Worked Examples12th GradeLevel 3 · Challenge

Free 12th Grade 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, optimization 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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Grade: 12th Grade
Subject: Math
Topic: Calculus
Difficulty: Level 3 · Challenge
Estimated time: 30 minutes
Curriculum: Texas grade levels (TEKS)
Questions: 10
Answers: Included, with working
Format: PDF (A4), print or complete online

Students will practice

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 optimization and motion problems

Curriculum: Texas grade levels (TEKS). 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

Study guide

How to use this worked examples

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

Common questions

Who is this worked examples for?

12th Grade 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 12th Grade Differentiation (Power Rule) Worksheet — Level 1.

Success TutoringWorksheets12th Grade · Mathematics · Calculus

Senior Calculus Practice with Worked Solutions

Name

Date

Score /

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

Section A: Differentiation

1.Differentiate y = x²e^3x. Give your answer in factored 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 = e^2x 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 minimize 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 meters. (a) Find the velocity when t = 2. (b) Find when the particle is at rest. (c) Find the acceleration when t = 2.[4 marks]

Preview — the PDF contains the full resource and answer sheet.

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For the particle in question 3, find the acceleration when t
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About this resource

Created by: Success Tutoring
Last reviewed: 1 October 2026
How it was made: Written by our team

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