Calc. 1( Limit of a Function and Intro Derivatives ( Tangent+ Velocity Problem

Goes through what a limit of a function is and is not , how to ind the slope of a tangent line, and a derivate at a point, and finding its function

What is the Difference Quotation Formula?

f(x+h)-f(x)/h

Common Behaviors Associated with Nonexistence of a Limit

1. f(x) approaches a different number from the right side of a than it approaches from the left side.
2. f(x) increases or decreases without bound as x approaches a.
3. f(x) oscillates between two fixed values as x approaches a.

Formula #1: Slope of the tangent line (Common Point)

Lim x➡️a f(x)-f(a)/x-a

Formula #2: Slope of Tangent Line+ Derivative at a Point( instantaneous rate of change at x=a

Lim h➡️0 f(a+h)-f(a)/h

Formula #3: Derivative Funiction( find a f(x) function

Lim h➡️0 f(x+h)-f(x)/h

Existence of a Limit Theorem

The limit of the right and left are the same value so the limit is same value.

Basic Limit Laws#1: The Sum/Difference Law

You can distribute your limit to each part of the expression( put the limit before each factor)

Basic Limit Laws#2: The Constant Multiple Law

The constant can be moved outside the limit and away from sitting next to the limit.

Basic Limit Laws#3: The Product Law

You can multiply each function by each other as long as they have the same limit

Basic Limit Laws#4: The Quotient Law

The value of the limit on num. and the limit of the denom. Equals the limit of the fun in num/ denom. As long as the denom. Does not equal zero.

Basic Limit Laws#5: The Power Law

As long as the n(interfere) is positive than the limit can be moved ext to then function and done together

Squeeze Theorem:

If the upper function( right) and lower function(left) are equal and the function is between the other two , than that middle function equals the equaled value of the outside functions

Definition of Continuous

Discontinuity: Removable

If you can fill in the circle and create a continuous function( has a function value and limit)

Discontinuity: Jump

Where you have a gap( break) between the line and you would have to go up or down to continue

Discontinuity: Infinite

An asymptote is usually a good indicator, but it’s when both the function value and limit does not exist in that limit

Continuous Domain

Be sure to write and not include anything discontinuous

Polynomials + Rational Functions

Polynomials are continuous everywhere(-,+) Rational, root, trig, inverse trig, exponential, logarithmic functions are continuous where defined( every number) in its domain

The Intermediate Value Therem(IVT)

Condition 1: Must be Continuous. 2: N must be between the intervals. 3: That function values don’t equal. To prove the function continuous at the interval.

Degree ratios

Tangent Line Equation

Y=Mx+b ( just find value of m through formulas and solve for b using the points for x and y.

Position and Velocity

S(t)= S( t2)-S(t1)/ t2-t1 : Distance/ time

Instantaneous Velocity

S’(t)=v(t)

Refer to Notes

Go to page 36( second page of 2.8) study how you make those graphs

Theorem

If f is differentiable at a, then f is continuous at a. NOT! The other way around