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
xyx = 2x = 5y = 1251

Evaluate I = ∫25 ∫01 (−xy² + 2) dy dx over the rectangle 2 ≤ x ≤ 5, 0 ≤ y ≤ 1.

f = −xy² + 22 ≤ x ≤ 50 ≤ y ≤ 1
I =
2.Easy
xy1−2P

For f(x, y) = −3xy³ + 3x² + 5y³ + xy, find the partial derivative ∂f/∂x at the point (1, −2).

f(x, y) = −3xy³ + 3x² + 5y³ + xyP = (1, −2)
fx(1, −2) =
3.Medium
xy11∇fP

For f(x, y) = 3xe3y, 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) = 3xe3yP = (1, 1)
θ = °
4.Medium
xy31P

The tangent plane to z = −x²y³ + 3x² + y³ − 3xy at the point where (x, y) = (3, 1) can be written as z = px + qy + c. Find c.

f(x, y) = −x²y³ + 3x² + y³ − 3xy(a, b) = (3, 1)
c =
5.Hard
xy02−1012AB

The marked points are the critical points of f(x, y) = x³ + y³ − 3xy. Click the local minimum.

f(x, y) = x³ + y³ − 3xy
  1. 1A (0, 0)
  2. 2B (1, 1)
6.Hard
xyy = xy = x²11

Find the area I = ∬R 1 dA of the region R between the parabola y = x² and the line y = x.

y = x²y = xf = 1
I =

Answer key

  1. 1. I = 2.5
  2. 2. fx(1, −2) = 28
  3. 3. θ = 71.57 °
  4. 4. c = 16
  5. 5. B (1, 1)
  6. 6. I = 0.1667

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