Day 16, Mat128: Calculus A

• Next time we finish off our major tool development with "the doozy": the chain rule.

• You've got some new homework assignments:
  1. Read section 2.5 (the Chain Rule), and carry out the preview.

     It's our last rule that we'll derive using the limit definition (and some hand-waving): but it's a very important rule!

  2. I've posted a couple of IMath assignments, to give you differentiation practice (including some with trig functions).

• Midterm exam:
  • Key, part 1 and part 2

  • Max: 98.5
  • Average: 77.5
  • Median: 82.5
  • Distribution of grades:

    90s	80s	70s	60s	< 60s
    5	4	2	4	2

  • Why don't you look over the key, and if you have any questions, we'll go over those next time. If you have any pressing questions, however, you can bring those up now.

  • Also, feel free to discuss your exam and/or your situation with me.

• Midterm grades:
  • They'll be available by tomorrow morning.
  • I drop your lowest IMath and Quiz scores ("reserving" a drop or two for the second half of the semester).
  • I assume that your midterm reflects what you'll do on the final.
  • At this point I don't do any curving based on class participation, etc. It's just what it is. So if you have an 89, it's a B+ (not an A-, which it might become if you have been actively participating, attending, etc.).

• We have been building tools, including tools for differentiating the sine and cosine functions.

  Note one more thing, that we haven't mentioned to this point: while it is true that \[ \sin'(x) = \cos(x) \] one can also see that \[ \sin'(x) = \sin\left(x+\frac{\pi}{2}\right) \]

  This is a trig identity ($\cos(x)=\sin\left(x+\frac{\pi}{2}\right)$ -- so we don't even really need two functions -- one is just a shift of the other); but it says something really interesting: differentiation is like a shift operator for sine.

  Sine looks down the road by $\frac{\pi}{2}$ for the slope of the tangent line at a point. Can you see that? And isn't that weird?:)

I'll start by handing out a 2.4. worksheet, and partnering you up. Then we'll go over the tangent together, using the quotient rule.

\[ \tan(x)=\frac{\sin(x)}{\cos(x)} \] So obviously there are some places where this function is going to "blow up" (or blow down).

Note that this figure shows that the derivative is consistent with the graph of the function:

• Tangent is always increasing wherever it is continuous, and its derivative is always positive there.
• Tangent is odd, and its derivative is even.
• Tangent and its derivative are periodic, with period $\pi$. (Note that this is only half the period of sine and cosine).
• It looks like the tangent has slope 1 at $x=0$.

I want to show you how we use tangent lines to find roots of functions -- that is, \(x\)-values where the function takes the value 0 (e.g., and especially, zeros of derivative functions), via a Mathematica demonstration of Newton's method

Suppose that you want to find the roots of the function \[ f(x)=x^2-2 \] We know that the roots are plus or minus \(\sqrt{2}\) -- in fact, this "root-finding" is a method for computing the square root of 2!

Suppose that we start with a "bad guess": \(x=3\). We shoot the tangent line (what is it?), and it points us to a new (and hopefully better) guess: 1.83333, as seen in this graphic:

Now we do what mathematicians love to do: we do it again! (Until satisfied).