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

For f(x, y) = −3xy² − y³ − 3xy, find the mixed second partial derivative ∂²f/∂x∂y at (−1, 2).

f(x, y) = −3xy² − y³ − 3xyP = (−1, 2)
fxy(−1, 2) =
2.Easy
xy−13P

For f(x, y) = −2xy² + 5x² − 4xy, find the partial derivative ∂f/∂y at the point (−1, 3).

f(x, y) = −2xy² + 5x² − 4xyP = (−1, 3)
fy(−1, 3) =
3.Medium
xy−2024−2024PQ

For f(x, y) = −4x³y² + 5x³ + y³ − 3xy, find the directional derivative Duf at P(1, 2) in the direction towards Q(5, −2).

f(x, y) = −4x³y² + 5x³ + y³ − 3xyP = (1, 2)Q = (5, −2)
Duf =
4.Medium
xy3−1P

The tangent plane to z = 5x³y³ + 3x² − 5y² − 3xy at the point where (x, y) = (3, −1) can be written as z = px + qy + c. Find c.

f(x, y) = 5x³y³ + 3x² − 5y² − 3xy(a, b) = (3, −1)
c =
5.Hard
xyy = 0.8y = x0.80.8

The integral I = ∫00.8 ∫x0.8 sin(y²) dy dx cannot be done in this order. Reverse the order of integration and evaluate it.

f = sin(y²)0 ≤ x ≤ y ≤ 0.8
I =
6.Hard
xy3

Use polar coordinates to evaluate I = ∬R e−(x² + y²) dA, where R is the disc x² + y² ≤ 9.

f = e−(x² + y²)0 ≤ r ≤ 30 ≤ θ ≤ 2π
I =

Answer key

  1. 1. fxy(−1, 2) = −15
  2. 2. fy(−1, 3) = 16
  3. 3. Duf = −22.63
  4. 4. c = 644
  5. 5. I = 0.09895
  6. 6. I = 3.141

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