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) = xy² + x³ + y³ + 2xy, find the partial derivative ∂f/∂y at the point (1, 2).

f(x, y) = xy² + x³ + y³ + 2xyP = (1, 2)
fy(1, 2) =
2.Easy
xy0.52P

For f(x, y) = 2y·e2xy, find the partial derivative ∂f/∂y at the point (0.5, 2).

f(x, y) = 2y·e2xyP = (0.5, 2)
fy(0.5, 2) =
3.Medium
xy22∇fP

For f(x, y) = 2y·exy, find the direction of steepest ascent at (2, 2) as the angle θ from the positive x-axis, in degrees between −180° and 180°.

f(x, y) = 2y·exyP = (2, 2)
θ = °
4.Medium
xy11P

The tangent plane to z = 2xe0.5y at the point where (x, y) = (1, 1) can be written as z = px + qy + c. Find c.

f(x, y) = 2xe0.5y(a, b) = (1, 1)
c =
5.Hard
xy−202−101(1, 0)(−1, 0)

Find the critical points of f(x, y) = x³ − 3x + y², classify them, and give the value fmin of f at its local minimum.

f(x, y) = x³ − 3x + y²
fmin =
6.Hard
xyx = 3y = 3x39

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

f = 2x − y0 ≤ x ≤ 30 ≤ y ≤ 3x
I =

Answer key

  1. 1. fy(1, 2) = 18
  2. 2. fy(0.5, 2) = 44.33
  3. 3. θ = 51.34 °
  4. 4. c = −1.649
  5. 5. fmin = −2
  6. 6. I = 13.5

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