Find a linear approximation of f(x)=3xe2x−10 at x = 5.

Question

taylor series?

Answer 1

f(5)=15, df/dx=3e²ˣ⁻¹⁰+6xe²ˣ⁻¹⁰=3e²ˣ⁻¹⁰(2x+1), at x=5, this is 3(11)=33.

So in slope-intercept form L(x)-15=33(x-5), L(x)=15+33(x-5)=15+33x-165=33x-150 or L(x)=33x+(-150).

The graph below shows how close the linear approximation (green) comes to f(x) (red):