(a) write the equation of the line that represents the linear approximation to the following function at the given point a. (b) use the linear approximation to estimate the given quantity. (c) compute the percent error in the approximation, 100 * [ | approximation - exact | over | exact | ], where the exact value is given by a calculator. f (x)= 3 - 3 x² at a = 1; f(0.9)

  (a) y = -6(x -1)

  (b) about 5.3%

Step-by-step explanation:

(a) The point used as the base for the linear approximation is (1, f(1)), where ...

  f(1) = 3 -3·1² = 0

The slope of the line at that point is ...

  f'(x) = 0 -3(2x) = -6x

  f'(1) = -6·1 = -6

So, in point-slope form, the equation of the approximating line is ...

  y = -6(x -1) +0

  y = -6(x -1)

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(b) The approximate value of f(0.9) is then ...

  y = -6(0.9 -1) = 0.6 . . . . approximate value of f(0.9)

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(c) The error in the approximation at x=0.9 is ...

  error% = (0.6 -f(0.9))/f(0.9) × 100%

where f(0.9) = 3(1 -0.9²) = 3·0.19 = 0.57

  error% = (0.6 -0.57)/0.57 × 100% = 0.03/0.57 × 100%

  error% ≈ 5.263% ≈ 5.3%

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