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Calculus IIIChapter 2: Partial DerivativesSection 2-3

Interpretations of Partial Derivatives — Question 1

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Question 1

For f(x,y)=x2+3xy−y2f(x,y)=x^2+3xy-y^2 at P=(1,2,3)P=(1,2,3):

Tasks

  1. Find fx,fyf_x,f_y at (1,2)(1,2).

  2. Interpret them as trace slopes.

  3. Write both trace tangent lines.

Original worksheet page 1: question and worked solution for 2-3-001
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Question 1 – Solution

Strategy. Hold one input fixed at a time.

See the diagram in the original worksheet below.

Step 1: Rates fx=2x+3yf_x=2x+3y, fy=3x−2yf_y=3x-2y, hence fx(1,2)=8\boxed{f_x(1,2)=8} and fy(1,2)=−1\boxed{f_y(1,2)=-1}.

Step 2: Meaning With y=2y=2, height rises 88 units per xx unit; with x=1x=1, it falls 11 unit per yy unit.

Step 3: Lines y=2,z−3=8(x−1)\boxed{y=2,\ z-3=8(x-1)} and x=1,z−3=−(y−2)\boxed{x=1,\ z-3=-(y-2)}.

Original worksheet page 2: question and worked solution for 2-3-001

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