f(x)=e^x; a=0; f(0.05)

Question

Please help ASAP thanks sir.

Answer 1

1. f(a)=f(0)=e⁰=1. So the slope of the linear approximation is 1 at x=a=0.

L(x)=x+b is the general form of the equation, where we need to find the constant b.

The line must pass through the point (0,1) which is (x, f(x))=(0,1).

Plug the coords in: 1=0+b so b=1 and L(x)=x+1.

2. So, when x=0.05, L(0.05)=1.05.

3. f(0.05)=e⁰⋅⁰⁵=1.05127 approx.

The error is 100(1.05-e⁰⋅⁰⁵)/e⁰⋅⁰⁵=-0.12% which means the approximation was 0.12% too low.