**Calculus**

Calculus was invented by Isaac Newton

for his studies in physics.

Background Information on Calculus:

Derivatives

*There are an infinite number of derivatives but mainly the first two are used.

* First derivative= f '(x)

Second = f ''(x)

etc...

* f '(x) = slope of a function

f ''(x) = slope of f '(x)

EXAMPLE:

f(x) = 2x+1

f '(x) = 2

Finding the Derivatives of Quadratics

f(x)=6x²+13x+21

*Multiply the exponent of the first term by the coefficient. Then drop off the constant and reduce the power by 1

Real Life Uses of Derivatives

f '(x)=12x+13

Quadratic Equation:

f "(x)=12

f "'(x)=0

f(x)= 6x² + 13x¹ + 21

The first Derivative:

* Find speed

* Find the vertex of parabolas

* Find the biggest area that can be made with a certain amount of something

The second derivative:

* Find acceleration

* Find slope of the first derivative

Multiply

First derivative!!

*Put an apostrophe in between the "f" and the (x). This is called a prime.

Area and Perimeter in Word Problems

A farmer is building a fence to stop his pigs from escaping. He only has 50 ft of wire to build his fence. What is the biggest field he can make with the wire?

Second derivative!!

Third derivative!!

*Do the same thing for the other derivatives but use the derivative before it instead of the function.

f '(x)=12x+13

*There are an infinite number of derivatives.

Example:

Derivatives of a Cubic

*To find the derivative of a cubic, you do the the same thing that you do with quadratics.

f(x)=5x³+8x²-2x+34

f '(x)=15x²+16x-2

f "(x)=30x+16

f "'(x)=30

Example:

Set Up two Equations (one for area and one for perimeter)

f(x) = 2x³ - 4x² + 6x + 8

f (x)= 4x²+6x+2

f '(x) = 6x² - 8x + 6

f '(x)= 8x+6

f "(x)=8

f ''(x) = 12X - 8

f "'(x)=0

f '''(x) = 12

I'm building a wall to so that when there are floods my house won't flood. I have a river flowing on one side of the area so I will only need to build a three sided wall. If I only have 30ft of wall how big is the biggest wall I can make? (in perimeter and area)

f ''''(x) = 0

2x+2y=50=perimeter

xy=Area

x

y

Isolate one of the variables

y=-x+25

2(x+y)=50

2

Now use substitution in the area equation, and turn it into a function!

= x (-x+25)

f(x)

x+y=25

Now get the first derivative.

f '(x)= -2x+25=0

f(x)=-x²+25x

12.5=x

Next plug x into y= -x+25. This is the equation we used to substitute for y.

y = 12.5

If you use those numbers as the sides like shown in our drawing, you can find the area!

Area=156.25ft²

Finding the Vertex of a Parabola

First you will have to find the first derivative of the function

f(x)= 4x² + 24x + 8

f '(x) = 8x + 24

Now you will have to take the first derivative and find where it is equal to zero

f '(x) = 8x + 24=0

x= -3

This will be the x coordinate of the vertex. Next you will have to take "x" and plug it in to the original function.

f(x)= 4x² + 24x + 8

4(-3)² + 24(-3) + 8

-28

This will be your y coordinate. Now you have your point for the vertex. This is the lowest point of this parabola because it's upwardly opening

(-3, -28)

Vertex

* We will be teaching you how to find the vertex of a parabola.

* The equation of a parabola is a quadratic

* The vertex of an upwards parabola is the lowest point

* The vertex of an downwards parabola is the highest point

* The slope of the parabola at the vertex is 0

x=12.5 and y=12.5

Now you will set the equation to zero

f '(x)= -2x+25

Vertex

y

x

river

Example

f ''''(x) = 0

HINT: Since this problem has a river the perimeter equations will be different

HOMEWORK!

If you get a 100% on the homework, you will get a piece of candy!

Why They Work

* Downward parabolas have the vertex as the highest point

* This is why the word problems work for finding the maximum area of certain things.

x = -24/8

Example

f(x) = 2x² + 16x + 19

f '(x) = 4x + 16

x = -4

y= -13

(-4, -13)

f(x)=y(50-2y)

f(x)=-2y +50y

2

x=25

y=12.5

Area=312.5ft²

2y+x=50