Day 8, Mat128: Calculus A

We were able to meet in this classroom, so plan on coming here if you come.

1. I asked you for the limit definition of the derivative, at a point (\(x=a)\).

   The limit is essential, and it's an operator: it acts on something.

   I think the problem is that we say "what does the limit equal?", and so people start writing \[ \lim_{h\rightarrow 0} = \ldots \]

   We need to write instead \[ \lim_{h\rightarrow 0}\left[\textrm{something} \right] = \ldots \] and say that "the limit of something as \(h \rightarrow 0\) equals...."

2. When you pass to the limit (because the expression has become determinate -- not indeterminate anymore) -- that is, when we let \(h \rightarrow 0\) -- the terms with an \(h\) vanish -- \(h\) becomes 0. \(h\) has to disappear, because it's a "dummy" variable.

3. This one is all about affine functions (no one remembered the name...:(); but we also frequently call them "linear" functions, since their graphs are lines.

   Remember that linear functions are known by this behavior: if you double \(x\), then \(y\) doubles. That doesn't happen for this function: \[ f(x)=3-2x \] But differences are linear (if we double a change in \(x\), we double the change in \(y\)): \[ f(x_2)-f(x_1)=(3-2x_2) - (3-2x_) = -2(x_2-x_1) \]

   and so the average rates of changes, e.g.

   \[ \frac{f(x_2)-f(x_1)}{x_2-x_1}= -2. \] will always be -2 for this function! And so will instantaneous rates of change.

   Since graphs of affine functions are lines, the tangent line is everywhere the same -- the given line, with its given slope -- and so the slope of the tangent line is everywhere the same -- the slope of the original line!

   And so the derivative function is simply \(f'(x)=-2\):

1. what the sign means
2. what zero values mean (or may mean!)

All we have is annual data. But we can use that data to get an approximation to the rate of change of plastic production. We could use forward, backward, or centered differences, and I generated a graphic illustration of each, in Mathematica.

Let's have a look (and here's the Mathematica source.)

We need to do that because you have some of these functions in your IMath homework.

Let's look at

• \(f(x)=\frac{1}{x}\)
• \(f(x)=\sqrt{x}\)