Question

find linear approximation f(x)=3xe^2x-10 at x=5

Answer 1

 

is the Taylor series expansion where a = 5 and f(x)=3xe²ˣ⁻¹⁰ and f(5)=15.

f'(x)=3e²ˣ⁻¹⁰+6xe²ˣ⁻¹⁰ and f'(5)=3+30=33.

So the linear approximation uses only f and f':

f(5)+f'(5)(x-5)=15+33(x-5)=15+33x-165=33x-150.