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
xyx = 1x = 3y = 5y = 21352

Evaluate I = ∫13 ∫25 (3y − 1) dy dx over the rectangle 1 ≤ x ≤ 3, 2 ≤ y ≤ 5.

f = 3y − 11 ≤ x ≤ 32 ≤ y ≤ 5
I =
2.Easy
xy−1−3P

For f(x, y) = −x²y³ + 5x² + 2y² + 3xy, find the partial derivative ∂f/∂x at the point (−1, −3).

f(x, y) = −x²y³ + 5x² + 2y² + 3xyP = (−1, −3)
fx(−1, −3) =
3.Medium
xy20.5∇fP

For f(x, y) = 2 ln(x² + y²), find the maximum rate of change of f at the point (2, 0.5), that is |∇f|.

f(x, y) = 2 ln(x² + y²)P = (2, 0.5)
|∇f| =
4.Medium
xy20.5P

The tangent plane to z = 3 sin(−x + 2y) at the point where (x, y) = (2, 0.5) can be written as z = px + qy + c. Find c.

f(x, y) = 3 sin(−x + 2y)(a, b) = (2, 0.5)
c =
5.Hard
xy−505−4−202(3, −1)(−3, −1)

f(x, y) = x³ − 27x + y² + 2y has a critical point at (−3, −1). Compute the discriminant D = fxxfyy − fxy² there.

f(x, y) = x³ − 27x + y² + 2yP = (−3, −1)
D =
6.Hard
xyx = 1.5y = x1.5

The integral I = ∫01.5 ∫y1.5 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 ≤ 1.5
I =

Answer key

  1. 1. I = 57
  2. 2. fx(−1, −3) = −73
  3. 3. |∇f| = 1.94
  4. 4. c = −0.9035
  5. 5. D = −36
  6. 6. I = 0.3462

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