Question

Use the differential to approximate the expression. Then use a calculator to approximate the quantity, and give the absolute value of the difference in the two results to four decimal places.
√
53

Answer 1

9514 1404 393

Answer:

   0.0056

Step-by-step explanation:

  f(x) = √(49 +x)

  f'(x) = 1/(2√(49 +x))

A linear approximation of f(x) expanded about x=0 is ...

  f(x) ≈ f(0) + f'(0)x = 7 +x/(2·7)

Then for √53, we have x=4

  f(4) ≈ 7 +4/14 = 7 2/7 . . . . . approximate √53 using differentials

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The calculator value of √53 is about 7.280110, so the difference in results is ...

  approx - actual ≈ 7.285714 -7.280110 = 0.005604 ≈ 0.0056