Finding the derivative at a given point is easy. You just calculate the derivative, then plug in the value that you’re given — the x-value that you need the derivative at. Just remember this is a two-step process, and the order is important. Ignore what’s inside the parentheses, and calculate the derivative — then plug in your value and you’re done.
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This was useful because we talked about it today in class. I forgot that to find the derivative at a point you have to go back and plug in your values for x to find the actual slope, not just some random formula.
doc has a few comments on this pencast:
1. Graphing the function: multiplying x^2 by 7 does 'squish' in the sense that it takes every y-value and multiplies it. A better and more useful way to think about it is to call it a 'vertical stretch' by 7. If you choose your words carefully, the whole thing will make more sense.
2. The derivative is NOT the tangent line itself, it is the SLOPE of the tangent line at a point. The derivative is either a number (at a point), or a function (over the domain of the original function.)
Remember the DEFINITION of the tangent line: it is the line through the pt (x,f(x)), with slope m = f'(x).
3. More useful language: you will learn much and understand more if you use the terminology
"f at the point a-4" rather than "f of a-4". This will keep in your mind the notion that f can be seen as a geometrical graph, and that graph has values at points.
4. Eric's recap of the process at the end was very useful, tying together the problem.
I agree that the analysis at the end was very useful. Also, the explanation that we must find the derivative first before we find the derivative at a certain point makes sense and seems obvious but is probably a common mistake people make. I liked seeing an example of finding the derivative at a point such as (a-4) rather than at 4 o 5. An undefined point always confuses me more than an arbitrary number.
This pencast was very helpful. All of the analysis explaining why we do what we do while finding derivatives makes the entire process easier to understand. I liked hearing about the parabola almost becoming a straight line- it's really cool to see how all of this really works out.
I enjoyed this video because he goes through a simple problem with minute details that help me to completely understand the concept of finding the tangent line at a point of an equation. These pencast really help with the homework as well if I am stuck on something i can usually find the answer i need on a pencast.
The concepts actually discussed in this pencast were covered long ago in class, but I found the idea of finding f'(x) before using substitution, a good thought in discussions much later in the class. It is MUCH easier to deal with f(x) than it is to deal with a large formula that has been monkeyed around with and had a lot of substitution. Simplicity is best!