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
xy−3−3P

For f(x, y) = −x²y³ − 5x² − y² − xy, find the partial derivative ∂f/∂y at the point (−3, −3).

f(x, y) = −x²y³ − 5x² − y² − xyP = (−3, −3)
fy(−3, −3) =
2.Easy

For f(x, y) = 3 ln(x² + 4y²), find the mixed second partial derivative ∂²f/∂x∂y at (1, 0.5).

f(x, y) = 3 ln(x² + 4y²)P = (1, 0.5)
fxy(1, 0.5) =
3.Medium
xy2−1P

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

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

Use the tangent plane of f(x, y) = 5x²y² − 2x³ − 5y³ at (3, 1) to approximate f(2.8, 1.2). Give the linear approximation L.

f(x, y) = 5x²y² − 2x³ − 5y³(a, b) = (3, 1)(x, y) = (2.8, 1.2)
L =
5.Hard
xy−202−4−202(1, −3)

Find the critical points of f(x, y) = −2x² + 3xy − 3y² + 13x − 21y + 5, classify them, and give the value fmax of f at its local maximum.

f(x, y) = −2x² + 3xy − 3y² + 13x − 21y + 5
fmax =
6.Hard
xyx = 4y = 0.5x42

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

f = 3x + 4y0 ≤ x ≤ 40 ≤ y ≤ 0.5x
I =

Answer key

  1. 1. fy(−3, −3) = −234
  2. 2. fxy(1, 0.5) = −6
  3. 3. c = 6.222
  4. 4. L = 5.8
  5. 5. fmax = 43
  6. 6. I = 42.67

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