Day 6, Mat128: Calculus A

What you may not do is use them for anything other than that. I'll pass around a sheet that you will sign, indicating that you are clear on that.

• Let's start with the preview activity: the preview activities are often the focus of the rest of the section, so they're worth doing.

• Key ideas are limits, from the left and from the right, and then a definition of instantaneous velocity. There's an important definition of limit in this section, as well (Definition 1.2.2). Let's take a look at that first:

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  Given a function \(f\) a fixed input \(x=a\) and a real number \(L\) we say that \(f\) has limit \(L\) as \(x\) approaches \(a\), and write \[ \begin{equation*} \lim_{x \to a} f(x) = L \end{equation*} \] provided that we can make \(f(x)\) as close to \(L\) as we like by taking \(x\) sufficiently close (but not equal) to \(a\). If we cannot make \(f(x)\) as close to a single value as we would like as \(x\) approaches \(a\) then we say that \(f\) does not have a limit as \(x\) approaches \(a\).

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  And because we want to talk about what's going on from the left of \(a\) and what's going on to the right of \(a\), we considered the so-called "one-sided limits": from the left:

  \[ \begin{equation*} \lim_{x \to a^-} f(x) = L \end{equation*} \]

  and from the right:

  \[ \begin{equation*} \lim_{x \to a^+} f(x) = L \end{equation*} \]

  So in this graph,

  

  we can say that \[ \begin{equation*} \lim_{x \to 1^-} g(x) = 3 \end{equation*} \]

  and from the right:

  \[ \begin{equation*} \lim_{x \to 1^+} g(x) = 2 \end{equation*} \]

• Then we'll revisit the average velocity/instantaneous velocity from last time:

  \[ \begin{equation*} IV_{t=a} = \lim_{b \to a} AV_{[a,b]} = \lim_{b \to a} \frac{s(b)-s(a)}{b-a}\text{.} \end{equation*} \]

  or, equivalently, letting \(b=a+h\) (that is, \(b\) is a point displaced from \(a\) by \(h\)),

  \[ \begin{equation*} IV_{t=a} = \lim_{h \to 0} AV_{[a,a+h]} = \lim_{h \to 0} \frac{s(a+h)-s(a)}{h}\text{.} \end{equation*} \]

  In the limit, the secant line method approaches the tangent line as \(h \to 0\):

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  The tangent ("touching") line osculates ("kisses") the curve at this point.

• Let's recall how this definition works in the case of that function we looked at last time, and the instantaneous velocity at the point $a=2$: \[ s(t)=64-16(t-1)^2 \]

  Then we'll have a look at Exercise 1.2.4

• One of the interesting ideas is that of a discontinuity in a function: not all functions are smooth, or even connected! What that means is that the definition above may not make sense in some cases. What can go wrong? Three things must happen:
  1. The limits from the right and left of the point \(x_0\) exist, as does the function at \(x_0\);
  2. The limits are equal, so that the limit exists at \(x_0\);
  3. The limit \(L\) is equal to the function value at \(x_0\): \[ \lim_{x\to x_0}f(x)=L=f(x_0) \]

  We interpret the derivative as the slope of the tangent line (the limit of secant lines!) at a point. You can frequently look at a graph and see where things go awry. For example, if there's no tangent line at a point, then there's no instantaneous velocity there.