- Homework returned: Graded #9, 12, 54
- Your quiz is returned.
- leading terms are all you need to consider: look at their ratio
- This function looks like $x$ as $x \to \pm \infty$.
- This function is odd.
- Your 3.5 homework is due today.
- You have two assignments to hand in at the beginning of next week. Sorry to keep you hopping....
- Whereas differentiation is essentially a mechanical exercise,
antidifferentiation may require insight, and broad exposure to
lots of different functions. One asks oneself this essential
question:
"Do I know a function whose derivative looks like this?" - Definition:
A function $F$ is called an antiderivative of $f$ on an open interval $I$ if $F^\prime(x) = f(x)$ for all $x$ in $I$. - All antiderivatives are the same up to a constant:
- I gave you a handout -- let's take a look at that.
- 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:
$F=-kx$ That is, the restorative force $F$ is proportional to the displacement $x$ from its equilibrium position. Proportionality constant $k$ is positive, so the negative sign says that the force is in the direction opposite the displacement (that's why the force is called "restorative" -- it seeks to restore the spring to equilibrium).
Newton said that
$F=ma$ "Force equals mass times acceleration".
The acceleration $a$ is the second derivative of $x$ with respect to time $t$. If we put that all together for the spring, 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 $h(t)$ of an object as a
function of time $t$:
$g=9.81m/s^2$ Thus, near the surface of the Earth, we get
$\frac{d^2h}{dt^2}=-g$ Once again, we should give initial position and initial velocity.
- A spring satisfies Hooke's law:
- 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 how to find anti-derivatives. We may find them
analytically, or we may find them numerically -- but we can generally
find one, one way or the other!
- Some examples:
- #3, p. 273 (find ALL anti-derivatives)
- #20 (find THE specific anti-derivative)
- #33 (anti-differentiate twice! First find the derivative, then $f$.)
- #41 (find THE specific anti-derivative of $2x+1$ passing through (1,6), and evaluate it at x=2)
- #70
- Suppose that you clock your speed
at each minute along a 10-minute trip, in mph:
Estimate the distance travelled.minute 0 1 2 3 4 5 6 7 8 9 10 speed (mph) 0 30 45 30 70 65 70 70 45 30 35 Note that there are 11 speeds: your initial speed (0), and your speed as you "cross the finish line" (35), at the 10 minute mark.
- Dirt!
$d=r*t$ - Assumes continuity of underlying speed function
- We might fit straight line segments between data points: this gives rise to the so-called "Trapezoidal rule".
- Other useful rules are rectangular: Left/Right/Midpoint rules:
- relax continuity! Use rectangles...
- also called rectangle rules: lrr, rrr
- Estimates:
- LRR: d = 45.5*10 mph*minutes = 7.58 miles
- RRR: d = 49*10 mph*minutes = 8.17 miles
- Trapezoidal rule = 47.25*10 mph*minutes = 7.88 miles
- Notes:
-
- Whenever you have two estimates, you have a third:
- the trapezoidal rule is the average of left and right rectangle rules.
- Each of these estimates corresponds to computing a rectangle height (an
average speed), and then multiplying by the total time (10
minutes). Hence we're using the "dirt" formula, and computing the
average (the "area under the curve", even though there's no curve!) in
several different ways.
- Basically, every function can be used to create a table like this, and
then we can use these strategies.
- Example:
- Often (as in the table above) we have data located at equal intervals ($\Delta{x}$). When you have a curve (or a formula) for f, you can take the time
intervals shorter and shorter, until you get as close to the area as you
need:
(notice that $a+N\Delta{x=b}$ that is, that $\Delta{x}=\frac{b-a}{N}$)
Then, in fact,
- Examples:
- #1, p. 293
- Generalizations?
- The area can be calculated by finding an average value on the
interval, times the length of the interval:
Area under the graph of f from a to b = $\bar{f}(b-a)$ where by $\bar{f}\mbox{}$. we mean an average value of f. - The area can be calculated by chopping the interval [a,b]
up into smaller and smaller chunks, and then using rectangular approximations
to the small areas.
- In our work so far, the intervals into which the interval is chopped have been equal-sized: no reason to stick with that.
- If the size of the intervals goes to zero, then the approximations get better and better, until they're perfect!
- The area can be calculated by finding an average value on the
interval, times the length of the interval:
- Some more examples:
- #6
- #15