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- Homework 3.5 is returned: graded #1 and 46
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An important example of optimization is in finding regression lines (or other functions).
In that problem, we try to find parameters $m$ and $b$ such that the data, $\{x_i,y_i\}_{i=1\ldots N}$, is best fit by a linear function (remember the most important functions in calculus!): \[ E(m,b)=\sum_{i=1}^{N}(y_i-(mx_i+b))^2 \] This is the same as minimizing the squared error, $E(m,b)$, between the data values $y_i$ and the model values, $mx_i+b$, at locations $x_i$.
You'll notice that there are two parameters, $m$ and $b$, that we need to find (creating a linear model), so this is a bi-variate (rather than uni-variate) problem -- but calculus finds the solution for us, nonetheless.
The most recent Charles David Keeling Mauna Loa CO2 Data is an example of even more complicated modeling (quadratic or exponential). Both cases are ultimately solved by minimizing an error function, using calculus.
- If we know the rate at which something proceeds, e.g. the rate at
which we're pumping carbon into the atmosphere, then we may want to
know how much we've actually pumped into the atmosphere over a given
period. This is the job of anti-differentiation: given the rate of
change of $p(t)$, $\frac{dp}{dt}$ from $t=0$ to $t=T$, find the amount
pumped in over the same period: $p(T)-p(0)$.
- Whereas differentiation is essentially a mechanical exercise,
anti-differentiation 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?" Note: I tend to hyphenate "anti-differentiation", whereas our author doesn't. I sometimes do, and sometimes don't. But I actually think that it deserves the hyphen....
- Definition:
A function $F$ is called an anti-derivative of $f$ on an open interval $I$ if $F^\prime(x) = f(x)$ for all $x$ in $I$. - 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 of those conditions.
- Examples:
- Two-percent of the women in a population (half men
and women) give birth each year (assume single
births). Hence, in the absence of deaths or migration,
$\frac{dP(t)}{dt}=.02P(t)$
Do we know a function which is its own derivative? Yes: 0! But that's not the answer (although you see that the differential equation is satisfied for that).
It turns out that exponential functions are proportional to their own derivatives, and so populations tend to grow exponentially in the absence of deaths (or other constraints).
In this case, $P(t)=P_0e^{.02t}$, where $P_0$ is the initial population, and $e \approx 2.718$ is arguably the most important constant in calculus.
- 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.
"Do I know a function whose derivative(s) make this true?" - The acceleration due to 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.
"Do I know a function whose derivative(s) make this true?"
- Two-percent of the women in a population (half men
and women) give birth each year (assume single
births). Hence, in the absence of deaths or migration,
- 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 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 -- the anti-derivative of a sum is the sum of the 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