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.

New worksheetPractise online with hints

For teachers: free to print, copy and use in class. Every new worksheet has new values and its own answer key. A link to ExamProblems on your course page is appreciated.

Name: ____________________   Date: ____________

1.Easy
xy12P

For f(x, y) = x²y − 3x³ + 4y² − xy, find the partial derivative ∂f/∂x at the point (1, 2).

f(x, y) = x²y − 3x³ + 4y² − xyP = (1, 2)
fx(1, 2) =
2.Easy
xy10.5P

For f(x, y) = 2y·e3xy, find the partial derivative ∂f/∂y at the point (1, 0.5).

f(x, y) = 2y·e3xyP = (1, 0.5)
fy(1, 0.5) =
3.Medium
xy33P

The tangent plane to z = −3xy² + 3x³ − 5y² − 3xy at the point where (x, y) = (3, 3) can be written as z = px + qy + c. Find c.

f(x, y) = −3xy² + 3x³ − 5y² − 3xy(a, b) = (3, 3)
c =
4.Medium
xy−1−1P

Use the tangent plane of f(x, y) = 3 ln(x² + 3y²) at (−1, −1) to approximate f(−0.95, −0.8). Give the linear approximation L.

f(x, y) = 3 ln(x² + 3y²)(a, b) = (−1, −1)(x, y) = (−0.95, −0.8)
L =
5.Hard
xy1.5

Use polar coordinates to evaluate I = ∬R e−(x² + y²) dA, where R is the quarter of the disc x² + y² ≤ 2.25 in the first quadrant.

f = e−(x² + y²)0 ≤ r ≤ 1.50 ≤ θ ≤ π/2
I =
6.Hard
xy−20−2−101(0, 0)(−1, −1)

f(x, y) = x³ + y³ + 3xy has a critical point at (0, 0). Compute the discriminant D = fxxfyy − fxy² there.

f(x, y) = x³ + y³ + 3xyP = (0, 0)
D =

Answer key

  1. 1. fx(1, 2) = −7
  2. 2. fy(1, 0.5) = 22.41
  3. 3. c = 72
  4. 4. L = 3.184
  5. 5. I = 0.7026
  6. 6. D = −9

Full step-by-step solutions: practise this topic on examproblems.com.