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
xy0.52P

For f(x, y) = 5 sin(−x + 2y), find the partial derivative ∂f/∂x at the point (0.5, 2).

f(x, y) = 5 sin(−x + 2y)P = (0.5, 2)
fx(0.5, 2) =
2.Easy

For f(x, y) = −2x²y³ − x² − 2y³ − 2xy, find the mixed second partial derivative ∂²f/∂x∂y at (−2, −1).

f(x, y) = −2x²y³ − x² − 2y³ − 2xyP = (−2, −1)
fxy(−2, −1) =
3.Medium
xy−4−202024vP

For f(x, y) = 3 ln(x² + 4y²), find the directional derivative Duf at (−1, 2) in the direction of v = (−1, −2).

f(x, y) = 3 ln(x² + 4y²)P = (−1, 2)v = (−1, −2)
Duf =
4.Medium
xy32P

The tangent plane to z = −xy² + 4x² − y² + 4xy at the point where (x, y) = (3, 2) can be written as z = px + qy + c. Find c.

f(x, y) = −xy² + 4x² − y² + 4xy(a, b) = (3, 2)
c =
5.Hard
xyy = 4xy = x²416

Evaluate I = ∬R y dA, where R is the region between the parabola y = x² and the line y = 4x.

y = x²y = 4xf = y
I =
6.Hard
xyx = 1.5y = x1.5

The integral I = ∫01.5 ∫y1.5 ex² dx dy cannot be done in this order. Reverse the order of integration and evaluate it.

f = ex²0 ≤ y ≤ x ≤ 1.5
I =

Answer key

  1. 1. fx(0.5, 2) = 4.682
  2. 2. fxy(−2, −1) = 22
  3. 3. Duf = −2.368
  4. 4. c = −32
  5. 5. I = 68.27
  6. 6. I = 4.244

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