### Precalculus and Calculus

```Students Mentoring Students Presents:
Ruben Sanchez
[email protected]
BEFORE WE START:
Do NOT be afraid to ask
questions.
• There are no dumb
questions!
• The only dumb thing to
do is not ask for help
when you are stuck.
Looking at Linear
Equations and Graphs
• There are many different
types of graphs but the
most common ones we
see, look something like
parent function graphs.
• To the right, there are
various examples of
some of the parent
functions that exist.
So what do these
functions mean?
•
All these functions can be used to
express information gathered or it
represents a relationship between
2 variables and then plotted on a
graph, in this case the Cartesian
Coordinate System.
•
Information that can be read off of
graphs include:
•
Distances
•
Velocity
•
Acceleration
•
Money/Interest Earned
Lets look at examples
on how to graph on the
coordinate system.
There are 2 main ways to
graph on a coordinate
plane.
1. Point- Slope form
2. Slope intercept form
We will look at both and
see how they are related.
We will also see how to
convert from point- slope
to slope intercept
Lets say they give us a point of ( 8, 6) and
that we are given a slope of 5.
Once we have our information, all we do is plug our
variables into the equation to the left.

 We are going to plug in the coordinates
Into the x and y with sub-script 1.
We just plug in our variables where it asks us to in the formula and that is all .
Once you do that we have our answer, a point-slope formula!
Point Slope Form
In Point slope Form we are given a point on the graph. And the slope of a line. When
we have these 2 main components we are able to plug into an equation to find an
equation for a line.
1)
Point (7,8)
Slope 6
3) Point (2, 8)
Slope 5
2)
Point (3,9)
Slope 2
4) Point (5,2)
Slope 8
5) Point (4,4)
Slope 9
Practice with Point-Slope Form
Practice putting these points/ slope examples into the point slope formula. Only set
up.
 Point-Slope
What do we have to do to make it look like this? 
Isolate Y! Y wants to be on its own!
So first we distribute the right side to
make things easier and we get
Then we add 6 on the left to leave Y by itself.
Remember the rule, whatever we do to one
side, we do to the other. Afterwards Y is left by
itself and we reach our answer.

Changing Point-Slope to Slope Intercept Form
Once we have our equation in point slope form, simple algebra is used to convert this
equation to slope intercept form, making it possible to graph on graph calculators.
Change from point-slope to slope-intercept
Using and understanding the Pythagorean Theorem
Pythagorean theorem deals with measurement of triangle sides and simple algebra
used to find lengths of sides. NOTE- this method only works when you have a right
triangle, a triangle with at least one angle measuring 90 degrees.
Take the Square root of both sides
Example:
Let us find the length of the hypotenuse for the triangle MLO above. We are given
the side lengths but missing the hypotenuse. All we have to do now is substitute into
the equation a2+b2=c2
Main Street
Commercial
Street
4 Miles
Park Avenue 3 Miles
Real Life Example with 90 Degree Triangles
If Park Ave. is 3 miles long and Commercial Street is 4 miles long, how long is Main
Street? (hint- set up Park Ave. as side A, and Commercial St. as side B)
Trowbridge Drive
6 Miles
Paisano
Drive
Montana Street
4 Miles
Real Life Example with 90 Degree Triangles
If Trowbridge is 6 miles long and Montana Street is 4 miles long, how long is Paisano
Street? (hint- set up Montana as side A, and Trowbridge Drive as side C )
Didn’t see this coming!
Let us say that the top bar (olive) of the bicycle is 3 feet, and the bar that goes down
from the seat to the pedals (violet) is 2 feet. How long is the bar that that connects
the handlebars to the pedals (blue)?
More real life applications with pre-calculus
We found out that right triangles could be used with distances or lengths but what
about other useful math terms or calculations? We can use perimeters and area and
relate it to real life applications. Fore example the length of track, perimeter of a
farm, etc…
circle is 25 Meters
Find the total distance around the track
What we need to do is find the “perimeter” of the track to find out the length of one
lap around the field. How do we do this problem?
Put a fence around this barn
The owner of this barn decides to put up a fence around his bar. Looking at it from a top
view it makes a rectangle measuring 35 feet long and 20 feet wide. If the farmer wants to
have “extra” space and put the fence 5 feet away from the barn on every side,
what will be the amount of fence the farmer will have to buy?
What will be the new total area of the enclosed land including the barn?
What is the Perimeter of the cardboard box?
If each can has a diameter of 2.5 inches, and the box is 4 cans wide and 6 cans long,
what is the perimeter of the cardboard? What is the area of the base of the
cardboard box?
If each side of the parking aisle can
Park 9 cars, how many cars can 8
Aisles hold?
Multiplication practice
How many cars can be parked if each parking aisle can park 9 cars on each side?
What if there are a total of 8 aisles?
How many desks will fit?
How many desks will fit if the desks are 3 feet wide and 4 feet long and the room
measures 16 feet by 21. The teacher wants to leave a gap of 3 feet in between each
row of desks.
How many test tubes do you need?
The doctors office you work for needs to order test tubes but the Doctor doesn’t
know how many test tubes to order for the 15 racks he ordered. Each rack’s
dimensions are 5 by 12 slots. How many test tubes does he need to order to fill each
slot on the racks?
How much does it cost to fill up?
Lets use the example with the man on the bicycle, if you are to pull up in a big V8
truck that requires you to fuel up with premium grade fuel, how much would it cost
to fill up the 24 Gallons the truck has? What if you are in a Prius and only have to fill
up 10 Gallons of Regular grade fuel?
Rates including speed
Nascar engines are built to last and endure very high speeds and accelerations. These
fine tuned engines can reach top speeds of 200+ miles per hour (MPH). How long
(time) would it take to complete 550 miles at a race? (set up as an equation)
Rates with distances
The Tour de France typically comprises 20 professional teams of nine riders each and
covers some 2,200 miles of flat and mountainous country, mainly in France, with
occasional and brief visits to Belgium, Italy, Germany, and Spain. How long would it
take to complete the race at a constant pace of 15 miles per hour?
Proportions
If 250 songs take up 1 gigabyte of space, how many songs can fit in modern day
electronics such as iPod’s/ iPad’s/ iPhone’s/ etc… that can hold 16 gigabytes? How
Rates
The average rate that water comes out from a residential outlet, or hose, is about
200 gallons per hour. At this rate, how long would it take to fill up a pool of 22,000
gallons?
Basic Calculus
Calculus can be used in many different applications. One key application that this
field of math is used in is Physics and Engineering. Others may use calculus to
calculate distances or positions. Whatever you wish to apply it to, we will cover the
basics to understand how to carry out processes in calculus.
Lets practice an example.
Derivatives
In our real life applications, derivatives can be understood as a rate of change. One way
we can think of a rate of change that we may not even notice is when we drive, miles per
hour. This is an example of a rate of change. Derivatives don’t always have a simple chart
that we can refer to but for integrals, we have charts that can help us set up the problem.
Let’s take the derivative of each function
Integrals
Here for integrals, there are some set integral guides that can help us set up the
problem and
In most cases and in textbooks, the constant shown here is is written as a “K” or a “C”
Integrals
Lets practice an example solving integrals.
Let us solve the definite integral
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