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Find the linearization L(x) of the function f(x) = ln(1 + x)

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 15, 2015 in CALCULUS by anonymous

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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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