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) = x²e2y, find the partial derivative ∂f/∂x at the point (−1, 1).

f(x, y) = x²e2yP = (−1, 1)
fx(−1, 1) =
2.Easy
xy3−1P

For f(x, y) = 5x³y² − x² − 4y² − 4xy, find the partial derivative ∂f/∂x at the point (3, −1).

f(x, y) = 5x³y² − x² − 4y² − 4xyP = (3, −1)
fx(3, −1) =
3.Medium
xy−2−1∇fP

For f(x, y) = 3x³y² − 4x³ + y² − 3xy, find the maximum rate of change of f at the point (−2, −1), that is |∇f|.

f(x, y) = 3x³y² − 4x³ + y² − 3xyP = (−2, −1)
|∇f| =
4.Medium
xy0.5−1P

Use the tangent plane of f(x, y) = 3x²e−y at (0.5, −1) to approximate f(0.7, −1.2). Give the linear approximation L.

f(x, y) = 3x²e−y(a, b) = (0.5, −1)(x, y) = (0.7, −1.2)
L =
5.Hard
xy−505−4−202(3, −3)(−3, −3)

f(x, y) = x³ − 27x + y² + 6y has a critical point at (3, −3). Compute the discriminant D = fxxfyy − fxy² there.

f(x, y) = x³ − 27x + y² + 6yP = (3, −3)
D =
6.Hard
xy10.25

Use polar coordinates to evaluate I = ∬R (x² + y²) dA, where R is the ring 0.0625 ≤ x² + y² ≤ 1.

f = x² + y²0.25 ≤ r ≤ 10 ≤ θ ≤ 2π
I =

Answer key

  1. 1. fx(−1, 1) = −14.78
  2. 2. fx(3, −1) = 133
  3. 3. |∇f| = 52.77
  4. 4. L = 4.077
  5. 5. D = 36
  6. 6. I = 1.565

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