Help with delta epsilon proofs...

**bemidjibasser** · January 6th 2009, 11:32 AM

I need to prove that lim (mx+b) as x approaches c = mc +b. Any tips or help would be greatly appreciated.
Also, What exactly is the relationship of the definite integral to the derivative?
Again, help and or tips would be greatly appreciated.

**Moo** · January 6th 2009, 12:09 PM

Hello,

Originally Posted by bemidjibasser

I need to prove that lim (mx+b) as x approaches c = mc +b. Any tips or help would be greatly appreciated.
Also, What exactly is the relationship of the definite integral to the derivative?
Again, help and or tips would be greatly appreciated.

For the second one, if f is a function and f ' is its derivative, we have :

$\int_a^b f'(x) ~dx=f(b)-f(a)$

Usually, we define an antiderivative of f, F and say $\int_a^b f(x) ~dx=F(b)-F(a)$

Since f is an antiderivative of f ', we have the formula above.

**galactus** · January 6th 2009, 12:12 PM

This actually is not that bad. I know that the epsilon-delta thing can be confusing.

$f(x)=mx+b, \;\ m\neq 0$

$\lim_{x\to c}(mx+b)=mc+b$

$|(mx+b)-(mc+b)|<{\epsilon}$

$|m(x-c)|<{\epsilon}$

$|x-c|<\frac{\epsilon}{|m|}={\delta}$

Therefore, for ${\epsilon}>0$ , let ${\delta}=\frac{\epsilon}{|m|}$

**bemidjibasser** · January 6th 2009, 03:41 PM

Thank you both. Is there a way to expand on the relationship of the derivative and the definite intergal? I think I can follow the formulas above, but how would you go about explaining more in words? Thanks again for help.

**Mathstud28** · January 6th 2009, 05:02 PM

An alternative to Moo's post would be that in a sense integration and differentiation are inverses. If $F(x)=\int_{a}^x f~dx$ and $f$ is continuous at some point $x_0$ then $F(x)$ is differentiable at $x_0$ and $F'(x_0)=f(x_0)$

**bemidjibasser** · January 6th 2009, 05:35 PM

Does the fundamental theorem have anything to do with this? Or the mean value theorem? Thank you again...

**Mathstud28** · January 6th 2009, 05:45 PM

Originally Posted by bemidjibasser

Does the fundamental theorem have anything to do with this? Or the mean value theorem? Thank you again...

These are the fundamental theorems of calculus (what me and Moo gave)

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