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.