# Given Derivative Thru a Point

• Dec 8th 2007, 03:37 PM
Super Mallow
Given Derivative Thru a Point
Another question..

f'(x)=1/XSQRED + 2X and goes through (-1,1).

I need to find a function that has that derivative and passes through that point
• Dec 8th 2007, 03:41 PM
Jhevon
Quote:

Originally Posted by Super Mallow
Another question..

f'(x)=1/XSQRED + 2X and goes through (-1,1).

I need to find a function that has that derivative and passes through that point

we want $\displaystyle f(x) = \int f'(x)~dx$ and $\displaystyle f(-1) = 1$ as an initial condition. who do you think you should proceed?
• Dec 8th 2007, 03:46 PM
Soroban
Hello, Super Mallow!

Quote:

$\displaystyle f'(x)\:=\:\frac{1}{x^2} + 2x$ and goes through (-1,1).

Find a function that has that derivative and passes through that point.

We have: .$\displaystyle f'(x) \;=\;x^{-2} + 2x$

Integrate: .$\displaystyle f(x) \;=\;-x^{-1} + x^2 + C \;=\;-\frac{1}{x} + x^2 + C$

We are told that, when $\displaystyle x = -1,\:f(\text{-}1) = 1$

. . So we have: .$\displaystyle -\frac{1}{\text{-}1} + (\text{-}1)^2 + C \;=\;1\quad\Rightarrow\quad C \:=\:-1$

Therefore: .$\displaystyle f(x)\;=\;-\frac{1}{x} + x^2 - 1$

• Dec 8th 2007, 03:50 PM
Super Mallow