Question

Find the linear approximation of f(x) at x=0

f(x) = ln(2+x^2)

Answer 1

The linear approximation uses the equation of a line tangent to the given function at \(x=0\):$$y-y_0=m(x-x_0)\\y-f(0)=m(x-0)\\y-\ln2=mx$$To determine the slope of this line, \(m\), we evaluate the derivative of \(f\) at \(x=0\):$$f'(x)=\frac{2x}{2+x^2}\\m=f'(x_0)=0$$Thus our approximation for \(f\) near \(x=0\) is:$$y=\ln2$$

Answer 2

Most helpful, thanks :)