_0³|2t-4| dt. v less than 0 on [0,2), v greater than 0 on (2,3]: int_0²(4-2t)+int_2³(2t-4)=4+1=5.

What to do with your score

The point of twenty questions is not the twenty. It is the pattern. Mark which family each miss came from, then spend your next session on that family alone. Two misses in integration means an evening on antiderivatives and the Fundamental Theorem. Two in applications means an evening on related rates and motion. None of these requires new theory, only a cleaner habit, so they are the fastest points to recover.

A single twenty-question set is a snapshot, not a study plan. To pass with margin you need volume: enough questions, in the exam's format, with explanations that turn a wrong answer into a correction. That is what Flying Prep's CLEP Calculus practice is built for: every question explained, with a free trial before you decide.