# Math Help - What's the difference? Need Help!

1. ## What's the difference? Need Help!

What is the difference in

a.) f(a+Δx)-f(a)/Δx

and

b.) limit as Δx->0 f(a+Δx)-f(a)/Δx

2. Originally Posted by lsp2010
What is the difference in

a.) f(a+Δx)-f(a)/Δx

and

b.) limit as Δx->0 f(a+Δx)-f(a)/Δx
I assume you're asking what is the difference between

$\frac{f(a + \Delta x) - f(a)}{\Delta x}$

and

$\lim_{\Delta x \to 0}\frac{f(a + \Delta x) - f(a)}{\Delta x}$.

The difference is that in the second you're asked to describe what happens as you make $\Delta x$ get close to $0$.

3. How would you sketch the difference?

4. Originally Posted by lsp2010
How would you sketch the difference?

5. Originally Posted by AllanCuz
yes. why?

6. Originally Posted by AllanCuz

7. Draw a picture showing the graph of a function. I presume you are allowed to use whatever function you want. (a, f(a)) is some point on that graph. $(a+ \Delta x, f(a+ \Delta x))$ is another point on that graph. $\frac{f(a+\Delta x)- f(a)}{\Delta x}$ is the slope of the line through those two points. Show that by drawing the line. Do that for several different values of $\Delta x$ so that you have several lines with one end at (a, f(a)). Do you see that the lines become closer and closer to the tangent line at (a, f(a)) as $\Delta x$ becomes smaller and smaller?

8. Originally Posted by lsp2010
On a set of axes label two point (a, f(a)) and (a+delta x, f(a+delta x))

The first is just the gradient formula between those two points (rise over run) so it represents the gradient of the line going through those points.

Now think about what happens as you make delta x smaller and smaller. What happens to the line through those two points?

9. Okay. So that would be for the second equation, right? WHat would I draw for the first equation so you can see a difference in the two?

10. Originally Posted by lsp2010
Okay. So that would be for the second equation, right? WHat would I draw for the first equation so you can see a difference in the two?
The first is the gradient of the line between the two points.
The second is ....... up to you.

11. i think that confused me more.

12. Originally Posted by lsp2010
i think that confused me more.
Let's go back a step. Do you get that the first expression (without the limit bit) is simply the gradient of the line between two points?

13. No. I don't think we ever use gradient. Or at least not the word. What is gradient?

14. Originally Posted by lsp2010
No. I don't think we ever use gradient. Or at least not the word. What is gradient?
another word for slope

15. oh. okay. so the first equation is the slope between the 2 points?

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