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
xy12P

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

f(x, y) = sin(−x + y)P = (1, 2)
fx(1, 2) =
2.Easy
xyx = 1x = 3y = 4y = 21342

Evaluate I = ∫13 ∫24 (3x·ey) dy dx over the rectangle 1 ≤ x ≤ 3, 2 ≤ y ≤ 4.

f = 3x·ey1 ≤ x ≤ 32 ≤ y ≤ 4
I =
3.Medium
xy−10.5P

The tangent plane to z = 2x³e−y at the point where (x, y) = (−1, 0.5) can be written as z = px + qy + c. Find c.

f(x, y) = 2x³e−y(a, b) = (−1, 0.5)
c =
4.Medium
xy31P

Use the tangent plane of f(x, y) = −3x³y² − x² − 2y² + xy at (3, 1) to approximate f(2.8, 0.8). Give the linear approximation L.

f(x, y) = −3x³y² − x² − 2y² + xy(a, b) = (3, 1)(x, y) = (2.8, 0.8)
L =
5.Hard
xy−202−2024AB

The marked points are the critical points of f(x, y) = −x³ + 3x − y² + 6y. Click the saddle point.

f(x, y) = −x³ + 3x − y² + 6y
  1. 1A (1, 3)
  2. 2B (−1, 3)
6.Hard
xy1

Use polar coordinates to evaluate I = ∬R y dA, where R is the upper half of the disc x² + y² ≤ 1 (y ≥ 0).

f = y0 ≤ r ≤ 10 ≤ θ ≤ π
I =

Answer key

  1. 1. fx(1, 2) = −0.5403
  2. 2. I = 566.5
  3. 3. c = 1.82
  4. 4. L = −39.2
  5. 5. B (−1, 3)
  6. 6. I = 0.6667

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