Which of the following represents the linear approximation to the function f (x) = x^3 + 5x^2 - 9x -5 in the vicinity of the point (1,-8)?

Question

a. y = -4x - 4

b. y = -2x - 6

c. y = -8

d. y = 2x -10

e. y = 4x -12

Answer 1

The linear approximation, L(x), of f(x) at (1 , -8) is just the equation of the tangent line to f(x) at that point. It is called that because for x-values close to x = 1, the y-value of the tangent line will be a decent approximation of the y-value of f(x).

To find the equation for L(x), we just need the slope of the tangent line at x = 1, which we find by taking the derivative of f(x), and evaluating it at x = 1:

f^'(x) = 3x² + 10x - 9

f^'(1) = 4

y + 8 = 4 (x - 1)

L(x) = 4x - 12