10.5 Parametric Equations

Parametric Equations Definition:

If f  and g are continuous  functions of t on an interval j, the set of ordered pairs (f(t), g(t)) is a plane curve B. The Equations

are parametric equations 

(taken from the text book)

What this means is that for a continuous function, where x, and y can be defined by a third variable, called a parameter and represented by as t. Then it is a parametric equation.

The Purpose of Parametric equations:

If you take for example the distance of a biker from home set at the origin. It will be written as a function of x(horizontal travel) but what if we needed to know where the biker was based on time. It is unnecessary for us to add in a third axis to get a third variable because its path can be modeled on a 2-D plane. This is were setting a parameter and making it a parametric equation gives us the tools necessary to properly model the flight path.

The rectangular equation for a biker moving at 4 ft/s east, and 3 ft/s north and starting at -12,-6 is:

With this equation though you get the bikers y location base on its x location. But this tells us nothing about time, and where the biker is at certain times, with this information we could not answer how long until the biker reaches its house from the location of -12,-9. But if we break up the x, and y vectors into functions of time then we can easily find how long it will take the biker.

The biker is moving at 4 ft/s east which is the x-axis, and moving at 3 ft/s north which is the y-axis. So the rate of change or slope for x is 4 and for y is 3. We can write the functions of x, and y in slope intercept form because they are straight lines.






Now to solve for be, we know at zero seconds the biker's location is at -12,-9. So we plug that point into the equation and solve for b. But we can only plug in the x coordinate into the equation for x and same for y.


 Simplify









Simplify





Now we have the parameters for the function based on time the has passed can solve for the original question, how long until the biker reaches the house. Simply set x to zero because that is the x coordinate of the house.

 If you want you can confirm by doing the same for y but it is not necessary.

Circles:

Another important role parametric equations play is in graphing circles. If you wanted to graph a circle you can always use.

 But if you wished to model a circle using parametric equations then you would set the parameters as follows

Now this is only for a circle that starts at (1,0) rotates counter clockwise, has radius of 1, and 1 full rotation is around 6.28 (two pi). With a few easy manipulations we can change the radius, period, and direction of rotation.


Changing k in the equation of x will change the x diameter and will do the same for y if changed in the y function.

The circle below has a radius of 5, because the value of k has been changed to 5 in both equations of x, and y. If it had been changed in only the x equation then the y diameter would be two.

Changing the sign of the function will change the rotation of the circle. + function will rotate counter clockwise, and a - function will rotate clockwise.

Switching sin and cos will change the starting location of the circle.

This is important for modeling a rotating wheel. With using parametric equations one can track the starting point on the wheel and the direction of rotation.

Not all wheels rotate at 1 rotation per two pi seconds. By changing what you multiply theta by can change the rotational speed of the wheel.

The variable h changes the period of one full rotation, where if h is one then the period is two pi.

Extra Resources:

Khan Academy

https://www.khanacademy.org/math/precalculus/parametric-equations

Phils Chatz

http://philschatz.com/precalculus-book/contents/m49409.html

9.2 Arithmetic Sequences and Partial Sums

9.2 Arithmetic Sequences and Partial Sums

A sequence whose consecutive terms have a common difference is called an arithmetic sequence

The sequence

         
is arithmetic if there is a number d such that

              

The number d is the common difference of the arithmetic sequence

Ex.















 





The nth term of an arithmetic sequence has the form



where d is the common difference between consecutive terms of the sequence


Ex.

Find a formula for the nth term of the arithmetic sequence whose common difference is 3 and whose first term is 2.




            


Therefore,




Find the 6th term of the above sequence.












The sum of a finite arithmetic sequence with n terms is




Find the sum of the integers from 1 to 100

9.7 Probability

Any happening whose result is uncertain is called an experiment, and its results are called outcomes. The experiment's sample space is the collection of every possible outcome it has, and an event is any subcollection of the sample space.

Probability

If an event has n(E) equally likely outcomes and its sample space S has n(S) equally likely outcomes, the probability of the event is:

 

Example 1

 A card is drawn from a standard 52 card deck. What is the probability that the card is a red 7?

n(E) = 2 cards (7 of hearts and 7 of diamonds)

n(S) = 52 cards

= 2/52 = 1/26 

Mutually Exclusive Events

Two events are mutually exclusive if they have no outcomes in common. The intersection of two sets A and B is . If A and B are mutually exclusive, then  = 0. 

If A and B are events in the same sample space, the probability of A or B occurring is 

P(A U B) = P(A) + P(B) -    

If A and B are mutually exclusive, then 

P(A U B) = P(A) + P(B)

Example 2

One card is selected from a deck of 52 playing cards. What is the probability that the card is either a spade (A) or a 3 (B)?   

P(A) = 13/52

P(B) = 4/52

  = 1/52

P(A U B) = 13/52 + 4/52 - 1/52 = 16/52 = 4/13

Independent Events 

Two events are independent if the occurrence of one has no effect on the occurrence of the other. To find the probability of both events happening, multiply the probabilities of each together. 

P(A and B) = P(A) * P(B)

Example 3

One bag contains 5 blue marbles and 1 white marble, and the other contains 4 red marbles and 5 white marbles. A marble is taken from the first bag A, and then a marble is taken from the second bag B. What is the probability of both marbles being white?

P(A) = 1/6

P(B) = 5/9

P(A and B) = (1/6) * (5/9)  = 5/54

The Complement of an Event 

The complement of an event is the collection of all the outcomes in the sample space that are not in the event. The complement of event A is denoted by A'. P(A') = 1 - P(A). 

 

  

9.6 Counting Principles

Before getting into probability, it's important to understand basic counting principles. This section focuses on such principles and how to apply them to future probability problems.

Fundamental Counting Principles: 
Let E1 and E2 be two events. The first event E1 can occur in m different ways. After E1 has occurred, E2 can occur in n different ways. The number of ways that the two events can occur is

m * n

The Fundamental Counting Principle can be extended to three or more events.

EXAMPLE

1. How many different trios of letters from the English alphabet are possible if the first letter cannot be Z?

  (Possible first letters) * (Possible second letters) * (Possible third letters)

  (25) * (26) * (26)

  16,900

PERMUTATIONS

Definition: a permutation of n different elements is an ordering of the elements such that one element is first, one is second, one is third, and so on.

EXAMPLE

1.How many permutations are possible for the letters A, B, C, D?

  First Position = any of the 4 letters

  Second Position = any of the remaining 3 letters

  Third Position = any of the remaining 2 letters

  Fourth Position = the one remaining letter

Total number of permutations = 4 * 3 * 2 * 1 = 4! = 24

Number of Permutations of n Elements: 

the number of permutations of n elements is

n * (n-1) ...4 * 3 * 2 * 1 = n!

In other words, there are n! different ways that n elements can be ordered.

Permutations of n Elements Taken r at Time:

the number of permutations taken r at a time is 

EXAMPLE

1. There are 10 horses in a race. In how many ways can there be 1st, 2nd, 3rd, 4th, and 5th place?

  10! / (10-5)! = 10*9*8*7*6 = 30,240

COMBINATIONS

Definition: permutations are when order does matter, but combination simply counts the number of permutations while order does NOT matter.

Combinations of n Elements Taken r at a Time: 

the number of combinations of n elements taken r at a time is 

EXAMPLE

1. You have a standard deck of 52 cards and will deal them out in sets of 5. How many different hands are possible?