Thread: Approximate function using first and second derivative?

1. Approximate function using first and second derivative?

I know how to approximate the value of a function by (a) plugging in values and (b) using a linear approximation, i.e. $\displaystyle f(x)\approx f(a)+f'(a)(x-a)$

But, how would one approximate the change in a function, i.e. $\displaystyle \Delta y$ of $\displaystyle y=30z+50z^2+40z^3$ when $\displaystyle z$ changes from 0.90 to 0.92 by using the first-order and second order derivatives?

Any insights would be much appreciated!

2. Originally Posted by horan
I know how to approximate the value of a function by (a) plugging in values and (b) using a linear approximation, i.e. $\displaystyle f(x)\approx f(a)+f'(a)(x-a)$

But, how would one approximate the change in a function, i.e. $\displaystyle \Delta y$ of $\displaystyle y=30z+50z^2+40z^3$ when $\displaystyle z$ changes from 0.90 to 0.92 by using the first-order and second order derivatives?

Any insights would be much appreciated!
You have:

$\displaystyle f(x+a)=f(x)+a f'(x)+ \frac{a^2}{2} f''(x) + ...$

Hence:

$\displaystyle f(x+a)-f(x)=a f'(x)+ \frac{a^2}{2} f''(x) + ...$

CB