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
xy12P

For f(x, y) = 5x²y + y² + xy, find the partial derivative ∂f/∂y at the point (1, 2).

f(x, y) = 5x²y + y² + xyP = (1, 2)
fy(1, 2) =
2.Easy
xyx = 1x = 2y = 4y = 11241

Evaluate I = ∫12 ∫14 (2x²y + 6) dy dx over the rectangle 1 ≤ x ≤ 2, 1 ≤ y ≤ 4.

f = 2x²y + 61 ≤ x ≤ 21 ≤ y ≤ 4
I =
3.Medium
xy−1−1∇fP

For f(x, y) = 4x²y² − 2x² − y³, find the direction of steepest ascent at (−1, −1) as the angle θ from the positive x-axis, in degrees between −180° and 180°.

f(x, y) = 4x²y² − 2x² − y³P = (−1, −1)
θ = °
4.Medium
xy−23P

The tangent plane to z = 3x³y² + 5x² − y³ at the point where (x, y) = (−2, 3) can be written as z = px + qy + c. Find c.

f(x, y) = 3x³y² + 5x² − y³(a, b) = (−2, 3)
c =
5.Hard
xyx = 1y = x11

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

f = 2x − y0 ≤ x ≤ 10 ≤ y ≤ x
I =
6.Hard
xy0202(1, 2)

f(x, y) = 2x² + 4xy + y² − 12x − 8y − 1 has a critical point at (1, 2). Compute the discriminant D = fxxfyy − fxy² there.

f(x, y) = 2x² + 4xy + y² − 12x − 8y − 1P = (1, 2)
D =

Answer key

  1. 1. fy(1, 2) = 10
  2. 2. I = 53
  3. 3. θ = −110 °
  4. 4. c = 898
  5. 5. I = 0.5
  6. 6. D = −8

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