We will cover the following topics in this class:
Last time we looked at finding derivatives by "guessing" what a sequence of numbers was getting closer to; the number we get closer and closer to is called the Limit.
Example:
This time we look briefly at how to be certain about what the limit is, using algebraic methods, which is shown in the last two lines of the above table.
Note, you will not be examined on finding a limit in this algebraic manner, but you will be expected to know what a limit is, and give an estimate of what the limit is from a sequence of values in a table. Also anyone who is going on to Math 121 or some other calculus course after Math 006 will need to know the above technique eventually.
You should understand how to find the derivative as the limit of the slopes of tangents, just as you should understand what multiplication really means. But you don't have to think about the meaning of multiplication every time you pick up your calculator, and in the same way, you don't always need to go through the process of writing a table to find a derivative. We'll list some rules you should learn how to use.
| Rule | Formula | Example | |||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| f(x) | f'(x) | ||||||||||||||||||||||
| constants | (d/dx)(c)=0 | 3 | 0 | ||||||||||||||||||||
| powers | (d/dx)xn = nxn-1 | x2 | 2x
| addition (sum rule) | (d/dx)(f(x)+g(x)) = f'(x) + g'(x) | x2 + 3 | 2x
| subtraction (difference rule) | (d/dx)(f(x)-g(x)) = f'(x) - g'(x) | 3-x2 | -2x
| multiplication (product rule) | (d/dx)(f(x)g(x))= f'(x)g(x) + f(x)g'(x) | (x2 + 3)(x2 - 3) | 2x(x2 - 3) + 2x(x2 + 3)
| division (quotient rule) | (d/dx)(f(x)/g(x)) = (f'(x)g(x) - f(x)g'(x))/(g(x)g(x)) | 3/x2 | -3*2x/x3*x3 = -6/x3
| composition (chain rule) | (d/dx)(f(g(x)) = g'(x)f'(g(x))
| (x2+3)2 | 2x*2(x2+3) = 4x(x2+3)
| |
(Note, due to limitations of html, the notation of derivative in the table is "(d/dx)", or f'(x), which means the derivative as usual. Note that the power rule works for all powers, not just whole numbers.)
In class I will give examples of application of these rules.
The following questions from the text book should be done for practice, but are not to be handed in for marking.
Note, Further questions from these sections will be set next week,
after we've covered these sections in more detail.
Assignment
Hand in answers to the following. Due date: Wednesday 11 February.
1. Find the derivatives of the following functions, using the rules given in above:
2. Find the equations for the tangent to the curve y= x2 -3x+1 at x=1, x=2, x=3. Sketch a graph of the curve between x=0 and x=4. (Plot points at x=1,2,3, and sketch the tangents).
Please send corrections to spelling mistakes, and any other comments, to me.
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