Calculus Practice Problems
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- Works on phones
- Shows the working
By Yasir Hafeez · Updated
These calculus practice problems come in two parts. First, an interactive quiz that checks each answer instantly and explains the solution. Then eight fully worked problems across derivatives, integrals and limits that you can solve on paper and check against the answers.
The problems are pitched at AP Calculus AB/BC and first-year university level. Everything is free and needs no sign-up.
How to use the practice quiz
- Choose a topic with the chips at the top: All topics, Derivatives, Integrals, Limits or Series.
- Read the question, work it out on paper, then tap one of the four answer choices.
- You’ll see right away whether you were correct, with a short worked explanation. The correct choice is highlighted either way.
- Tap Next question to continue. The score and progress bar track the round; at the end you get a percentage and can start another round. The questions are shuffled each time.
How to get the most out of practice
Multiple choice rewards recognizing the right answer. Exams usually ask you to produce it. For each quiz question, cover the choices. Solve it fully before you look. When you miss one, find the rule behind it. Often this is the chain rule or a sign error. Redo a similar calculus practice problem from the list below without notes.
A good session mixes topics. Doing ten chain-rule problems in a row feels productive, but choosing which method to use is half of every exam question.
Worked practice problems
Try each one before reading the solution. You can confirm any derivative, integral or limit with the calculators on this site, such as the derivative calculator and the integral calculator.
Derivatives
1. Find \( f'(x) \) for \( f(x) = (x^2 + 1)^5 \), then evaluate \( f'(1) \).
Solution. Chain rule with outer function \( u^5 \) and inner \( u = x^2 + 1 \):
$$f'(x) = 5(x^2+1)^4 \cdot 2x = 10x(x^2+1)^4$$
So \( f'(1) = 10 \cdot 2^4 = 160 \).
2. Differentiate \( g(x) = e^x\cos x \).
Solution. Product rule: \( g'(x) = e^x\cos x - e^x\sin x = e^x(\cos x - \sin x) \). At \( x = 0 \), the slope is \( 1 \).
3. Find the slope of \( h(x) = \frac{x}{x+1} \) at \( x = 1 \).
Solution. By the quotient rule, \( h'(x) = \frac{(x+1) - x}{(x+1)^2} = \frac{1}{(x+1)^2} \), so \( h'(1) = \frac14 \).
Integrals
4. Evaluate \( \int_0^2 (3x^2 - 2x + 1)\,dx \).
Solution. Antiderivative \( x^3 - x^2 + x \), so the answer is \( (8 - 4 + 2) - 0 = 6 \).
5. Evaluate \( \int_0^{\pi/2} \sin^2 x\cos x\,dx \).
Solution. U-substitution with \( u = \sin x \), \( du = \cos x\,dx \); the limits become 0 and 1:
$$\int_0^1 u^2\,du = \frac13$$
6. Evaluate \( \int_1^e x\ln x\,dx \).
Solution. Integration by parts with \( u = \ln x \), \( dv = x\,dx \), so \( du = \frac{dx}{x} \) and \( v = \frac{x^2}{2} \):
$$\int x\ln x\,dx = \frac{x^2\ln x}{2} - \frac{x^2}{4} + C$$
At \( e \) this is \( \frac{e^2}{2} - \frac{e^2}{4} \); at 1 it’s \( -\frac14 \). The answer is \( \frac{e^2 + 1}{4} \approx 2.097 \).
Limits
7. Find \( \lim_{x \to 2} \frac{x^2 - 4}{x - 2} \).
Solution. Direct substitution gives \( \frac00 \). Factor: \( \frac{(x-2)(x+2)}{x-2} = x + 2 \) for \( x \neq 2 \), so the limit is \( 4 \).
8. Find \( \lim_{x \to 0} \frac{e^{3x} - 1}{x} \).
Solution. Also \( \frac00 \). By L’Hôpital’s rule, differentiate top and bottom: \( \frac{3e^{3x}}{1} \to 3 \).
Bonus: \( \lim_{x \to \infty} \frac{4x^3 - x}{2x^3 + 5} = 2 \), the ratio of the leading coefficients, since the degrees match.
Mistakes these problems are designed to catch
- Dropping the inner derivative in problem 1 gives \( 5(x^2+1)^4 \), missing the factor \( 2x \).
- Forgetting to change the limits in problem 5. If you keep \( 0 \) and \( \frac{\pi}{2} \) after substituting, you get the wrong number.
- Choosing \( u = x \) in problem 6 leaves you needing \( \int \ln x\,dx \) inside the formula. Pick \( u = \ln x \) instead.
- Plugging in too early in problem 7. A \( \frac00 \) result means “simplify first,” not “the limit doesn’t exist.”
More worked problems
Limit. \( \lim_{x \to 3} \frac{x^2 - 9}{x - 3} \). Factor: \( \frac{(x - 3)(x + 3)}{x - 3} = x + 3 \to 6 \).
Derivative. \( \frac{d}{dx}(x^3\ln x) \). Product rule: \( 3x^2\ln x + x^3 \cdot \frac{1}{x} = 3x^2\ln x + x^2 \).
Integral. \( \int xe^{x^2}\,dx \). Let \( u = x^2 \), \( du = 2x\,dx \): \( \frac{1}{2}\int e^u\,du = \frac{1}{2}e^{x^2} + C \).
Related rates. A balloon’s radius grows at 2 cm/s. How fast is its volume growing when \( r = 5 \) cm? For calculus practice problems, \( V = \frac{4}{3}\pi r^3 \). So, \( \frac{dV}{dt} = 4\pi r^2\frac{dr}{dt} = 4\pi(25)(2) = 200\pi \approx 628.3 \) cm³/s. See related rates.
Optimisation. A farmer has 100 m of fence for a rectangular pen. Area \( A = x(50 - x) \) is largest where \( A' = 50 - 2x = 0 \), at a 25 m × 25 m square with area 625 m².
A weekly study plan
- Day 1: limits and continuity, ten problems.
- Next, Day 2: derivative rules, with the derivative calculator to check each answer.
- Day 3: applications of derivatives (tangents, extrema, related rates).
- Also, Day 4: antiderivatives and substitution.
- Day 5: definite integrals and areas.
- Then, Day 6: a mixed timed set; mark every mistake.
- Day 7: redo only the problems you got wrong.
How to check your own answers
Differentiate an antiderivative to see if you get back the integrand; plug a value into a limit from both sides; and graph functions in the graphing calculator to see whether a maximum or root is where you found it.
Related guides
Further reading
- Calculus I (Paul’s Online Math Notes) — full notes on limits, derivatives and integrals, with practice problems for each section.
- Calculus II (Paul’s Online Math Notes) — integration techniques, applications and series for when you move beyond Calculus I.
FAQ
Where can I find calculus practice problems with answers?
On this page: the quiz explains every answer as soon as you choose, and each of the eight worked problems has a full solution. For more by topic, each linked guide includes its own practice problems.
Are these problems at AP Calculus level?
Yes. They cover the core AP Calculus AB and BC skills: differentiation rules, substitution, integration by parts, limits and series. They also suit a first college calculus course.
How many questions does the quiz have?
A full round has 16 questions across derivatives, integrals, limits and series. Picking one topic gives a shorter, focused round, and the order is shuffled each time.
Can I choose only one topic?
Yes. Tap Derivatives, Integrals, Limits or Series above the question to practice just that area. Tap All topics to mix them again.