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
xy11P

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

f(x, y) = 3xe−yP = (1, 1)
fx(1, 1) =
2.Easy
xy10.5P

For f(x, y) = 3y·e3xy, find the partial derivative ∂f/∂y at the point (1, 0.5).

f(x, y) = 3y·e3xyP = (1, 0.5)
fy(1, 0.5) =
3.Medium
xy1−1P

Use the tangent plane of f(x, y) = −xy³ − 3x³ − 2y³ − xy at (1, −1) to approximate f(1.2, −1.1). Give the linear approximation L.

f(x, y) = −xy³ − 3x³ − 2y³ − xy(a, b) = (1, −1)(x, y) = (1.2, −1.1)
L =
4.Medium
xy−2024024vP

For f(x, y) = 2 sin(2x − y), find the directional derivative Duf at (2, 1) in the direction of v = (−4, 4).

f(x, y) = 2 sin(2x − y)P = (2, 1)v = (−4, 4)
Duf =
5.Hard
xy−505−202(3, 0)(−3, 0)

f(x, y) = −x³ + 27x − y² has a critical point at (−3, 0). Compute the discriminant D = fxxfyy − fxy² there.

f(x, y) = −x³ + 27x − y²P = (−3, 0)
D =
6.Hard
xyx = 4y = 0.5x42

Evaluate I = ∬T (3x − 2y) dA over the triangle T bounded by y = 0, x = 4 and y = 0.5x.

f = 3x − 2y0 ≤ x ≤ 40 ≤ y ≤ 0.5x
I =

Answer key

  1. 1. fx(1, 1) = 1.104
  2. 2. fy(1, 0.5) = 33.61
  3. 3. L = 0.6
  4. 4. Duf = 4.2
  5. 5. D = −36
  6. 6. I = 26.67

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