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Find the linearization L(x) of the function

Find the linearization L(x) of the function f(x) = ln(1 + x) at a = 0 and use it to approximate the numbers ln 1.2 and ln 1.01. Compare the estimates from the linear approximation with the values given by a calculator. Which one of the two estimates is more accurate, the estimation of ln 1.2 or the estimation of ln 1.01?

asked Mar 20, 2015 in CALCULUS by anonymous
reshown Mar 20, 2015 by bradely

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2 Answers

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Step 3:

Now use the linear approximation to find the $\ln \ (1.2)$ and $\ln \ (1.01)$ .

Rewrite $\ln \ (1.2) = \ln \ (1+0.2)$ .

Now compare it with the function $\ln \ (1+x)$ .

Here $x=0.2$ .

Now approximate for $x=0.2$ .

$\\L(x)=x \\ \\L(0.2)=0.2$

Step 4:

Rewrite $\ln \ (1.01) = \ln \ (1+0.01)$ .

Now compare it with the function $\ln \ (1+x)$ .

Here $x=0.01$ .

Now approximate for $x=0.01$ .

$\\L(x)=x \\ \\L(0.01)=0.01$

Step 5:

Now calculate $\ln \ (1.2)$ and $\ln \ (1.01)$ using calculator.

$\ln \ (1.2)=0.18232$

$\ln \ (1.01)=0.00995$

The second approximation is significantly better, since it is significantly closer to $0$ than $1.2$ .

Solution :

(1) The linear approximation is $\\L(x)=x$ .

(2) $\\L(0.2)=0.2$ .

(3) $\\L(0.01)=0.01$ .

(4) The second approximation is more accurate.

answered Mar 20, 2015 by yamin_math Mentor

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