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
xy−11P

For f(x, y) = −3xy³ + x² − 3xy, find the partial derivative ∂f/∂x at the point (−1, 1).

f(x, y) = −3xy³ + x² − 3xyP = (−1, 1)
fx(−1, 1) =
2.Easy
xy12P

For f(x, y) = −4x³y³ + 5x² − 5y³ + 3xy, find the partial derivative ∂f/∂y at the point (1, 2).

f(x, y) = −4x³y³ + 5x² − 5y³ + 3xyP = (1, 2)
fy(1, 2) =
3.Medium
xy−2−1∇fP

For f(x, y) = 5x³y + 2x³ − 4y² + 2xy, find the maximum rate of change of f at the point (−2, −1), that is |∇f|.

f(x, y) = 5x³y + 2x³ − 4y² + 2xyP = (−2, −1)
|∇f| =
4.Medium
xy3−1P

The tangent plane to z = 4x³y + 3x³ − 4y² + 2xy at the point where (x, y) = (3, −1) can be written as z = px + qy + c. Find c.

f(x, y) = 4x³y + 3x³ − 4y² + 2xy(a, b) = (3, −1)
c =
5.Hard
xy05−2024(0, 0)(3, 3)

f(x, y) = x³ + y³ − 9xy has a critical point at (3, 3). Compute the discriminant D = fxxfyy − fxy² there.

f(x, y) = x³ + y³ − 9xyP = (3, 3)
D =
6.Hard
xyx = 1y = 2x12

Evaluate I = ∬T (2x + y) dA over the triangle T bounded by y = 0, x = 1 and y = 2x.

f = 2x + y0 ≤ x ≤ 10 ≤ y ≤ 2x
I =

Answer key

  1. 1. fx(−1, 1) = −8
  2. 2. fy(1, 2) = −105
  3. 3. |∇f| = 52.35
  4. 4. c = 172
  5. 5. D = 243
  6. 6. I = 2

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