• 60 questions

Calculus Practice Test

Limits, derivatives and integrals in five topics, worked after every answer, to take on screen or print.

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Find the limit as x → 0 of sin(3x)/x.

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Unofficial practice questions, not affiliated with any school, college or exam body. Bank checked October 7, 2026.

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Limits and continuity 12

  • f(x) = x² + k for x < 2 and f(x) = 3x + 1 for x ≥ 2. Which k makes f continuous at x = 2? Answer: k = 3. Source: California Calculus standard 2.0 (continuity)
  • f(x) = x³ + x − 1 is continuous. Between which two whole numbers does the intermediate value theorem guarantee a zero of f? Answer: 0 and 1. Source: California Calculus standard 3.0 (the intermediate value theorem)
  • Find the limit as x → 0 of (1 − cos x)/x. Answer: 0. Source: California Calculus standard 1.3 (special limits)
  • Find the limit as x → 0 of sin(3x)/x. Answer: 3. Source: California Calculus standard 1.3 (special limits)
  • Find the limit as x → 1 of (x³ − 1)/(x − 1). Answer: 3. Source: California Calculus standard 1.1 (limits of quotients)
  • Find the limit as x → 2 of (x² − 4)/(x − 2). Answer: 4. Source: California Calculus standard 1.1 (limits of quotients)
  • Find the limit as x → 3 of 2x² − 5x + 1. Answer: 4. Source: California Calculus standard 1.1 (limits of sums and products)
  • Find the limit as x → 4 of (√x − 2)/(x − 4). Answer: 1/4. Source: California Calculus standard 1.1 (limits of quotients)
  • Find the limit as x → ∞ of (3x² + 1)/(5x² − 2x). Answer: 3/5. Source: California Calculus standard 1.0 (limits at infinity)
  • Find the limit as x → ∞ of (4x + 7)/(x² + 1). Answer: 0. Source: California Calculus standard 1.0 (limits at infinity)
  • Find the limit of 1/(x − 2) as x → 2 from the right. Answer: ∞. Source: California Calculus standard 1.0 (infinite limits)
  • Find the limit of |x|/x as x → 0 from the left. Answer: −1. Source: California Calculus standard 1.0 (one-sided limits)
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Derivatives 12

  • Find the derivative of f(x) = (3x + 1)⁵. Answer: 15(3x + 1)⁴. Source: California Calculus standard 5.0 (the chain rule)
  • Find the derivative of f(x) = 4x³ − 5x + 2. Answer: 12x² − 5. Source: California Calculus standard 4.4 (derivative formulas)
  • Find the derivative of f(x) = arctan x. Answer: 1/(1 + x²). Source: California Calculus standard 4.4 (derivatives of inverse trigonometric functions)
  • Find the derivative of f(x) = cos(x²). Answer: −2x sin(x²). Source: California Calculus standard 5.0 (the chain rule)
  • Find the derivative of f(x) = e^(2x). Answer: 2e^(2x). Source: California Calculus standard 4.4 (derivatives of exponential functions)
  • Find the derivative of f(x) = ln(x² + 1). Answer: 2x/(x² + 1). Source: California Calculus standard 4.4 (derivatives of logarithmic functions)
  • Find the derivative of f(x) = x/(x + 1). Answer: 1/(x + 1)². Source: California Calculus standard 4.4 (derivative formulas)
  • Find the derivative of f(x) = x² sin x. Answer: 2x sin x + x² cos x. Source: California Calculus standard 4.4 (derivative formulas)
  • Find the limit as h → 0 of [(2 + h)² − 4]/h. Answer: 4. Source: California Calculus standard 4.0 (the definition of the derivative)
  • Find the second derivative of f(x) = x⁴ − 2x². Answer: 12x² − 4. Source: California Calculus standard 7.0 (derivatives of higher orders)
  • For f(x) = √x, what is the slope of the tangent line at x = 9? Answer: 1/6. Source: California Calculus standard 4.1 (the derivative as the slope of the tangent line)
  • For the circle x² + y² = 25, find dy/dx at the point (3, 4). Answer: −3/4. Source: California Calculus standard 6.0 (implicit differentiation)

Using derivatives 12

  • A particle has position s(t) = t³ − 6t² + 9t meters. What is its velocity at t = 2 seconds? Answer: −3 m/s. Source: California Calculus standard 4.2 (the derivative as a rate of change)
  • A rectangle is made from 40 m of fencing. What is the largest area it can enclose? Answer: 100 m². Source: California Calculus standard 11.0 (optimization)
  • At which x does f(x) = x³ − 6x² have an inflection point? Answer: x = 2. Source: California Calculus standard 9.0 (inflection points)
  • f(x) = x³ − 3x has a local maximum at which x? Answer: x = −1. Source: California Calculus standard 9.0 (maxima and minima)
  • Find the limit as x → 0 of (e^x − 1)/x. Answer: 1. Source: California Calculus standard 8.0 (L'Hôpital's rule)
  • For f(x) = x² on [1, 3], which c satisfies the mean value theorem? Answer: c = 2. Source: California Calculus standard 8.0 (the mean value theorem)
  • Newton's method is applied to f(x) = x² − 2 starting from x₀ = 1. What is x₁? Answer: 1.5. Source: California Calculus standard 10.0 (Newton's method)
  • On which interval is f(x) = x³ − 12x decreasing? Answer: −2 < x < 2. Source: California Calculus standard 9.0 (intervals of increase and decrease)
  • The radius of a circle grows at 2 cm per second. How fast is the area growing when the radius is 5 cm? Answer: 20π cm² per second. Source: California Calculus standard 12.0 (related rates)
  • What is the largest value of f(x) = x³ − 3x on the interval [0, 2]? Answer: 2. Source: California Calculus standard 9.0 (maxima and minima)
  • What is the minimum value of f(x) = x² − 6x + 5? Answer: −4. Source: California Calculus standard 9.0 (maxima and minima)
  • Which line is tangent to y = x² at x = 3? Answer: y = 6x − 9. Source: California Calculus standard 4.1 (the tangent line)

Integrals 12

  • Approximate the integral from 0 to 2 of x² dx with a left Riemann sum of 4 equal subintervals. Answer: 1.75. Source: California Calculus standard 13.0 (Riemann sums)
  • Evaluate the integral from 0 to 1 of (4x³ + 2x) dx. Answer: 2. Source: California Calculus standard 15.0 (the fundamental theorem of calculus)
  • Evaluate the integral from 0 to 2 of 3x² dx. Answer: 8. Source: California Calculus standard 15.0 (the fundamental theorem of calculus)
  • Evaluate the integral from 0 to π of sin x dx. Answer: 2. Source: California Calculus standard 20.0 (integrals of trigonometric functions)
  • Evaluate the integral from 1 to e of (1/x) dx. Answer: 1. Source: California Calculus standard 15.0 (the fundamental theorem of calculus)
  • Find the indefinite integral of 1/(1 + x²). Answer: arctan x + C. Source: California Calculus standard 18.0 (inverse trigonometric functions as integrals)
  • Find the indefinite integral of 2x(x² + 1)³. Answer: (x² + 1)⁴/4 + C. Source: California Calculus standard 17.0 (substitution)
  • Find the indefinite integral of 6x² − 4x + 1. Answer: 2x³ − 2x² + x + C. Source: California Calculus standard 17.0 (integrals computed by hand)
  • Find the indefinite integral of e^(3x). Answer: e^(3x)/3 + C. Source: California Calculus standard 17.0 (substitution)
  • Find the indefinite integral of x e^x. Answer: x e^x − e^x + C. Source: California Calculus standard 17.0 (integration by parts)
  • The integral of f from 0 to 3 is 5 and from 3 to 5 is −2. What is the integral of f from 0 to 5? Answer: 3. Source: California Calculus standard 13.0 (the definite integral)
  • Let F(x) be the integral from 0 to x of √(1 + t²) dt. What is F′(x)? Answer: √(1 + x²). Source: California Calculus standard 15.0 (the fundamental theorem of calculus)

Using integrals 12

  • A force of F(x) = 3x newtons moves an object from x = 0 to x = 4 meters. How much work is done? Answer: 24 J. Source: California Calculus standard 16.0 (work)
  • A particle has acceleration a(t) = 6t m/s² and velocity 2 m/s at t = 0. What is its velocity at t = 3 seconds? Answer: 29 m/s. Source: California Calculus standard 16.0 (acceleration)
  • A particle has velocity v(t) = 3t² − 6t m/s. What is its displacement from t = 0 to t = 3 seconds? Answer: 0 m. Source: California Calculus standard 16.0 (velocity)
  • A particle has velocity v(t) = 3t² − 6t m/s. What total distance does it travel from t = 0 to t = 3 seconds? Answer: 8 m. Source: California Calculus standard 16.0 (velocity)
  • Evaluate the improper integral from 1 to ∞ of 1/x² dx. Answer: 1. Source: California Calculus standard 22.0 (improper integrals)
  • Find the area between y = 4 − x² and the x-axis. Answer: 32/3. Source: California Calculus standard 16.0 (area)
  • Find the area between y = x and y = x² from x = 0 to x = 1. Answer: 1/6. Source: California Calculus standard 16.0 (area)
  • Find the area under y = e^x from x = 0 to x = ln 3. Answer: 2. Source: California Calculus standard 16.0 (area)
  • Find the area under y = x² from x = 0 to x = 3. Answer: 9. Source: California Calculus standard 16.0 (area)
  • The region under y = √x from x = 0 to x = 4 is revolved about the x-axis. What is the volume? Answer: 8π. Source: California Calculus standard 16.0 (volume of a solid)
  • Solve dy/dt = 0.5y with y(0) = 4. Answer: y = 4e^(0.5t). Source: California Calculus standard 27.0 (growth and decay)
  • What is the average value of f(x) = x² on the interval [0, 3]? Answer: 3. Source: California Calculus standard 14.0 (the integral as a model)

How this list was compiled: Our own questions. The topic list is the Calculus standards (1.0 to 27.0) in California Common Core State Standards: Mathematics, pages 116 and 117, California Department of Education, https://www.cde.ca.gov/be/st/ss/documents/ccssmathstandardaug2013.pdf, read 2026-10-07. Every answer is mathematics on the numbers and functions printed in the question: it is worked in the explanation, computed a second time, and each other choice is tested to be wrong (2026-10-07). Licence: Questions, choices and explanations are original work of this site.

How it works

This calculus practice test is unofficial: 60 questions of our own on limits, derivatives and integrals, each worked step by step. It is a course test, tied to no exam.

Read the full guide

Five topics

Limits and continuity, derivatives, using derivatives, integrals, and using integrals: 12 questions each. The list follows the Calculus standards in California's mathematics standards, pages 116 and 117, read October 7, 2026. Sequences and series, standards 23 to 26, are not in this bank.

How the math is written

Whole-number powers are written raised, as in x² or (3x + 1)⁵; other exponents use a caret, as in e^(2x). √ is a square root, ln is the natural logarithm, and C is the constant of integration.

  • When a limit gives 0/0, try factoring or rationalizing before L'Hôpital's rule.
  • Differentiate your antiderivative to check it: you should get the integrand back.
  • In a rate problem, write the formula that links the quantities first, then differentiate with respect to time.

Questions and answers

Is there a printable calculus practice test PDF?

Yes. Print the question sheet puts 20 questions on paper, and Add all adds the rest of the 60, with the answer key if you tick it. Pick Save as PDF in the print window and you have the file.

What topics are on a calculus test?

A first course covers limits and continuity, derivatives and their uses, and integrals and their uses. California's standards also list series and differential equations, and note that many high school programs cannot cover everything in one year.

Do I need a calculator for this calculus practice test?

No. Every answer comes out by hand as a whole number, a fraction, a short decimal or an expression. Your own class test may allow a calculator; your teacher decides.

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