use linear approximation to estime ln(e+0.1)

Question

L(x)=f(a)-f'(a)(x-a)

 

f(x)=lnx

f'(x)= (1/x)(1)

 

Now I am stuck with putting into the linear approximation equation.

I think its L(x)=lne-(1/e)(x-e) but am not sure. 

 

thus how to continue from there. 

Answer 1

ln(e +.1)

 

We can use Linearization to solve.

 

L(x) = f(a) + f ' (a)(x-a)

 

our a value will be conveniently the value of e

 

our function f(x) will be ln(x)

 

so f(a) = f(e) = ln(e) = 1

 

L(x) = 1 + f '(e)(x-e)

 

f '(e) is the slope of the lnx function at x = e

 

The derivative of lnx = 1/x  so our slope we need is 1/e

 

L(x) = 1 +1/e(x-e)  This is our linearization equation. We want to approximate ln(e+.1) so we use e+.1 as our input

 

L(e+.1) = 1 +1/e(e+.1-e) = 1+1/e(.1) = 1+.1/e  approximately 1.036787994

 

Compare to ln(e+.1) which is 1.036127419