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Calculus IIIChapter 2: Partial DerivativesSection 2-3

Interpretations of Partial Derivatives — Question 9

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Question 9

At one point a concentration field has Cx=4C_x=4, Cy=0C_y=0, Ct=−1.5C_t=-1.5 in appropriate mg/L per meter or minute units.

Tasks

  1. Interpret each.

  2. Explain what Cy=0C_y=0 does not imply.

  3. Predict separate changes for .2.2 m in +x+x, .2.2 m in +y+y, and 22 minutes.

Original worksheet page 1: question and worked solution for 2-3-009
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Question 9 – Solution

Strategy. Preserve held-fixed conditions and units.

Step 1: Meaning Concentration rises 44 mg/L per xx meter, has zero instantaneous yy slope, and falls 1.51.5 mg/L per minute.

Step 2: Caution Cy=0C_y=0 at one point does not mean independence from yy nearby.

Step 3: Predictions ΔCx≈.8mg/L\boxed{\Delta C_x\approx.8\ \mathrm{mg/L}}, ΔCy≈0\boxed{\Delta C_y\approx 0}, and ΔCt≈−3.0mg/L\boxed{\Delta C_t\approx-3.0\ \mathrm{mg/L}}.

Original worksheet page 2: question and worked solution for 2-3-009

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