# How do I use the linear approximation of a function given a value, a, and change in x?

My book gives a few definitions/formulas for obtaining linear approximation, but I'm having trouble understanding how to use them.

Heres the question:

a.) Use the Linear Approximation for f(x) = ln($x^2$ +3) at a = 1 with $\Delta x$ = 0.7 to approximate $\Delta f \approx$ ______________

b.) That Linear Approximation to the value of f(a + $\Delta x$) has error __________ (use calculator to get decimal value)

c.)That Linear Approximation in f(a + $\Delta x$) has percentage error __________

My book says something like this: $\Delta f$ = f(a + $\Delta x$) - f(a), where $\Delta x$ is small. By definition the derivative is the limit.

f '(a) = $\frac{lim}{\Delta x\rightarrow0}\Delta f$ = $\frac{lim}{\Delta x\rightarrow0}(\frac{f(a + \Delta x) - f(a)}{\Delta x})$ = $\frac{lim}{\Delta x\rightarrow0}\frac{\Delta f}{\Delta x}$ So when $\Delta x$ is small, we have $\frac{\Delta f}{\Delta x}\approx$ f '(a) and thus, $\Delta f \approx$ f '(a) $\Delta x$

I thought I was applying the formula correctly, but I guess not. Here's what I've done: I would have thought (a) required $\Delta f = f(a + \Delta x) - f(a) \approx f'(a)\; \Delta x$ and asking for the last of these expressions
so (b) is $f(a + \Delta x) - (f(a) + f'(a)\; \Delta x)$
and (c) might be $\dfrac{ f(a + \Delta x) - (f(a) + f'(a)\; \Delta x)}{f(a + \Delta x) }$ perhaps multiplying by $100$, though the desired denominator might be $\Delta f = f(a + \Delta x) - f(a)$ rather than $f(a + \Delta x)$.
• Ok so for b it would be ln(5.89) $\approx$ 1.773? – Cozen Oct 29 '13 at 0:15