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−32P

For f(x, y) = 2x³y + 2x³ + 5y² − xy, find the partial derivative ∂f/∂y at the point (−3, 2).

f(x, y) = 2x³y + 2x³ + 5y² − xyP = (−3, 2)
fy(−3, 2) =
2.Easy
xy1−1P

For f(x, y) = 5 sin(−3x − 2y), find the partial derivative ∂f/∂x at the point (1, −1).

f(x, y) = 5 sin(−3x − 2y)P = (1, −1)
fx(1, −1) =
3.Medium
xy3−2P

The tangent plane to z = −3x²y² + 3x³ + 4y² + 3xy at the point where (x, y) = (3, −2) can be written as z = px + qy + c. Find c.

f(x, y) = −3x²y² + 3x³ + 4y² + 3xy(a, b) = (3, −2)
c =
4.Medium
xy−1012024vP

For f(x, y) = ln(x² + 3y²), find the directional derivative Duf at (0.5, 1) in the direction of v = (−1, 4).

f(x, y) = ln(x² + 3y²)P = (0.5, 1)v = (−1, 4)
Duf =
5.Hard
xyx = 1.2y = x1.2

The integral I = ∫01.2 ∫y1.2 cos(x²) dx dy cannot be done in this order. Reverse the order of integration and evaluate it.

f = cos(x²)0 ≤ y ≤ x ≤ 1.2
I =
6.Hard
xy1.5

Use polar coordinates to evaluate I = ∬R e−(x² + y²) dA, where R is the disc x² + y² ≤ 2.25.

f = e−(x² + y²)0 ≤ r ≤ 1.50 ≤ θ ≤ 2π
I =

Answer key

  1. 1. fy(−3, 2) = −31
  2. 2. fx(1, −1) = −8.105
  3. 3. c = 164
  4. 4. Duf = 1.716
  5. 5. I = 0.4957
  6. 6. I = 2.81

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