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← Calculus I
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Practice deliberately

Work through Calculus I unit by unit, then test yourself under time.

Practice mode gives one hint and a worked explanation after every question. Test mode mixes all 7 units, holds the explanations until you submit, and then reports which units need another pass. The 8 questions printed further down this page are fixed, so you can read them and their worked answers without starting a set.

Calculus I coverage and review map

Connect every question to the exact skill it rehearses. Work through the units in order, then return to any topic that still needs a hint or a second attempt.

This page follows the Calculus I syllabus unit by unit. For shorter mixed retrieval outside the course sequence, use calculus practice questions with answers.

01

Limits and continuity

Define the central objects, connect at least two representations, solve direct and unfamiliar applications, and explain how the result can be checked.

Goal: recognise a limits and continuity problem from its wording, carry out the governing method, and check that the result is reasonable.

02

Derivative definition

Relate local change to slope, apply the appropriate derivative rule, and use sign, units, and graph behavior to interpret the result.

Goal: recognise a derivative definition problem from its wording, carry out the governing method, and check that the result is reasonable.

03

Differentiation rules

Relate local change to slope, apply the appropriate derivative rule, and use sign, units, and graph behavior to interpret the result.

Goal: recognise a differentiation rules problem from its wording, carry out the governing method, and check that the result is reasonable.

04

Implicit differentiation

Relate local change to slope, apply the appropriate derivative rule, and use sign, units, and graph behavior to interpret the result.

Goal: recognise a implicit differentiation problem from its wording, carry out the governing method, and check that the result is reasonable.

05

Related rates

Relate local change to slope, apply the appropriate derivative rule, and use sign, units, and graph behavior to interpret the result.

Goal: recognise a related rates problem from its wording, carry out the governing method, and check that the result is reasonable.

06

Optimization

Relate local change to slope, apply the appropriate derivative rule, and use sign, units, and graph behavior to interpret the result.

Goal: recognise a optimization problem from its wording, carry out the governing method, and check that the result is reasonable.

07

Antiderivatives

Relate local change to slope, apply the appropriate derivative rule, and use sign, units, and graph behavior to interpret the result.

Goal: recognise a antiderivatives problem from its wording, carry out the governing method, and check that the result is reasonable.

Calculus I questions with worked answers

These 8 questions are printed in full on this page and are drawn across all 7 course units. Nothing here is generated on the fly. Cover the options, solve each one on paper, and only then open the worked answer to compare your method with the one shown.

Question 1: Find the limit of (3x^2 - x + 2) / (5x^2 + 7) as x approaches infinity.

Calculus I · Limits and continuity

  1. 3/5
  2. 0
  3. infinity
  4. 1
Show the worked answer
  1. Both the top and the bottom grow like x^2 for very large x.
  2. Divide every term by x^2: (3 - 1/x + 2/x^2) / (5 + 7/x^2).
  3. As x grows, 1/x and 1/x^2 head to 0.
  4. The expression settles at 3 / 5.

Answer: 3/5

Question 2: Let f(x) = (x^2 - 25) / (x - 5) when x is not 5, and f(5) = k. What value of k makes f continuous at x = 5?

Calculus I · Limits and continuity

  1. k = 10
  2. k = 0
  3. k = 5
  4. no value works
Show the worked answer
  1. For continuity, k must equal the limit of f as x approaches 5.
  2. Factor the top: x^2 - 25 = (x - 5)(x + 5).
  3. Cancel (x - 5), leaving x + 5 for every x other than 5.
  4. As x approaches 5, x + 5 approaches 10, so k = 10.

Answer: k = 10

Question 3: Use the limit definition to find f'(2) for f(x) = 3x^2 - x.

Calculus I · Derivative definition

  1. 11
  2. 10
  3. 12
  4. 5
Show the worked answer
  1. The definition is the limit of [f(a + h) - f(a)] / h as h approaches 0.
  2. f(2 + h) = 3(2 + h)^2 - (2 + h) = 12 + 12h + 3h^2 - 2 - h, and f(2) = 10.
  3. The difference is 11h + 3h^2, so the quotient is 11 + 3h.
  4. As h approaches 0 this becomes 11, so f'(2) = 11.

Answer: 11

Question 4: If f(x) = (2x + 1) / (x^2 + 3), what is f'(1)?

Calculus I · Differentiation rules

  1. 1/8
  2. 1/4
  3. -1/8
  4. 2/16
Show the worked answer
  1. Use the quotient rule: (bottom x derivative of top - top x derivative of bottom) / bottom squared.
  2. At x = 1 the top is 3, the bottom is 4, the derivative of the top is 2, and the derivative of the bottom is 2x = 2.
  3. So the numerator is 4(2) - 3(2) = 8 - 6 = 2.
  4. The denominator is 4^2 = 16, so f'(1) = 2 / 16 = 1/8.

Answer: 1/8

Question 5: For the curve x^2 + xy + y^2 = 7, what is dy/dx at the point (1, 2)?

Calculus I · Implicit differentiation

  1. -4/5
  2. -5/4
  3. 4/5
  4. -1
Show the worked answer
  1. Differentiate every term with respect to x, treating y as a function of x.
  2. That gives 2x + y + x(dy/dx) + 2y(dy/dx) = 0.
  3. At (1, 2): 2 + 2 + (1 + 4)(dy/dx) = 0, so 4 + 5(dy/dx) = 0.
  4. dy/dx = -4/5.

Answer: -4/5

Question 6: A 13 ft ladder leans on a wall. The base slides away at 2 ft/s. How fast is the top falling when the base is 5 ft from the wall?

Calculus I · Related rates

  1. -5/6 ft/s
  2. -6/5 ft/s
  3. -2 ft/s
  4. -13/5 ft/s
Show the worked answer
  1. The ladder gives x^2 + y^2 = 169. When x = 5, y^2 = 169 - 25 = 144 so y = 12.
  2. Differentiate with respect to time: 2x(dx/dt) + 2y(dy/dt) = 0.
  3. Solve for dy/dt: dy/dt = -(x / y)(dx/dt) = -(5 / 12)(2).
  4. That is -10/12 = -5/6 ft/s. The negative sign means the top is moving down.

Answer: -5/6 ft/s

Question 7: An open box is made by cutting squares of side x from the corners of a 24 in by 24 in sheet and folding up the sides. What is the largest volume?

Calculus I · Optimization

  1. 1,024 cubic in
  2. 1,152 cubic in
  3. 864 cubic in
  4. 576 cubic in
Show the worked answer
  1. The base measures (24 - 2x) by (24 - 2x) and the height is x, so V = x(24 - 2x)^2.
  2. Differentiate and factor: V' = (24 - 2x)(24 - 6x).
  3. V' = 0 at x = 12, which gives no box, and at x = 4.
  4. At x = 4 the base is 16 by 16, so V = 4 x 256 = 1,024 cubic inches.

Answer: 1,024 cubic in

Question 8: Evaluate the definite integral of 6x^2 - 4x + 5 from 0 to 2.

Calculus I · Antiderivatives

  1. 18
  2. 22
  3. 14
  4. 26
Show the worked answer
  1. Antidifferentiate term by term: 2x^3 - 2x^2 + 5x.
  2. At x = 2: 2(8) - 2(4) + 10 = 16 - 8 + 10 = 18.
  3. At x = 0 every term is 0.
  4. Subtract: 18 - 0 = 18.

Answer: 18

Which unit each printed question belongs to

Use this map after marking your work. If two misses share a unit, review that unit before starting a generated set.

The 8 printed Calculus I questions, the unit each one tests, and its answer.
QuestionUnitWhat it asksAnswer
1Limits and continuityFind the limit of (3x^2 - x + 2) / (5x^2 + 7) as x approaches infinity.3/5
2Limits and continuityLet f(x) = (x^2 - 25) / (x - 5) when x is not 5, and f(5) = k. What value of k makes f continuous at x = 5?k = 10
3Derivative definitionUse the limit definition to find f'(2) for f(x) = 3x^2 - x.11
4Differentiation rulesIf f(x) = (2x + 1) / (x^2 + 3), what is f'(1)?1/8
5Implicit differentiationFor the curve x^2 + xy + y^2 = 7, what is dy/dx at the point (1, 2)?-4/5
6Related ratesA 13 ft ladder leans on a wall. The base slides away at 2 ft/s. How fast is the top falling when the base is 5 ft from the wall?-5/6 ft/s
7OptimizationAn open box is made by cutting squares of side x from the corners of a 24 in by 24 in sheet and folding up the sides. What is the largest volume?1,024 cubic in
8AntiderivativesEvaluate the definite integral of 6x^2 - 4x + 5 from 0 to 2.18

How the generated Calculus I sets are created

The 8 questions above are fixed and checked. The generator at the top of the page is different: it writes fresh questions with AI from the course and unit information shown here, then the application checks each one for a complete prompt, four choices, one keyed answer, and an explanation. Generated questions are original practice—not official or released exam questions—and AI can still make mathematical mistakes. Verify a disputed answer with the stated method, your course materials, or the MathGPT solver, and follow the site's academic-integrity guidance.

Questions about this Calculus I practice page

What does this page cover?

It covers all 7 Calculus I units listed above. Choose one unit for focused work, or mixed review to test method selection, and switch to test mode when you want all units mixed under time.

When can I see correct answers and explanations?

The 8 printed questions on this page keep their worked answers behind a toggle, so you can check any one of them straight away. In the generator above, practice mode explains each question as soon as you answer it, while test mode holds every explanation until you submit.

What should I do with a missed question?

Classify the miss as a definition, setup, calculation, interpretation, or timing error. Re-solve it from a blank page, then use the MathGPT tutor for a hint or method check.