Verify Calculus Answers: Ensuring Accuracy

[linuxwolf]

Let me start off with, I took calculus in high school 11 years ago. I'm now studying again to clep out of the class and want to verify that I'm remembering / learning things properly.

Homework Statement

Evaluate the difference quotient for the given function. Simplify your answer.

Homework Equations

20) f(x) = x^3, [f(a+h) - f(a)] / h

22) f(x) = (x+3) / (x+1), [f(x) - f(1)] / (x-1)

The Attempt at a Solution

20) [(a+3h)^3 -a^3] / h
(a^3 + 3a^2h + 3ah^2 + h^3 - a^3) / h
(3a^2h +3ah^2 + h^3)/h
3a^2 + 3ah + h^2

22) [ (x+3) / (x+1) - (1+3) / 1+1) ] / (x-1)
Focusing on top first:
(x+3) / (x+1) - 4/2
(x+3) / (x+1) - 2/1
[1 * (x+3)] /[1 * (x+1)] - [2 * (x+1)] / [ 1 * (x+1) ]
[(x+3) - (2x+2) ] / (x+1)
(1-x) / (x+1)

Now add the bottom back in
[(1-x) / (x+1)] * 1/(x-1)
[-1(x-1) / (x+1)] * 1/(x-1)

ANSWER:
-1/(x+1)I just want to verify I did that all right. I'm also having trouble remembering how, if possible, to simplify problem number 20. Don't have a teacher to go to so I can verify my answers to non odd problems. I appreciate any feedback.

 

[fss]

The Attempt at a Solution

20) [(a+3h)^3 -a^3] / h
(a^3 + 3a^2h + 3ah^2 + h^3 - a^3) / h
(3a^2h +3ah^2 + h^3)/h
3a^2h + 3ah + h^2

Take another look at what happened between those two steps.

 

[linuxwolf]

I realized that, and edited original post..

Originally last line in problem 20 said: 3a^2h + 3ah + h^2

should have read:

3a^2 + 3ah + h^2

I have it written in my paper that way, just typoed. Sorry. Still need to see if can simplify that any further, I'm thinking not, but want to verify. Also want to verify that my thoughts / work is all correct to this step.

 

[Char. Limit]

I realized that, and edited original post..

Originally last line in problem 20 said: 3a^2h + 3ah + h^2

should have read:

3a^2 + 3ah + h^2

I have it written in my paper that way, just typoed. Sorry. Still need to see if can simplify that any further, I'm thinking not, but want to verify. Also want to verify that my thoughts / work is all correct to this step.

Well, if you're actually taking the derivative, you can take the limit as h approaches 0. Are you doing that?

 

[linuxwolf]

No. as stated in the original post... Directions say to evaluate the difference quotient

 

[Char. Limit]

No. as stated in the original post... Directions say to evaluate the difference quotient

Don't really see much else you can do here then.