University multivariable calculus test to download as PDF

This University multivariable calculus test lasts 60 minutes and is marked out of 20. It covers the whole chapter for pupils aged 18-19, and the full answers are on a separate page.

⏱️ Duration : 60 min
🎯 Total : 20 marks
❓ 5 questions

Mark/20

Question 1 – Partial derivatives and the tangent plane
Let \(f(x,y)=x^2e^y\).
1. Find \(f_x\), \(f_y\) and \(f_{xy}\). (2 marks)
2. Find the tangent plane to \(z=f(x,y)\) at \((2,0,4)\). (2 marks)
Question 2 – Critical points
Find and classify the critical points of \(f(x,y)=x^3-12x+y^2-4y\).
Question 3 – Lagrange multipliers
Find the maximum of \(f=xy^2\) subject to \(x+y=3\), with \(x,y\ge0\).
Question 4 – Double integrals
1. Evaluate \(\displaystyle\int_0^2\!\!\int_0^1(x+2y)\,dx\,dy\). (2 marks)
2. Use polar coordinates to find \(\iint_D\dfrac{dA}{1+x^2+y^2}\) over the unit disc. (2 marks)
Question 5 – Problem: errors in a box
A cuboid has measured sides \(x=4\) m, \(y=3\) m and \(z=2\) m, each with possible error \(\pm0.05\) m. Its volume is \(V=xyz\).
1. Find \(V\) and the partial derivatives at these values. (2 marks)
2. Estimate the maximum error in \(V\) using the total differential. (2 marks)

How to use this University multivariable calculus test

First, revise with the multivariable calculus lesson and the worksheet. Then sit the test in one go, with a timer set to 60 minutes. After that, mark your work with the multivariable calculus test answers. If you lose marks on a question, redo it the next day without help.

Finally, you can download this University multivariable calculus test as a PDF and print it like a real exam paper. Moreover, all the pages of this chapter work together, so you can move between them at any time. For example, many pupils read the lesson on Monday, practise on Wednesday and test themselves on Friday.