- I hope that you had a nice, peaceful, restful holiday.
- Your exam is returned. You
probably want to think about it as much as I do....:(
- Curve: Add 4 to your score.
- Nonetheless, results were atrocious. We'll try to fix things this way:
Your final will replace your lowest hourly test score. For most of you, this was your lowest test score. But a few of you did well.
That puts more pressure on the final, but it allows us to essentially eliminate the impact of this exam, if you make improvement down the road.
Now: how to do that?
- Are you reading your book? Even if you don't read
it all, at least look over the example problems, to see
what kinds of problems the technique(s) of a section
can help us to solve.
- Take notes
You might have noticed that the fourth problem on the exam was covered in its entirety the Friday before the exam. I thought that it would be fun to see who followed along.
5 of you got it correct; 12 out of 22 received a 50% or below.
Take notes, then review the notes before an exam.
- It's late in the game, but many of you are still
afflicted with horrible algebra issues. Two that are
worth mentioning are
- $(a+b)^2 = a^2+2ab+b^2$
(and, in particular, $(a+b)^2 \ne a^2+b^2$). This one's a killer, and several of you fall victim to it. I've seen it time and again this semester. You've got to get this out of your system, because it's sand in your gears.
- Common denominator: $\frac{a}{b}+\frac{c}{d} = \frac{ad+bc}{bd}$
- $(a+b)^2 = a^2+2ab+b^2$
- An odd algebraic thing is that some of you don't
seem to know that $-1 \cdot a = -a$. That is, that
multiplying by -1 produces the additive inverse (that
thing which, added to $a$, produces 0). So $-1 \cdot
\sqrt{3} = -\sqrt{3}$.
- Many of you froze when given the equation of a
circle: you didn't even plot a few points. Stick in
some $x$-values, and produce some $y$-values. It's that
easy. Pretty soon you'd recognize a circle, perhaps.
Also, being given the fact that the tangent line passed through the point (1,-4), some of you didn't plot THIS point! It's on both the circle and the tangent line. Seems like you'd want to graph that point -- it's pretty important!
And if your tangent line doesn't go through that point? Comment on that fact. Let me know that you know that you've made a mistake -- even if you can't find it.
Then label your graph.
- Only 7 of 22 could produce the volume of a cube
(as a function of $s$). Some of you came up
with $V=lwh$ -- but it's a cube! So
$V(s)=s^3$. A cube has six sides, each a square
of area $s^2$ -- so the surface area of a cube
of side length $s$ is $A(s)=6s^2$.
You need to have a reservoir of mathematics facts at your fingertips. Another on this exam was the equation of a circle.
I've emphasized the importance of knowing classes of functions (polynomials, rational, trig, etc.); you also need to have some basic mathematical facts at your fingertips.
- Are you reading your book? Even if you don't read
it all, at least look over the example problems, to see
what kinds of problems the technique(s) of a section
can help us to solve.
- Take a look at the key, and come with questions next time.
- Curve: Add 4 to your score.
- You have a homework due from 3.4. How are you doing on asymptotes?
So now we know how to handle closed, bounded intervals; now let's talk about what happens when we let $x$ become unbounded. How will a function behave as $x$ races off to $\infty$ or $-\infty$?
- Examples of functions without finite limits at infinity:
- non-constant polynomials (go to infinity, in the positive or negative sense)
- the usual trig functions (while sine and cosine are bounded, but don't settle down to some nice limiting value)
- But rational functions may have horizontal asymptotes, as do
exponential functions.
Another example with a horizontal asymptote is knowledge as a function of time -- #51, p. 222. We might guess that accumulated knowledge in studying for an exam looks something like this:
We might imagine that this physical process becomes less productive from hour to hour as the evening wears on (the law of diminishing returns).
Other Examples:
- The hyperbola: $f(x)=\frac{1}{x}$
More generally, If $r>0$ is a rational number, then
$\lim_{x \to \infty} \frac{1}{x^r}=0$ If, furthermore, $x^r$ is defined for negative numbers, then$\lim_{x \to -\infty} \frac{1}{x^r}=0$ - The most general form of an hyperbola is this: $f(x)=\frac{ax+b}{cx+d}$
- What is its horizontal asymptote?
- Where is its vertical asymptote?
- A rational function will have a horizontal asymptote
any time that the degree of the denominator equals or
exceeds that of the numerator: if
$r(x)=\frac{p(x)}{q(x)}$ then if the degree of q exceeds that of p, there is a horizontal asymptote, and the value of the asymptote is given by examining the approximating function given by the ratio of leading terms alone.
- Interestingly enough, if the degree of the numerator ($n$)
exceeds that of the denominator ($m$), then the rational function
ultimately looks like a polynomial of degree $n-m$: for
example, consider the function
$f(x)=\frac{x^2-1}{x}$. Its graph will ultimately look a lot
like the function $g(x)=x$ (which one can determine by
(gasp!) long division....
This is called a "slant asymptote" (not a horizontal asymptote, for obvious reasons! If you get far from the origin, then the difference between the two functions falls away.
We can then replace the more complicated with the simpler.
We use this idea in physics all the time: we assume that gravity is constant at the surface of the Earth. In fact, it varies as distance to the center of the Earth, but we're so far away that we can take this as a constant (its value at about 4000 miles -- our distance to the center of the Earth). We're far enough from the origin (the center of the Earth) that we treat acceleration due to gravity as locally constant.
- The usual rules of limits apply, with the caveat that one
must be careful when dealing with $\infty$. It
is not a number, and cannot be added to anything else (e.g. it
is not valid to write that $\infty-\infty=0$ -- for example,
$x^2$ and $x$ both go to infinity, but their difference,
$x^2-x$, also goes to infinity).
We can say, however, that $\infty*\infty=\infty$, that $\infty*1=\infty$, that $\infty/1=\infty$, etc. So some of the usual rules apply (and hopefully make sense!).
- The hyperbola: $f(x)=\frac{1}{x}$
- Examples:
- #7, p. 234
- #11
- #16
- #44 (re-express the function first, and use symmetry!)
- #49 (factor first, and use symmetry!)
- Whereas differentiation is essentially a
mechanical exercise, anti-differentiation may require insight, and broad
exposure to lots of different functions. One asks oneself:
"Do I know a function whose derivative looks like this?" - Definition:
- All anti-derivatives are the same up to a constant:
- Differential Equations:
- A differential equation is an equation linking a function and its derivative(s).
- The simplest differential equations look like this:
$\frac{dy}{dx}=g(x)$ - Usually an initial condition (or conditions) will
also be provided -- so that the solution ($y(x)$) has
value $y=y_0$ at $x=x_0$.
- Our job then is to find the function $y(x)$ which
satisfies both
- the differential equation, and
- the initial condition(s).
- Examples:
- A simple example:
- $\frac{dy}{dx}=2x$
- $y(0)=1$
- A spring satisfies Hooke's law:
$F=-kx$ where $k$ is the spring constant (characteristic of the particular spring). Newton said that
$F=ma$ If we put those two laws together, we get
$\frac{d^2x}{dt^2}=\frac{-k}{m}x$ For initial conditions, we should give initial position and initial velocity.
- Gravity is essentially constant at the Earth's
surface: we can use a differential equation to deduce
the formula for the height h of an object as a
function of time.
The acceleration due to gravity at the earth's surface is often taken as approximately
$g=-9.81m/s^2$ where the minus sign indicates that gravity's inclination is to make your height decrease (to 0 at the center of the earth, if gravity could).
Your force on the earth (or the earth's force on you!) is your weight, $m g$.
Since
$F=ma$ If height is given by $h(t)$, then
$m\frac{d^2h}{dt^2}=mg$ or
$\frac{d^2h}{dt^2}=-9.81$ Once again, we should give initial position and initial velocity.
Now you ask yourself this: do I know a function whose second derivative is a constant? And the answer is YES! -- a quadratic function. The specific quadratic function is
$h(t)=\frac{-9.81}{2}t^2+v_0t+h_0$ (Check and make sure!). Then, once you've specified $h_0$ and $v_0$, the motion of the projectile is fully determined.
- A simple example:
- A differential equation is an equation linking a function and its derivative(s).
- This process turns out to be crucial for deciding what $f$ looks
like, given $f'$ (or higher derivatives). Remember when we were looking
at a graph of $f'$, and trying to guess what $f$ looks like? Well,
we're not going to need to do that, once we know the anti-derivative.
- Some examples:
- #3, p. 273
- 20
- 33
- 41
- 64