Write the equation of the line that represents the linear approximation to the following function at the given point a. b. graph the function and the linear approximation at a. c. use the linear approximation to estimate the given function value. d. compute the percent error in your approximation, 100 |approx - exact| / |exact|, where theexact value is given by a calculator. f(x) = 7 - x^2 at a = 2; f(1.9) l(x) =

  a, b.  see below

  c.  L(1.9) = 3.4

  d.  f(1.9) = 3.39; error ≈ 0.29%

Step-by-step explanation:

a. A graphing calculator is useful for graphing these functions.

b. The linear approximation at x=2 is ...

  L(x) = f'(2)(x -2) +f(2)

The derivative is ...

  f'(x) = -2x

so the linear equation is ...

  L(x) = (-2(2))(x -2) +(7 -2^2) = -4x +8 +(7 -4)

  L(x) = -4x +11

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c. L(1.9) = -4(1.9) +11 = 3.4

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d. f(1.9) = 7 -(1.9^2) = 7 -3.61

  f(1.9) = 3.39

  percent error = ((3.4 -3.39)/3.39)×100% ≈ 0.29%