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
xy2−2P

For f(x, y) = −x³y + 2x³ + y³ − 2xy, find the partial derivative ∂f/∂x at the point (2, −2).

f(x, y) = −x³y + 2x³ + y³ − 2xyP = (2, −2)
fx(2, −2) =
2.Easy
xy12P

For f(x, y) = 3x³y² − 2x³ − y² + xy, find the partial derivative ∂f/∂y at the point (1, 2).

f(x, y) = 3x³y² − 2x³ − y² + xyP = (1, 2)
fy(1, 2) =
3.Medium
xy−13P

The tangent plane to z = −4xy³ + x³ − y³ − 2xy at the point where (x, y) = (−1, 3) can be written as z = px + qy + c. Find c.

f(x, y) = −4xy³ + x³ − y³ − 2xy(a, b) = (−1, 3)
c =
4.Medium
xy0505vP

For f(x, y) = 5xy² + 5x² − 2xy, find the directional derivative Duf at (3, 3) in the direction of v = (−3, −4).

f(x, y) = 5xy² + 5x² − 2xyP = (3, 3)v = (−3, −4)
Duf =
5.Hard
xy2

Use polar coordinates to evaluate I = ∬R √(x² + y²) dA, where R is the disc x² + y² ≤ 4.

f = √(x² + y²)0 ≤ r ≤ 20 ≤ θ ≤ 2π
I =
6.Hard
xy−20202AB

The marked points are the critical points of f(x, y) = x³ − 3x + y² − 4y. Click the local minimum.

f(x, y) = x³ − 3x + y² − 4y
  1. 1A (1, 2)
  2. 2B (−1, 2)

Answer key

  1. 1. fx(2, −2) = 52
  2. 2. fy(1, 2) = 9
  3. 3. c = −274
  4. 4. Duf = −108.6
  5. 5. I = 16.76
  6. 6. A (1, 2)

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