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.

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1.Easy
xy2−1P

For f(x, y) = −4xy + 5x² + 3y², find the partial derivative ∂f/∂x at the point (2, −1).

f(x, y) = −4xy + 5x² + 3y²P = (2, −1)
fx(2, −1) =
2.Easy

For f(x, y) = 5xy² − x³ + 4y² + 4xy, find the mixed second partial derivative ∂²f/∂x∂y at (−1, 2).

f(x, y) = 5xy² − x³ + 4y² + 4xyP = (−1, 2)
fxy(−1, 2) =
3.Medium
xy3−1P

The tangent plane to z = −4xy + 5x³ − 5y² at the point where (x, y) = (3, −1) can be written as z = px + qy + c. Find c.

f(x, y) = −4xy + 5x³ − 5y²(a, b) = (3, −1)
c =
4.Medium
xy33P

The tangent plane to z = 2x²y³ − 3x² + 2y³ at the point where (x, y) = (3, 3) can be written as z = px + qy + c. Find c.

f(x, y) = 2x²y³ − 3x² + 2y³(a, b) = (3, 3)
c =
5.Hard
xy−202−202ABC

The marked points are the critical points of f(x, y) = x⁴ + y⁴ − 16xy. Click the saddle point.

f(x, y) = x⁴ + y⁴ − 16xy
  1. 1A (2, 2)
  2. 2B (−2, −2)
  3. 3C (0, 0)
6.Hard
xyx = 4y = 0.5x42

Evaluate I = ∬T (3x) dA over the triangle T bounded by y = 0, x = 4 and y = 0.5x.

f = 3x0 ≤ x ≤ 40 ≤ y ≤ 0.5x
I =

Answer key

  1. 1. fx(2, −1) = 24
  2. 2. fxy(−1, 2) = 24
  3. 3. c = −277
  4. 4. c = −2025
  5. 5. C (0, 0)
  6. 6. I = 32

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