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Transcript of Calculus
Calculus was invented by Isaac Newton
for his studies in physics.
Background Information on Calculus:
*There are an infinite number of derivatives but mainly the first two are used.
* First derivative= f '(x)
Second = f ''(x)
* f '(x) = slope of a function
f ''(x) = slope of f '(x)
f(x) = 2x+1
f '(x) = 2
Finding the Derivatives of Quadratics
*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)= 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
*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?
*Do the same thing for the other derivatives but use the derivative before it instead of the function.
*There are an infinite number of derivatives.
Derivatives of a Cubic
*To find the derivative of a cubic, you do the the same thing that you do with quadratics.
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) = 12X - 8
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
Isolate one of the variables
Now use substitution in the area equation, and turn it into a function!
= x (-x+25)
Now get the first derivative.
f '(x)= -2x+25=0
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!
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
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
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
* 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
f ''''(x) = 0
HINT: Since this problem has a river the perimeter equations will be different
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
f(x) = 2x² + 16x + 19
f '(x) = 4x + 16
x = -4