- Your 3.3 homework is returned
- #9 -- straightforward -- but you need to know how to factor that quadratic!
- #20 -- there are infinitely many functions, but yours needs to meet the criteria. Mallory?
- #52 -- Thomas
- Your exam on Monday will cover sections 2.6, 2.7-3.1.
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, antidifferentiation 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 antiderivatives 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 of those conditions.
- Examples:
- 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 spring satisfies Hooke's law:
- 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 $f$. 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