Slope of Tangent line is driving me nuts kind of question

irishpump

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Oct 25, 2011
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I recently took an exam, and one of the questions I got wrong was:

The slope of the secant line to the graph of g(x) between x=5 and x=5+h is 10 - 6h + 2h^2. What is the slope of the tangent line at x=5?

I did the 4 step process for the definition of a derivative. He told me its easier than that when I tried to show him the work. I keep looking at this and I cant understand how to do it. He said I need to think about the definition of a derivative and relationship between a secant and tangent line. So since the full 4 step process was unecessary for this kind of problem, I've since tried taking the limit of g(x) approaching 0 and I've tried it approaching 5. I've now gotten the answer of 6 but I have no idea how I can check this, and quite frankly I need some help for this kind of question coming up at the end semester finals.

Any help is very much appreciate!
 
The "four step process"? Do tell.

Slope of secant: \(\displaystyle \frac{g(5+h)-g(5)}{h}\;=\;10-6h+2h^{2}\)

It may be confusing that we are NOT given a definition of g(x). We have been given the result of the secant slope.

The tangent slope is defined in terms of the secant slope as the limit as h approaches zero (0). This makes the slope of the tangent 10.

I'd still like to see the "four step process".
 
Thank you very much! I noticed with the mess I made on my paper that was one of my answers when I set h to zero. The 4 step process as my Professor calls it, really is just the definition of a derivative broken down. So he would have it as:

Step 1: f(x)= X+h
Step 2: f(x)= (X+h) - f(x)
Step 3: f(x)= (X+h) - f(x)/h
Step 4: lim h->0

I don't know if he taught it to this way to make it easier to remember by breaking it down, but I'll be honest and say after awhile I quit labeling each step and just wrote out the entire definition to solve a problem. Thank you again. I kept trying to reintroduce the equation back into the definition of a derivative. I had no idea the result was given, and just simply needed to plug in zero. I know out of frustration I did it at some point of my messy paper lol. Thank you again very much!
 
Oddly, you seem not to have made the connection between the process and the secant. Oh well. Some methods work better than others. This one seems a bit excessive, to me.
 
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