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Calculus IIIChapter 2: Partial DerivativesSection 2-5

Differentials — Question 4

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Question 4

A rectangular plate has measured sides x=30.0x=30.0 cm and y=20.0y=20.0 cm, each accurate to within 0.10.1 cm. Its area is A=xyA=xy.

Tasks

  1. Use differentials to estimate the maximum absolute area error.

  2. Estimate the maximum relative and percentage errors.

  3. Compare with the exact worst-case outward error.

Original worksheet page 1: question and worked solution for 2-5-004
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Question 4 – Solution

Strategy. For a worst-case differential bound, replace signed increments by their maximum absolute values.

See the diagram in the original worksheet below.

Step 1: Absolute error dA=ydx+xdy.dA=y\,dx+x\,dy. Thus |dA|≤20(0.1)+30(0.1)=5.0cm2.|dA|\le 20(0.1)+30(0.1)=\boxed{5.0\ \mathrm{cm^2}}.

Step 2: Relative error Since A=600A=600 cm2^2, |dA|A≈5600=1120≈0.00833,\frac{|dA|}{A}\approx\frac 5{600}=\boxed{\frac 1{120}\approx 0.00833}, or 0.833%\boxed{0.833\%}.

Step 3: Exact comparison The largest outward area is (30.1)(20.1)=605.01,(30.1)(20.1)=605.01, so the exact positive error is 5.015.01 cm2^2. The differential misses only the product dxdy=0.01dx\,dy=0.01 cm2^2.

Original worksheet page 2: question and worked solution for 2-5-004

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