Several variables
Partial derivatives, gradients, and double integrals.
Pen and paper is fine · no calculator needed why?
Opens at level 51.
Lesson and untimed practice
Read the full lesson and try each skill without a clock. About 3 minutes to read.
skills · practice stats
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Partial derivatives not tried yet
worked example
f(x, y) = 6y³ − 6x². What is ∂f/∂x at (1, −2)?
Answer: -12
- Treat y as a constant: ∂f/∂x = −12x.
- At (1, −2): −12(1) = −12.
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Double integrals not tried yet
worked example
What is the double integral of f(x, y) = −y over the rectangle 0 ≤ x ≤ 1, 2 ≤ y ≤ 3? Enter a whole number or a decimal.
Answer: -2.5
- Integrate over y first, from 2 to 3, treating x as a constant: that gives −5/2.
- Then integrate that over x from 0 to 1: −5/2 = −2.5.
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Gradient not tried yet
worked example
f(x, y) = −2y² + 12x + 13y. What is the length of the gradient ∇f at (1, 1)?
Answer: 15
- ∇f = (∂f/∂x, ∂f/∂y) = (12, −4y + 13), which is (12, 9) at (1, 1).
- Its length is √(12² + 9²) = √(144 + 81) = √225 = 15.
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Classify a critical point not tried yet
worked example
f(x, y) = 3x² − 2xy + 3y² − 4x + 12y. What kind of point is (0, −2) for f?
- saddle point
- local maximum
- not a critical point
- local minimum
Answer: local minimum
- f_x = 6x − 2y − 4 and f_y = −2x + 6y + 12; both are 0 at (0, −2), so it is a critical point.
- f_xx = 6, f_yy = 6 and f_xy = −2, so D = f_xx·f_yy − f_xy² = 6 × 6 − (−2)² = 32.
- D > 0 and f_xx > 0: a local minimum.
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