Day 34, Mat128: Calculus A

• It will cover from 2.2, and 2.4 through 2.7. Sorry, I said 2.3 last time: that's
  • 2.2: The sine and cosine functions
  • 2.4: Derivatives of other trigonometric functions
  • 2.5: The chain rule
  • 2.6: Derivatives of Inverse Functions
  • 2.7: Derivatives of Functions Given Implicitly

  As you can see from that list, we're basically finding derivatives, in all kinds of cases and different ways.

  The rest of the semester is going to be about applications of those derivatives. Until now, about the only application we've seen is using a local linearization in place of (and as a good approximation of) a function.

• section 2.7 homework (here's a key with the solutions, but without hand work -- let me know if you have questions):

  You can use Desmos to plot the implicit curves:

  • Desmos for #6
  • Desmos for #7

• You may expect to see another limit definition problem on the exam. Do not forget the limit definition! It's the key to everything else.

• You may also expect a "proof/demonstration" of something.

• You should also expect some problems that look just like your homework.

This is a technique which helps resolve indeterminate limits, of the form \(\frac{0}{0}\), for example:

Suppose $h(x)=\frac{f(x)}{g(x)}$, where $f$ and $g$ are differentiable and $g'(x) \ne 0$ near $a$ (except possibly at $a$). Suppose that \[ \lim_{x \to a}f(x)=0 \;\;{\text{and}}\;\; \lim_{x \to a}g(x)=0 \] Then \[ \lim_{x \to a}h(x)= \lim_{x \to a}\frac{f(x)}{g(x)} = \frac{f'(a)}{g'(a)} \]

• Here's a key. Let's have a look at it!
• Median: 7.5
• I saw some strange things (this one on multiple papers): \[ \frac{1}{\sqrt{1-(-x)^2}} \longrightarrow \frac{1}{\sqrt{1+(x)^2}} \]
• Some thought that \((\ln(x))^2\) is equal to \( 2(\ln(x))\), which is false; what is true is that \[ \ln(x^2) = 2 \ln(x) \] (so long as \(x\) is positive). What is more generally true is that \[ \ln(x^2) = 2 \ln(|x|) \]

  This is really an illustration of the importance of understanding compositions, and understanding the nature of the beasts (classes of functions) that we work with. You need to become familiar with these classes: feel them in your soul; paint them onto your ceilings, or eyelids, or....

  • Polynomials
  • Rational functions (ratios of polynomials)
  • Power functions (e.g. roots)
  • Exponential functions (and their inverses, the logs)
  • Trigonometric functions (and their inverses)
  What if you were a small animal vet, and didn't understand
  • dogs and cats
  • birds
  • rodents
  • reptiles
  • parasites?

• 2.2: The sine and cosine functions
  • You should be able to graph these in a minute or less; zeros, peaks and pits, etc.
  • For the rules to apply to \(\sin(x)\), etc., \(x\) must be measured in radians (and not degrees). Degree measure represents a function composition: if you want to compute sines in terms of degrees \(\theta\), then we can rewrite \(\sin(x)\), where \(x\) is measured in radians, as \[ \sin(x) = \sin\left(\frac{2\pi}{360}\theta\right) \] When your calculator is in "degree mode", this is the conversion it is doing. And sadly, it is still called "sine". We should have given it a different name, because they're basically hiding that conversion factor of \(\frac{2\pi}{360}\) from you. (Were these the evil mathematicians of old? Or just lazy mathematicians... yeah, probably just lazy!)

  • These functions are their own fourth derivatives, and their second derivatives are their negatives (key to oscillations, because force is a function of acceleration, which is related to the second derivative of position!).
  • Remember your three essential trig identities: these were critical for deriving the derivatives of sine and cosine.
  • Sine and cosine are just shifts of each other, right? So you really only need to know one, in fact, but it's handy to have both at the ready.

    Just like it's not really necessary to create a whole new function (cosecant) for what is just the multiplicative inverse of sine, but it's sometimes convenient....

• 2.4: Derivatives of other trigonometric functions
  • These were just applications of the quotient rule, essentially.

• 2.5: The chain rule
  • This is a critical rule, which we did not prove, but merely motivated (using the local linearization, and crossing fingers!).
  • It's all about differentiating compositions, and compositions are so important!

• 2.6: Derivatives of Inverse Functions
  • First of all, a function is only invertible if it passes both the vertical and horizontal line tests.
  • But we create inverses even when the HLT fails; we simply restrict the domains, and try to remember that we've thrown away a lot of stuff, which may be important some day....
  • We used the chain rule and a trick to get the general form of the derivative of an inverse function: \[ \frac{d}{dx}\left(f^{-1}(x)\right) = \frac{1}{f'(f^{-1}(x))} \]

• 2.7: Derivatives of Functions Given Implicitly
  • Our first and most important example is a circle, which fails the vertical line test -- the curve is not the graph of a function.
  • But we can compute derivatives using an implicit formula, where we don't have an explicit formula for \(y\), but rather an implicit one, e.g. \[ x^2+y^2=1 \]
  • And when things are getting ugly, it's sometimes useful to recall that we may only be looking for the slope at a single point, and so substitute the point in ASAP.