For the function f(x) =xe^-(x^2 -x)/2

Question

1) Determine the linearization L(x) of f(x) at x=1.

2) Use the linearization of f(x) at x=1 to estimate the value of f(1.2).

Answer 1

f(x) =xe^-(x^2 -x)/2

I assume you meant:

f(x) =xe^{-(x^2 -x)/2}

First, find the slope by taking the 1st derivative. For f'(x), I get:

f'(x) = e^{-(x^2 -x)/2} + x*(-e^{-(x^2 -x)/2} * -(2x - 1)/2

= (1+x-4 +1/2)* e^{-(x^2 -x)/2}

= (x - 2.5)*e^{-(x^2 -x)/2}

Find f'(1), which the the slope of L(x). Then you can generate the equation for L(x) using the slope and the given point ( (x,y) where x = 1, and y = f(1) )