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Question 1
CalculusFoundation

Calculus practice question 1

Calculus practice topics

Question sets mix direct skills with unfamiliar applications so method selection becomes part of the practice.

01

Limits and continuity

Evaluate limits that start as 0/0, use standard trigonometric limits, and choose constants that make a piecewise function continuous.

02

Derivatives and applications

Differentiate with the power, product, quotient, and chain rules, then use the derivative to find maxima, minima, and rates of change.

03

Integrals and accumulation

Find antiderivatives, evaluate definite integrals with the fundamental theorem, and read an integral as accumulated change.

04

Series and approximation

Test a series for convergence, sum a geometric series, and estimate values with tangent-line and Taylor approximations.

Calculus practice questions with worked answers

These 8 questions are printed in full on this page, with every step of the arithmetic written out. Cover the options, solve the question on paper first, and only then open the worked answer to compare your method with the one shown.

Question 1: Find the limit of sin(5x)/x as x approaches 0.

Foundation level · limits and continuity

  1. 5
  2. 1
  3. 0
  4. 1/5
Show the worked answer
  1. The standard result is that sin(u)/u approaches 1 as u approaches 0.
  2. Rewrite the expression to build that pattern: sin(5x)/x = 5 x sin(5x)/(5x).
  3. As x approaches 0 the piece sin(5x)/(5x) approaches 1.
  4. So the limit is 5 x 1 = 5. Check with x = 0.001: sin(0.005)/0.001 is about 4.99998.

Answer: 5

Question 2: f(x) = x^2 when x is at most 2, and f(x) = ax + 1 when x is greater than 2. What value of a makes f continuous at x = 2?

Exam level · limits and continuity

  1. a = 3/2
  2. a = 2
  3. a = 5/2
  4. a = 1
Show the worked answer
  1. Continuity at x = 2 means the two pieces must give the same value there.
  2. The left piece gives 2^2 = 4.
  3. The right piece gives a(2) + 1 = 2a + 1, so set 2a + 1 = 4.
  4. 2a = 3, so a = 3/2 = 1.5. Check: 1.5 x 2 + 1 = 4.

Answer: a = 3/2

Question 3: Differentiate f(x) = 4x^3 - 6x^2 + 2x - 9.

Foundation level · derivatives and applications

  1. 12x^2 - 12x + 2
  2. 12x^2 - 12x + 2 - 9x
  3. 4x^2 - 6x + 2
  4. 12x^2 - 6x + 2
Show the worked answer
  1. The power rule says the derivative of ax^n is a x n x^(n-1).
  2. 4x^3 becomes 4 x 3 x^2 = 12x^2.
  3. -6x^2 becomes -6 x 2 x = -12x, and 2x becomes 2.
  4. The constant -9 has derivative 0, so f'(x) = 12x^2 - 12x + 2.

Answer: 12x^2 - 12x + 2

Question 4: A ball's height is h(t) = 40t - 5t^2 meters, with t in seconds. At what time does it reach its highest point?

Exam level · derivatives and applications

  1. 4 seconds
  2. 8 seconds
  3. 5 seconds
  4. 2 seconds
Show the worked answer
  1. The high point is where the upward speed drops to zero, so set h'(t) = 0.
  2. h'(t) = 40 - 10t.
  3. 40 - 10t = 0 gives 10t = 40, so t = 4 seconds.
  4. Check the height there: 40(4) - 5(16) = 160 - 80 = 80 meters.

Answer: 4 seconds

Question 5: Evaluate the definite integral of (2x + 1) from x = 1 to x = 3.

Exam level · integrals and accumulation

  1. 10
  2. 12
  3. 8
  4. 14
Show the worked answer
  1. Find an antiderivative: the integral of 2x is x^2 and the integral of 1 is x, so F(x) = x^2 + x.
  2. Evaluate at the top limit: F(3) = 9 + 3 = 12.
  3. Evaluate at the bottom limit: F(1) = 1 + 1 = 2.
  4. Subtract: 12 - 2 = 10.

Answer: 10

Question 6: A car's velocity is v(t) = 3t^2 meters per second. How far does it travel between t = 0 and t = 4 seconds?

Challenge level · integrals and accumulation

  1. 64 meters
  2. 48 meters
  3. 96 meters
  4. 16 meters
Show the worked answer
  1. Distance is the accumulated velocity, which is the integral of v(t) over the time interval.
  2. The antiderivative of 3t^2 is t^3.
  3. Evaluate from 0 to 4: 4^3 - 0^3.
  4. 4^3 = 64, so the car travels 64 meters.

Answer: 64 meters

Question 7: Does the series 1/2 + 1/4 + 1/8 + ... converge, and if so what is its sum?

Foundation level · series and approximation

  1. Yes, it converges to 1
  2. Yes, it converges to 2
  3. Yes, it converges to 1/2
  4. No, it diverges
Show the worked answer
  1. Each term is half the one before, so this is geometric with first term 1/2 and ratio 1/2.
  2. A geometric series converges when the ratio is between -1 and 1, and 1/2 is in that range.
  3. The sum is first term / (1 - ratio) = (1/2) / (1 - 1/2) = (1/2)/(1/2).
  4. That equals 1. Check the partial sums: 0.5, 0.75, 0.875, 0.9375, closing in on 1.

Answer: Yes, it converges to 1

Question 8: Use a linear approximation of f(x) = square root of x at x = 25 to estimate the square root of 26.

Challenge level · series and approximation

  1. 5.1
  2. 5.2
  3. 5.05
  4. 5.5
Show the worked answer
  1. The tangent line at x = 25 gives f(25) + f'(25) x (26 - 25).
  2. f(25) = 5, and f'(x) = 1/(2 root(x)), so f'(25) = 1/10 = 0.1.
  3. The estimate is 5 + 0.1 x 1 = 5.1.
  4. The true value is about 5.099, so the tangent line is accurate to two decimal places.

Answer: 5.1

How the three Calculus difficulty levels differ

The difficulty buttons above change what a question asks of you, not just the size of the numbers. Each example below is taken from the question set on this page.

Difficulty levels for Calculus practice, with an example question from this page.
LevelWhat it testsExample questionTime target
FoundationOne skill at a time, with the numbers kept small enough to check in your head.Find the limit of sin(5x)/x as x approaches 0.About 1 minute
ExamThe wording of a real test paper: pick the method first, then carry out two or three steps.f(x) = x^2 when x is at most 2, and f(x) = ax + 1 when x is greater than 2. What value of a makes f continuous at x = 2?2 to 3 minutes
ChallengeTwo ideas combined, or a result you have to interpret after the calculation ends.A car's velocity is v(t) = 3t^2 meters per second. How far does it travel between t = 0 and t = 4 seconds?4 to 5 minutes

How to practice Calculus effectively

Begin without notes and explain your choice before checking. For every miss, identify whether the cause was a definition, setup, calculation, interpretation, or time decision. Re-solve the question from a blank page, then return to the same skill in a mixed set tomorrow.

What your Calculus answer review should show

A useful review shows more than the correct option. Compare the method with your first attempt, locate the earliest incorrect decision, and write one rule that would prevent the same error in a new Calculus problem.

Move from mixed Calculus questions to a complete course

This page targets flexible Calculus question practice. When you need a syllabus-aligned sequence with unit selection, use Calculus I practice by unit and return here later for mixed retrieval.

Questions about Calculus practice

When should I change the difficulty?

Move up after you can solve several questions accurately without hints and explain the method. Move down for one short set when errors show that a definition or setup is still uncertain.

How often should I practice?

Short sessions on several days usually build stronger recall than one long session. Revisit missed Calculus skills the next day, then mix them with older topics later in the week.

Where can I review the lessons in order?

Use Calculus I practice by unit for a syllabus-aligned sequence with unit selection, practice, and a complete answer review.

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