More Derivatives

Derivatives do not exist in four types of functions: Corners, cusps, jumps, and asymptotes.  

Implicit Differentiation is the process of finding the derivative of a function with two variables. The derivative dy/dx is taken out of the entire equation. Wherever the derivative of x is found, a dx/dx is placed next to that piece, simplifying to 1. Wherever the derivative of y is found, dy/dx is multiplied by that piece. The last step is to solve for dy/dx.

The most important thing to remember in a related rate problem is to make sure to multiply dy/dx in the places where it is needed.

Blog Post #5

1.  F'(0) Represents a critical point. It is a point on the graph in which there is no slope, where a change in direction occurs. It represents a possible maximum/minimum on the graph.

2.  To find where a function increases and decreases, first find the derivative of the equation. If the derivative (slope) is positive, then the function is increasing. If the slope is negative, the function is decreasing. You can also find a critical point. As you approach the critical point from the left and right, the y value will be increasing or decreasing. This will tell you whether the function is increasing or decreasing as it approaches the critical point.

3. The Chain Rule is the process of finding the derivative of a composite function.
Ex: Find the Derivative of the equation f(x)= (4-2x)^3, and the equation of the tangent line at x=3

Chain Rule: f(x)=g(h(x))                          f '(x)= g '(h(x))h '(x)
f(x)= (4-2x)^3          f '(x)= 3(4-2x)^2*(-2) = -6(4-2x)^2
Find the Tangent Line: f(3)=(4-2(3))^3=-8
(3,-8)

Slope: -6(4-2(3))^2= -24

y+8=-24(x-3)

4.  h(x)=f(g(h)) where g(-4)=5, g'(-4)=2, and f'(5)=20. Find h'(-4).
h'(-4)=f'(5)(2)
h'(-4)=20(2)
h'(-4)=40

Continuity:
f(x)= {-3+x,    x<-4  
         {x^-2,       -4<x<2
         {x+1,      x>2

Step 1: x--> -4-     (-3+(-4))  = -7
            x--> -4+    (-4^2 - 2 = 14
The function is not continuous because it does not come together at x=-4 from the left and right sides.

Intermediate Value Theorem: f(x)= x^2-2x - 3  on [2,6]
f(2) = -3
f(6) = 21
The IVT allows us to say that along the interval [2,6] there is a solution N.

f(x)= x^2 + 4x -22 on [1,3]
f(1)= -17
f(3)= -1
The IVT doesn't prove that there is a solution along the interval [2,6]

Derivatives: Two types of derivatives.

One way to solve: f(x) - f(a)
                         ----------------
 x-->a                        x-a


Second Way to Solve: Difference Quotient  f(x+h) -f(x)
                                                             -----------------
                                                                      h                  h--> 0

Ex: f(x) = 3x+7


3(x+h) +7 -(3x+7)
----------------------     h --> 0        
             h

3x+3h +7 -3x - 7          h(3x+3 + 7 -3x -7
--------------------  -->                                  
            h
x=3 The slope of the function as a whole (derivative) is 3.


Instantaneous vs Average Velocity: Instantaneous Velocity is slope at one certain point. Average Velocity is the average slope over an equation as a hole.

Limits

A limit is the value that a function  approaches as the input some value. The limit (can never be reached) but the limit values we find are as close as possible.

To solve a limit: Try first to plug in the limit you’re approaching to the equation. If this doesn’t work solve the equation by factoring and cancelling out, rationalizing,  or combining fractions.

Types of Limits done in class: Polynomial/Quadratic, Square Root, Fractional

Fractional: Get a common denominator to simplify.

Polynomial/Quadratic: Factor to simplify then solve

Square root: Rationalize- multiply the numerator and denominator by the conjugate and simplify to solve.

Piecewise:

X--> -2             

As x approaches -2 from the left (-3-x) , x= -1

As x approaches from the right, (2x), x= -4.

Because these are not equal, the Limit does not exist at -2.

Infinite Limit:

Lim      x - 1/x

x--> 0

as x approaches 1, when solved you find the answer is undefined. This means that there is an “infinite limit,” in which the closer you get to zero, the limit gets very large.

Calculus Review

f(x)= 7x^3 + 12x^2 -109x + 30

Y-Intercept: The constant term in the polynomial equation is the Y-intercept. The y-intercept for this equation is (0,30). This is one of the points that is easily spotted on the graph.

Descarte's Rule of signs: This rule allows you to how many positive, negative, or imaginary solutions there are.
First, count the number of sign changes (+/-) are in the original equation. This will tell you how many positive solutions there are. To find out how many negative solutions there are, multiply the equation by -1 and count the sign changes. An odd degree will change the sign to opposite, an even degree will leave it the same.

f(x)= 7x^3 + 12x^2 -109x + 30
Positive: + , + , - , +                                  2 sign changes
Negative: -7x^3 - 12x^2 + 109x + 30      1 sign change
Imaginary : There are three roots, and 2 are positive and one is negative, so there are no imaginary roots.

Rational Root Theorem:  Find the factors of the constant term and the leading coefficient.
30: +/- for each... 1, 2, 3, 5, 6, 10, 15, 30
7: +/- for each... 1, 7
Then put the factors of the top number over the bottom.
Other possible factor could include: +/- 2/7. 3/7, 5/7, 6/7, 10/7, 15/7, 30/7

Synthetic Division:   - 5  | 7    12    -109    30
                                        |___ -35__ 115_-30_____
                                    3  | 7    -23       6       0
                                        |____21___-6_____________
                                          7     -2       0
Place the coefficients of your equation inside of this figure, and leave space below. Place one of the possible solutions outside of the figure. Bring down the leading coefficient to start.
Step 1: Multiply the solution by the coefficient. Place your answer under the next coefficient, then add. Repeat for all coefficients. If your last part adds to zero, the solution is correct, and the equation is now narrowed down a degree, repeat until all solutions are shown.
The solutions for this polynomial are -5, 3, 2/7

You can also plug your equation into a calculator to see the graph of it and know your solutions are correct.

1. The hardest topic in math so far was unit circles because there is lots to memorize.

2. My favorite topic in math was solving equations because it was easiest for me.

3. I want to be a physical therapist or trainer.

4. A school goal I have is to graduate in the top 15 in the class. I can try to maintain a 4.0 for this year and do extra to keep A's.

5. An out of school goal I have is to win districts at least in baseball this year. I can be a team leader, put in the extra time, and play with aggression and confidence to go get every ball in the outfield.

6. A function is an equation in which there is one output for every input. Can be put into an equation. 

 Example: f(x)=3x+2