1) Functions can be used to predict where a thrown ball will be at any given time. A function relates an input variable like time (x) to an output variable like the ball's position or speed (f(x)). 2) A function has boundaries for its input (domain) and output (range). For a function describing a ball's speed over time, the domain is all time after being thrown and the range is the minimum and maximum speeds. 3) Sometimes a function's output gets infinitely close to but never exactly reaches a number, known as the limit of the function.

1. 1)
When you throw a ball into the air, how do you know when and where to catch it?
If you threw the ball the same way every time, you could predict exactly where
the ball would be at any given second. You express this prediction with math, by
using something called a function. Functions are written in the form f(x), where
x represents the number you’re measuring (in this case, how many seconds since
you threw the ball), and f represents what happens whenever x changes. What else
can you predict using a function?
2)
Any given function has boundaries to both x and f(x). The boundary of x is
called the domain, and the boundary of f(x) is called the range. For instance,
if f(x) represents the speed of a ball after you throw it, and x represents time
since you threw the ball, then the domain begins with the time you throw the
ball and extends forever into the future, and the range is the maximum and
minimum speed the ball attains.
3)
Sometimes, a function’s value gets very close to a certain number, but never
reaches it exactly. This is known as the limit of the function at a particular
value. Let’s say that you’re walking to a wall. You go halfway to the wall, and
then you stop. Then you go halfway from your new position to the wall, and then
you stop again. Think about it; will you ever get to the wall?
4)

2. Pretend I start walking along a track. After one second exactly, I pause for an
instant. Nobody sees me pause, because I immediately start to accelerate faster.
My speed in miles per hour can be represented by a function with three
equations. At x less than 1, f(x) = x. At x greater than 1, f(x) = 2x – 1. At x
equals 1, f(x) = 0, because I pause. However, the limit at x equals 1 second is
f(x) = 1, because I’m walking at one mile per hour at every time approaching one
second.
5)
You can compute limits even if they seem to get infinitely small. You do this by
assigning letter terms to values that would otherwise be difficult to consider.
For instance, if you have a ship that’s accelerating forever through space, you
can say, “I’m going to call the distance that ship travels “x.” If, for example,
the space ship unexpectedly slows to half its speed, then you can call the
resulting distance “x/2.” Mathematics is about more than numbers; it’s about
creative use of letter terms like this.
6)
You can also look at the limit of an increment of change of any sort. In the
previous problems, we looked at the limit of increments of movement in relation
to total distance moved. We considered the value of a function as the increments
of movement approached infinity. However, we can also look at the limit of
increments of time.
More commonly in the world, we look at an amount of time that is infinitely

3. close to zero seconds, in which an infinitely small change occurs. In calculus,
we would call that time ‘dx,” and the change that happened “dy.”
7)
The average speed of an object is equal to the distance it covers over time. The
instantaneous speed of an object is equal to dy/dx, where dx represents an
infinitely small span of time, and dy represents an infinitely small distance
traveled. At any given time, dy/dx may be different. A function that represents
all the possible values of dy/dx of a given function is called the derivative of
that given function. You write the derivative by putting an apostrophe in front
of the f, so the derivative of f(x) is f’(x)
8)
The acceleration due to gravity is ten meters per second per second. This means
that when something is falling through the air, its downward speed increases ten
meters per second every second. Speed is a change in distance over time, and
acceleration is a change in speed over time. This means that acceleration is a
change in distance over time, over time.
9)
Acceleration can be any value that you want it to be. When you’re speeding up or
slowing down in a car, you’re accelerating. Similarly, when you kick a ball,
you’re making the ball accelerate. You make it stop moving toward you and start
moving away from you. If you apply a constant force to an object, it will
accelerate at a constant rate, dependent upon how much force you apply to it and

4. how heavy the object is. This relationship is ordered by the function
Acceleration equals Force / Mass, and is known as Newton’s Second Law. The Law
is also written Force = Mass * Acceleration.
10)
While the function for speed over time due to gravity is f(x) = 10x (because it
increases ten meters per second every second), the function for distance
traveled over time due to gravity is 5x^2. You can confirm this with the values
from problem 8 from a couple days ago.
The derivative of distance is speed. Similarly, the derivative of the function
5x^2 is 10x. You get that by multiplying the function by the number in the
exponent (in this case 2) and decreasing the exponent by 1. You must do this
with every x-term in the function, and you can do it with any exponent, as long
as you follow the basic rule. This is known as the Power Rule in calculus.
11)
If you’re trying to find the rate of change of multiple distance variables, but
you only know their relationship to one another, you can take the derivative of
those distance variables with respect to time. For instance, if you know a given
distance x will always be twice the distance of a given distance y, you know
that the given speed dx/dt will also always be twice the speed of dy/dt.
Whenever you are given a rate of change for a particular variable in a problem,
you know the quantity of the derivative of said variable.

5. 12)
Even if the problem seems slightly more difficult, the basic strategy is the
same. Identify the variables, identify the rates of change you need to find, and
find a way to connect them. If you’re trying to find the rate of change of
height given a changing volume, for instance, you need to find the relationship
between h, dh/dt, V, and dV/dt. Don’t be scared away by constants! Just look for
the relationship between variables, and differential calculus will be a cinch.
13)
What if you know the rate of change of a quantity, but you want to find the
quantity itself? Let’s use a more real-life example – say you know how quickly
something accelerates over a given span of time, but you want to find how far it
goes in that time. If a car accelerates from zero to sixty in three seconds, for
instance, how far has it gone? To do this, we need to take the derivative in
reverse. This is known as the antiderivative, or definite integral of a
function.
14)
We can approximate integrals if we only have a data set for the function we’re
trying to integrate. For instance, if a car is accelerating sporadically, we can
take its speed at a variety of points to determine more or less how far it’s
gone. We take the average of the speed between measurements, and then multiply
it by the time between measurements to get the distance traveled in that time.
Then we add all the distances. Incidentally, this turns into the integral as the
time between measurements approaches zero, and number of measurements approaches

6. infinity.
15)
Integrals are the addition of an infinite number of rates of change multiplied
by infinitely small increments of change. We write integrals with a ∫ sign,
placing the boundary numbers at the top and bottom of the curve. After the
curve, we write the rate-of-change function we’re integrating multiplied by a
differential (like dt or dx – the d simply indicates that it’s an
infinitesimally small increment of whatever variable we’re using). Use the idea
of the integral to divide a problem into more manageable chunks.
16)
For integral word problems, the function you’re given is for the rate of change
of a certain variable, and you want to find the total quantity of that variable
over a given period of time. Just remember this, and you’ll understand the
fundamentals of basic integrals.
17 and 18)
When you think about integrals, remember to consider what the d value means at
the end. The integral symbol is like saying, “add the value of every real number
in this following function between these two bounds multiplied by an infinitely
small increment.” This can be used to deal with ideas like density, which
represents how much stuff is in any given space. To get density, you divide a
total space into an infinite number of parts. Then add up all the amounts of
stuff in each infinitely small part multiplied by the size of each part.

7. 19)
We write dy/dt as more of a symbol rather than a relationship between two
variables. However, if we consider that dy just means an infinitesimal change in
the y variable, and dt just means an infinitesimal change in the t variable,
depending on how small the change we make to a given variable, we can use
derivatives to easily approximate how that will affect a related variable. When
we separate a derivative dy/dt into its component parts, those parts are known
as differentials.
20)
If the rate of change of a function is determined by the value of the function
itself, we can write dy/dt = ky where k is a numerical constant and y is the
value of the function. Because we know how to use differentials in integrals, we
can rearrange the values to get k dt = 1/y dy. If we integrate both sides of
this equation, we find the total y value change over a total time value. It’s
too difficult to explain here, but just know that the derivative of ln y = 1/y.
So if you’re integrating the function 1/y, know that it transforms into ln y.
21)
Note that y can be replaced by any function. According to Newton’s law of
cooling, objects lose heat proportional to the difference between their own
environment and their own temperature. In the language we know, this means that
the rate of change of heat (f(t)) equals a constant k times the difference
between an object’s heat (given by f(t) and the temperature of the environment,

8. or df(t)/dt = k(f(t) – E), where E is the temperature of the environment.
Note that to reverse the natural log “ln” function, simply put e to the power of
the natural logarithm function. You probably also have an ‘e’ button on you
calculator. For instance, “e” to the power of “ln 9” equals 9.
22)
A philosopher named Zeno in ancient Greece came up with a number of paradoxical
situations. They were designed to prove that people never get where they’re
going, but only get really, really close. The most famous of these is known as
the paradox of “Achilles and the turtle.” Achilles was a famous Greek hero, so
he let a turtle get a head start in a race. Zeno argued that Achilles could
never catch up, because every time he got to where the turtle used to be, the
turtle would be a few steps ahead, and so forth. We can use the idea of limits
to prove Zeno wrong.
( http://www.youtube.com/watch?v=RcCYshlgCZY , http://www.britannica.com/EBchecked/topic-
art/442556/321/Zenos-paradox-illustrated-by-Achilles-racing-a-tortoise)
23)
Any time you’re trying to figure out how fast a number is changing – whether
that number represents something’s speed, whether it represents something’s
size, or anything else – you’re trying to find a derivative. If a number is
changing, you can devise a function to show that number’s value depending on the
time. You apply the derivative to that function to get the derivative function.
Conversely, any time you’re trying to figure out how much a number has changed

9. over a period of time, or if you are given the information for how much a number
changes over a period of time, you need to take the integral. Integrals and
derivatives are two sides of the same coin. One goes up, one goes down.
( http://www.mathworks.com/matlabcentral/fx_files/15310/1/Snaptraj.png ,
http://people.mech.kuleuven.be/~bruyninc/blender/pictures/trapacc_pos.png ,
http://rlv.zcache.com/a_definition_of_jerk_physics_keychain-p146001755603171307qjfk_400.jpg
)
24)
Almost every concept in Newtonian mechanical physics, which governs the way
everyday objects behave, stems from Newton’s Second Law. An object’s speed
changes proportional to how much force is put on the object, inverse to how much
inertia the object holds. Put more simply, force equals mass times acceleration.
Multiplication and division don’t necessarily mean numbers – they mean
proportional conceptual physical relationships. We invent units of measurement
to fit nicely with these relationships. For instance, what you think of as your
“weight” is more than just a number on a scale. It is your inertia in the world.
How much effort does it take to move you?
Nearly every other concept in Newtonian mechanics involves taking derivatives or
integrals of these three basic concepts: effort, inertia, and movement, and
finding relationships between new concepts to form even more.
(http://www.sparknotes.com/physics/workenergypower/workpower/section4.rhtml )

10. 25)
Two of the most important concepts in physics are momentum and impulse. Momentum
is the integral of force – that is, it is the sum total of the force added to an
object over a given period of time. You find momentum in one of two ways –
multiply an object’s mass by its velocity, or, more complicatedly, multiply the
force on an object by the amount of time the object feels that force. You can
use the second way to find the change in momentum, also known as the impulse.
When considering collisions, consider Newton’s Third Law: change in momentum is
always conserved.
( http://screwattack.com/blogs/Wandering_Swordsmans-blog/The-Physics-of-Gaming-Momentum-and-
Impulse
, http://springfield.news-leader.com/specialreports/hammonsfield/nie/hitting-physics.jpg)
26)
Real-life applications of differential problems are almost always in the realm
of exponential growth. And since we live in a world in which things can’t grow
forever, rates of change of a variable aren’t just determined by that variable’s
quantity. They are also bounded by another quantity. For instance, in the case
of people getting sick due to virus, the more people that have the virus one
day, the more people get it the next day. But as time wears on, people develop
antibodies or succumb; the same person that got the virus can’t get it again.
Hence, rate of change of infected people is also bounded by the total number of

11. people.
We write this equation as dy/dt = ky(n – y), where y is the variable in question
and n is the upper bound. If we integrate this and do some algebraic work, we
get
ln y – ln (n – y) = nkt + C
To solve for y, we do a little more work and get
f(y) = n/(1 + ce^-nkt)
(c = e^-C)
You can do the algebra yourself, but it’s long and difficult. Simply remember
this formula, use it creatively, and you will know how to use differential
equations to solve real life problems like the following bonus question.
(http://www.youtube.com/watch?v=DjlEJNfsOKc)