## Functions and Their Applications

When you buy your favorite type of chocolate you are charged by how much you want. The cost is therefore a function, because it depends on the number of ounces of chocolate you buy. That is, for each input (weight in ounces), there is one output (cost, say in dollars). By contrast the price of a house is not a function of the lot size on which a house is built because houses on the same size lot may sell for many different prices.Describe an everyday situation between variables that is a function. Clearly explain why the relationship a function.Describe an everyday situation between variables that is not a function. Clearly explain why the relationship is not a function.

## Time Perception as We Age

1. Our theory is that your perception of a length of time is inversely proportional to your age. Using the variables above, how would you state the theory in a mathematical equation?2. Using this theory, how long will a year seem when you are two times older than you are now? To investigate, let ptime1 be how long a year seems now, and let ptime2 be how long a year will seem when you are twice as old. Pick a number for ptime1 (maybe a year feels like 157 days to you, for instance), and then calculate ptime2. Pick a number for ptime1 that is different from other students.3. Compare your ptime1 and ptime2. What is the relationship between them? Would this relationship change depending on how old you are now? Take a look at other student answers. Do you see a pattern? Can you prove that pattern mathematically?4. Let’s say that ptime1 is how long a year seemed when you were one year old. Let’s say ptime1 = 365 “time units.” Using age as our x variable and ptime as our y variable, use this online graphing tool to graph our theoretical mathematical model of how your perception of time changes over the course of your life. Here is a short video on how to use the basics of the graphing tool.5. Use “Settings” within the graphing tool to create a window that covers a typical human lifespan (for age) and a reasonable span of “time units” (for ptime). Attach a screen shot of your graph with your post. (You can use your Print Screen key to capture a screen shot and paste it into a Word Document).6. Do you think our theory is an accurate model of what you experienced over the course of your lifetime so far? If so, why? If not, why not?

## If you want to put a 4 x 8 piece of plywood through a 3 foot square opening in your ceiling by turning it diagonally, is the opening big enough? Use a 45-45-90 triangle since it is a square

If you want to put a 4 x 8 piece of plywood through a 3 foot square opening in your ceiling by turning it diagonally, is the opening big enough? Use a 45-45-90 triangle since it is a square

## How do I use DeMoivre’s theorem to solve z3-1=0?

If z3-1=0, then we are looking for the cubic roots of unity, i.e. the numbers such that z3=1.
If you’re using complex numbers, then every polynomial equation of degree k yields exactly k solution. So, we’re expecting to find three cubic roots.
De Moivre’s theorem uses the fact that we can write any complex number as ρeiθ=ρ(cos(θ)+isin(θ)), and it states that, if
z=ρ(cos(θ)+isin(θ)), then
zn=ρn(cos(nθ)+isin(nθ))
If you look at 1 as a complex number, then you have ρ=1, and θ=2π. We are thus looking for three numbers such that ρ3=1, and 3θ=2π.
Since ρ is a real number, the only solution to ρ3=1 is ρ=1. On the other hand, using the periodicity of the angles, we have that the three solutions for θ are
θ1,2,3=2kπ3, for k=0,1,2.
This means that the three solutions are:
ρ=1,θ=0, which is the real number 1.
ρ=1,θ=2π3, which is the complex number -12+√32i
ρ=1,θ=4π3, which is the complex number -12-√32i

## How do I find the sine of the angle between two vectors?

I’m assuming you either have the components of the vectors, or their magnitude and angle. You will need to apply the cross product to get the sine.
Honestly, it’d be much easier if someone explains it to you on paper, it’s too long to type here and may seem hard, but it’s pretty easy.

## How do you write the notation for end behavior?

You write the notation using the limit notation.
So first of all before you write the limit notation you need to look at the degree of the polynomial and determine if the graph is odd or even.
Let’s take a look at the below polynomial
f(x)=x3-3×2-x+2
The degree of the polynomial is 3 which is odd number. So the rule here is that if the degree is odd then the will be opposite and if it is even it will be even.
limf(x)x→∞=∞
limf(x)x→-∞=-∞
The notation is read as, “The limit of f(x) as x goes to [infinity] is …[infinity]”
Here is a graph of a the function and as you can see the goes the opposite direction.
graph{x^3-3x^2-x+2 [-10, 10, -5, 5]}
Now let’s do even one.
x2+1
The degree is 2 so the function is even. and you write the limit function as below.
limf(x)x→∞=∞
limf(x)x→-∞=∞
Let’s graph the function and you can see the in the parabola so it doesn’t matter what number you pick for x it will always be positive thus the end behavior will go the same direction.
graph{x^2+1 [-25.66, 25.65, -12.84, 12.83]}

## What is the equation of a sphere in standard form?

This is because the sphere is the locus of all
points P(x,y,z) in the space whose distance from C(xc,yc,zc) is equal to r.
So we can use the formula of distance from P to C, that says:
√(x-xc)2+(y-yc)2+(z-zc)2=r and so:
(x-xc)2+(y-yc)2+(z-zc)2=r2,
x2+2(x)(xc)+x2c+y2+2(y)(yc)+y2c+z2+2(z)(zc)+z2c=r2,
x2+y2+z2+ax+by+cz+d=0,
in which
a=2xc;
b=2yc;
c=2zc;
d=x2c+y2c+z2c-r2;
So:
C(-a2,-b2,-c2)
and r, if it exists, is:
r=√x2c+y2c+z2c-d.
If the center is in the Origin, than the equation is:
x2+y2+z2=r2,