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
xy20P

For f(x, y) = 3x³ey, find the partial derivative ∂f/∂x at the point (2, 0).

f(x, y) = 3x³eyP = (2, 0)
fx(2, 0) =
2.Easy

For f(x, y) = −4x³y³ − x³ + 3y² − xy, find the second partial derivative ∂²f/∂y² at (1, −2).

f(x, y) = −4x³y³ − x³ + 3y² − xyP = (1, −2)
fyy(1, −2) =
3.Medium
xy21P

The tangent plane to z = 2x³e3y at the point where (x, y) = (2, 1) can be written as z = px + qy + c. Find c.

f(x, y) = 2x³e3y(a, b) = (2, 1)
c =
4.Medium
xy1−2∇fP

For f(x, y) = −4x²y + x³ − 3y³ + xy, find the direction of steepest ascent at (1, −2) as the angle θ from the positive x-axis, in degrees between −180° and 180°.

f(x, y) = −4x²y + x³ − 3y³ + xyP = (1, −2)
θ = °
5.Hard
xy−20202(1, 2)(−1, 2)

Find the critical points of f(x, y) = x³ − 3x + y² − 4y, classify them, and give the value fmin of f at its local minimum.

f(x, y) = x³ − 3x + y² − 4y
fmin =
6.Hard
xyy = xy = x²11

Evaluate I = ∬R xy dA, where R is the region between the parabola y = x² and the line y = x.

y = x²y = xf = xy
I =

Answer key

  1. 1. fx(2, 0) = 36
  2. 2. fyy(1, −2) = 54
  3. 3. c = −1607
  4. 4. θ = −66.45 °
  5. 5. fmin = −6
  6. 6. I = 0.04167

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