I have a question on Calculus, where I have to find the "first order approximation", on a function where:
sqrt[(C^2 + sin(h))], where C > 0, and h is small.
Please may someone help me to start this question, as I don't know how to start it. I'm unsure on how to approach this question.
Thank you.
3 Answers
If $f$ is a differentiable function then for small $h$ you know that the derivative is near the difference quotient: $$ f'(x) \approx \frac{f(x+h) - f(x)}{h} $$ so $$ f(x+h) \approx f(x) + f'(x) \times h . $$ That last equation is referred to as a "first order approximation". A second order approximation would add an $h^2$ term involving the second derivative. You will learn about that later in your course.
If $h=0$ the expression equals $C$. You are looking for the next term in the Taylor series near $h=0$. We have $f(h) \approx f(0)+hf'(0)$ so you want to take the derivative with respect to $h$. Your first order approximation will then look like $C+h($something).
The 1st order linear approximation is: $L(x) = f(0) + f'(0)x= C + \dfrac{x}{2C}$ .
