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
xy1−3P

For f(x, y) = 3xy² + 4x² − 3y² + xy, find the partial derivative ∂f/∂y at the point (1, −3).

f(x, y) = 3xy² + 4x² − 3y² + xyP = (1, −3)
fy(1, −3) =
2.Easy
xy23P

For f(x, y) = 5x²y + 4x³ − 4y³ − 3xy, find the partial derivative ∂f/∂y at the point (2, 3).

f(x, y) = 5x²y + 4x³ − 4y³ − 3xyP = (2, 3)
fy(2, 3) =
3.Medium
xy12∇fP

For f(x, y) = −4x²y² + 3x³ − 3y² − xy, find the maximum rate of change of f at the point (1, 2), that is |∇f|.

f(x, y) = −4x²y² + 3x³ − 3y² − xyP = (1, 2)
|∇f| =
4.Medium
xy10.5P

The tangent plane to z = 3 sin(3x + y) at the point where (x, y) = (1, 0.5) can be written as z = px + qy + c. Find c.

f(x, y) = 3 sin(3x + y)(a, b) = (1, 0.5)
c =
5.Hard
xy−505−4−202(3, −3)(−3, −3)

Find the critical points of f(x, y) = −x³ + 27x − y² − 6y, classify them, and give the value fmax of f at its local maximum.

f(x, y) = −x³ + 27x − y² − 6y
fmax =
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. fy(1, −3) = 1
  2. 2. fy(2, 3) = −94
  3. 3. |∇f| = 38.29
  4. 4. c = 8.78
  5. 5. fmax = 63
  6. 6. I = 0.04167

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