Question

Find the slope of the curve at the indicated point. Use the limit of the difference quotient.
y=7/(5+x) , x=8

I think the setup is ([7/(5+(8+h))]-[7/(5+8)])/h
Once i get there and add up the denominators, I'm not sure how to go about finishing it.

Answer 1

Okay, so I think I'm getting somewhere (maybe).

I tried my best to continue and simplify and maybe get common denominators in my numerator. This is what I have;

([91/(169+13h)]-[(91+7h)/(169+13h)])/h

Answer 2

I know the answer is: m=-7/169

Answer 3

your work looks correct so far.
Just add the numerators together.
Factor out an "h"
This will cancel with "h" in denominator.

Answer 4

The difference quotient is
\[\frac{y(x+h)-y(x)}{h}=\frac{\dfrac{7}{5+(x+h)}-\dfrac{7}{5+x}}{h}\]and so the difference quotient when \(x=8\) is
\[\frac{y(x+h)-y(x)}{h}=\frac{\dfrac{7}{13+h}-\dfrac{7}{13}}{h}\]The slope is then
\[\lim_{h\to0}\frac{\dfrac{7}{13+h}-\dfrac{7}{13}}{h}\]which you can compute by finding the LCD of the numerator and simplifying:
\[\frac{\dfrac{7}{13+h}-\dfrac{7}{13}}{h}=\frac{\dfrac{7\times13}{13(13+h)}-\dfrac{7(13+h)}{13(13+h)}}{h}=\frac{-7h}{13h(13+h)}=-\frac{7}{13(13+h)}\]

Answer 5

So I subtracted my numerators; 91-91+7h, so my numerator is 7h.
I have (7h/(169+13h))/h
How do I factor an h out of that equation?

Answer 6

Also, how is it going to end up negative? The answer we're given is -7/169

Answer 7

use @HolsterEmission work as reference.
Notice the bottom "h" can be put in denominator
\[\rightarrow \frac{\frac{a}{b}}{c} = \frac{a}{bc}\]

Then the h will cancel with h in term "-7h" in numerator

Answer 8

its negative, you forgot to distribute the negative

---> 91 - (91+7h) = 91 - 91 - 7h = -7h

Answer 9

The last step is applying the limit .... h =0

Answer 10

Oh okay! I didn't realize you had to distribute the negative. I thought that since they had common denominators, you just combined them as in (91-91+7h). Thank you.

So in @HolsterEmission's last step, you just plug 0 in for h?

Answer 11

yep

Answer 12

Oh, awesome. Thank you!

Answer 13

yw

fyi use parenthesis around multiple terms to help remind you to distribute

Answer 14

Okay. Thank you!