Training Calculus Placement Test Practice — Calculus
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Placement Test Practice — Calculus

29 min Calculus

Placement Test Practice — Calculus

This practice test brings together everything you have learned across the four calculus lessons — limits, derivatives, derivative applications, and integrals — in a single placement-style assessment. The twenty questions below mirror the format and difficulty you would encounter on a college mathematics placement exam, covering direct computation, conceptual understanding, and applied problem-solving.

Work through each problem without a calculator first, then check your answers against the key. If you find yourself stuck on a particular topic, revisit the corresponding lesson to reinforce that skill before attempting the problem again. Consistent practice with mixed-topic sets like this one is one of the most effective ways to build fluency and confidence with calculus fundamentals.

Practice Test — 20 Questions

1. $$\lim_{x \to 2}(x^2+1)$$
2. $$\lim_{x \to 4} \frac{x^2-16}{x-4}$$
3. Differentiate $x^5$.
4. Differentiate $3x^2-7x+1$.
5. Differentiate $\sin x$.
6. Differentiate $e^x$.
7. Find the critical point of $x^2-2x$.
8. Is that point a max or min?
9. If $s(t)=t^2+4t$, find $v(t)$.
10. If $v(t)=9t-2$, find $a(t)$.
11. $$\int x^2\,dx$$
12. $$\int (4x+3)\,dx$$
13. $$\int_0^1 x\,dx$$
14. $$\int_1^2 3\,dx$$
15. State the meaning of continuity at a point.
16. Which rectangle of fixed perimeter has maximum area?
17. Differentiate $x^2\cos x$ using the product rule.
18. $$\lim_{x \to -1}(2x^3)$$
19. Find an antiderivative of $\cos x$.
20. What theorem connects derivatives and definite integrals?
Show Answer Key

1. $5$

2. $8$

3. $5x^4$

4. $6x-7$

5. $\cos x$

6. $e^x$

7. $x=1$

8. Minimum

9. $2t+4$

10. $9$

11. $\frac{x^3}{3}+C$

12. $2x^2+3x+C$

13. $\frac{1}{2}$

14. $3$

15. Function value exists, the limit exists, and they are equal.

16. A square

17. $2x\cos x-x^2\sin x$

18. $-2$

19. $\sin x+C$

20. The Fundamental Theorem of Calculus

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