Verify the given linear approximation at a = 0. Then determine the values of x for which the linear approximation is accurate to within 0.1. (Enter your answer using interval notation. Round your answers to three decimal places.)

Question

ln(1 + x) ≈ x

Answer 1

Hello, this is Pro-Truth-Efficient speaking to you:
Consider the function:
f(x) = ln(1+x)
The linear approximation of f at a is given by,
f(x) ~ L(x) = f(a) + f'(a)(x-a)
The derivative of function f with respect to x is,
f'(x) = 1/(1+x)
The value of f and f' at x=0 is:
f(0) = 0
f'(0) = 1
The linear approximation of f at a=0 is given by,
f(x)~L(x)=f(0) + f'(0)(x-0)
=0 +(1)x
=x
We have
ln(1+x) ~ x

Mod[ln(1+x) - x] < 0.1

ln(1+x) - 0.1By using graphing calculator,
- 0.383 < x < 0.516
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