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Worked ExamplesYear 13Level 3 · Challenge

Free Year 13 Senior Calculus Practice with Worked Solutions

Ten exam-style calculus questions covering differentiation (product, quotient and chain rules with exponentials, logarithms and trigonometric functions), tangents, definite integrals, area between curves, stationary points, optimisation and straight-line motion. Every question has marks allocated and a full step-by-step solution, so students can mark their own working the way an examiner would.

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Year
Year 13
Subject
Mathematics
Topic
Calculus
Difficulty
Level 3 · Challenge
Estimated time
30 minutes
Curriculum
New Zealand Curriculum
Questions
10
Answers
Included, with working
Format
PDF (A4) + print

Students will practise

  • differentiating products, quotients and composite functions involving eˣ, ln x and trigonometric functions
  • finding the equation of a tangent to a curve
  • evaluating definite integrals and finding the area between two curves
  • classifying stationary points and solving optimisation and motion problems

Curriculum: New Zealand Curriculum. We show specific outcome codes only where they have been verified against the official curriculum document.

What's next?

Completed: Senior Calculus Practice with Worked Solutions

  1. 1Year 13 Differentiation (Power Rule) Worksheet — Level 3
  2. 2Year 13 Differentiation (Power Rule) Worksheet — Level 3 (Set 2)
  3. 3Year 13 Integration (Polynomials) Worksheet — Level 2
  4. 4Year 13 Integration (Polynomials) Worksheet — Level 3
Study guide

How to use this worked examples

  1. Print the PDF or complete it on screen. Write the student's name and date at the top.
  2. Read the instructions for each part together before starting. Encourage working to be shown, not just answers.
  3. Mark with the answer sheet. Compare the method, not only the final answer, and give credit for correct working.
  4. Record the score, celebrate what went well, then follow the "What's next" suggestion.

Common questions

Who is this worked examples for?

Year 13 students (typically ages 17–18) working on calculus. It is pitched at level 3 · challenge.

Are the answers included?

Yes. The PDF includes a full answer sheet with the working for every question. You can also view the answers online.

How long does it take?

About 30 minutes for 10 questions. Short, regular sessions work best: two or three a week beats one long one.

Do I need to sign up to download?

No. Click Download Free PDF and it opens immediately. It is free for personal, classroom and homeschool use.

What should we do next?

Try Year 13 Differentiation (Power Rule) Worksheet — Level 3.

Preview

Year 13 · Mathematics · Calculus

Senior Calculus Practice with Worked Solutions

Success Tutoring

Student name:

Date:

Score: /

Answer every question, showing all working. Marks are shown in brackets. Total: 33 marks.

Section A: Differentiation

1.Differentiate y = x2e3x. Give your answer in factorised form.[2 marks]
2.Differentiate y = ln(x2 + 1).[2 marks]
3.Differentiate y = .[3 marks]
4.Find the equation of the tangent to the curve y = e2x at the point where x = 0.[3 marks]

Section B: Integration and area

5.Evaluate ∫ from 1 to 4 of 3√x dx.[3 marks]
6.Evaluate ∫ from 0 to of cos 2x dx. Give an exact answer.[3 marks]
7.Find the area of the region enclosed between the parabola y = x2 and the line y = 2x.[4 marks]

Section C: Applications

8.Find the coordinates of the stationary points of y = x3 − 6x2 + 9x + 1 and determine their nature.[4 marks]
9.An open box has a square base of side x cm and height h cm. Its volume is 32 cm3. Show that its external surface area is S = x2 + , and find the dimensions that minimise S.[5 marks]
10.A particle moves in a straight line so that its displacement from the origin after t seconds is x = t3 − 6t2 + 9t metres. (a) Find the velocity when t = 2. (b) Find when the particle is at rest. (c) Find the acceleration when t = 2.[4 marks]
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Created by
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Last reviewed
1 October 2026
How it was made
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