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 worksheetPractise 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
xy0.5−1P

For f(x, y) = sin(2x + y), find the partial derivative ∂f/∂y at the point (0.5, −1).

f(x, y) = sin(2x + y)P = (0.5, −1)
fy(0.5, −1) =
2.Easy
xy−11P

For f(x, y) = 3 ln(x² + 4y²), find the partial derivative ∂f/∂y at the point (−1, 1).

f(x, y) = 3 ln(x² + 4y²)P = (−1, 1)
fy(−1, 1) =
3.Medium
xy1−2∇fP

For f(x, y) = −x³y² + 3x³ + 5y² + 2xy, find the direction of steepest ascent at (1, −2) as the angle θ from the positive x-axis, in degrees between −180° and 180°.

f(x, y) = −x³y² + 3x³ + 5y² + 2xyP = (1, −2)
θ = °
4.Medium
xy3−1P

The tangent plane to z = x²y² − 4x³ + y³ + 3xy at the point where (x, y) = (3, −1) can be written as z = px + qy + c. Find c.

f(x, y) = x²y² − 4x³ + y³ + 3xy(a, b) = (3, −1)
c =
5.Hard
xy3

Use polar coordinates to evaluate I = ∬R (x² + y²) dA, where R is the upper half of the disc x² + y² ≤ 9 (y ≥ 0).

f = x² + y²0 ≤ r ≤ 30 ≤ θ ≤ π
I =
6.Hard
xyy = xy = x²11

Find the area I = ∬R 1 dA of the region R between the parabola y = x² and the line y = x.

y = x²y = xf = 1
I =

Answer key

  1. 1. fy(0.5, −1) = 1
  2. 2. fy(−1, 1) = 4.8
  3. 3. θ = −116.6 °
  4. 4. c = 200
  5. 5. I = 63.62
  6. 6. I = 0.1667

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