Multivariable calculus worksheet with answers

Free printable Multivariable calculus worksheet: 6 problems from easy to exam level, with an answer key. Print it or save it as a PDF.

New worksheet Practise online with hints

For teachers: free to print, copy and use in class. Every new worksheet has new values and its own answer key. A link to ExamProblems on your course page is appreciated.

Name: ____________________ Date: ____________

1.Easy

For f(x, y) = −4x²y² + 3x² − y³, find the partial derivative ∂f/∂x at the point (−3, 3).

f(x, y) = −4x²y² + 3x² − y³P = (−3, 3)

f_x(−3, 3) =

2.Easy

For f(x, y) = 2y·e^−xy, find the partial derivative ∂f/∂x at the point (−1, 1).

f(x, y) = 2y·e^−xyP = (−1, 1)

f_x(−1, 1) =

3.Medium

The tangent plane to z = −xy³ + 5x³ − 3xy at the point where (x, y) = (1, −2) can be written as z = px + qy + c. Find c.

f(x, y) = −xy³ + 5x³ − 3xy(a, b) = (1, −2)

c =

4.Medium

z = 2 sin(2x − 3y) with x = 3t − 2 and y = t² − 2. Use the chain rule to find dz/dt at t = 0.

z = 2 sin(2x − 3y)x = 3t − 2y = t² − 2t = 0

dz/dt =

5.Hard

The integral I = ∫₀² ∫_x² sin(y²) dy dx cannot be done in this order. Reverse the order of integration and evaluate it.

f = sin(y²)0 ≤ x ≤ y ≤ 2

I =

6.Hard

The integral I = ∫₀² ∫_y² 1/(1 + x²)² dx dy cannot be done in this order. Reverse the order of integration and evaluate it.

f = 1/(1 + x²)²0 ≤ y ≤ x ≤ 2

I =

Answer key

1. f_x(−3, 3) = 198
2. f_x(−1, 1) = −5.437
3. c = −40
4. dz/dt = −4.994
5. I = 0.8268
6. I = 0.4

Full step-by-step solutions: practise this topic on examproblems.com.