Slope, Distance, Functions, and

ContinuityJeremy, Vince, Sean, Ray, Nick

Period 4

Slope

• Definition:• Slope intercept equation:

y=mx+b• Vertical lines have no slope• Horizontal lines have a

slope of 0• Parallel lines have the same

m value and different b value• Perpendicular lines’ slopes

are the negative reciprocal of one another

Practice: 1.Slope between (1,3) (-1,5) Answer: -12.Slope of the line x=1 Answer: No slope3. Slope of a line parallel to y=2x+1 that goes through the point (5,4) Answer: y=2x-6

Real Life Example:On the left is the side view of a house. The house is 10 ft tall and the roof starts at 7 ft. The top of the roof is 4 ft. left of the right side of the house. Find the slope of the roof. (The origin is the bottom right corner of the house)Answer: The slope is -3/4

Absolute Value and Coordinate Plane DistanceBut first, a math joke:Why don’t they serve alcohol at a math party?• You can’t drink and derive

•Absolute Value ( x ), is defined as the formula that follows:x= x if x>0, and –x if x<0

•So, if x= -3 , x=3. Likewise, if x = 3 , x=3, -3

Real-Life Example: You are at a city that is at sea-level. You decide to take a scuba diving trip and go 200 feet below sea level. You then decide after that to go parasailing and are 200 feet in the air. Despite the scuba diving being 200 below sea level (-200), you are the same distance from the surface in both trips and have the same absolute value.

Distance in the Coordinate Plane and practice

• Practice: • x=-3. 1. │x│= ?

32. │x-5│=?

83. Distance between (3,3) and (-4,-1)?

d=√65=8.062

Real-Life Example: You are 1 mile north and 3 miles east (3, 1) of your friends house. Your friends house is 2 miles west and 4 miles south (-2,-4) of the town hall (0,0). You need to make sure your helicopter has enough gas to get from your roof to your friend’s roof. You get 1 mile per gallon. How many gallons do you need?Using the formula, you can figure out that it is approximately 7.071 miles, so you need a little more than 7 gallons, unless you’re adventurous...

Relations, Functions and their inverses• Circles• (x-h)2 + (y-k)2=a2

• Center is ( h, k )• Radius is a

• Inverses• The inverse of a function is a reflection over the line y=x.• The domain of a function is the range of its inverse and vice versa.• The inverse of a function is only a function if and only if it is a one-to-one function( Passes horizontal line test ).

Relations, Functions and their Inverses

AN ALGORITHM

Continuous functions • Continuity Test- the function f(x) is continuous at x=c if and only if all

three of the following statements are true:1. F(c) exists c is in the domain of f2. lim x->c f(x) exist f has a limit as x approaches c3. lim x->c f(x) = f(c) the limit value equals the function value

Types of Discontinuity:

Removable Jump Oscillating Infinite

*Here is a chant that helps you remember the functions that are always continuous at all points in their domain

Red Parrots continuously Repeat Everything They Learn

RationalPolynomalRadicalExponentialTrigonometricLogarithmic

Intermediate Value Theorem for Continuous Functions

• The Intermediate Value Theorem states a function y=f(x) that is continuous on a closed interval [a,b] takes on every value between f(a) and f(b).

• In other words, if L is any number between f(a) and f(b), then there exists at least one number c contained within [a,b] such that f(c)=L

• This theorem was first proved by Bernard Bolzano in 1817

Example 1: Is there a solution to x5 - 2x3 - 8 = 0 on the contained interval [-1,3] ?

Solution: f(-1) = -7f(3) = 181Since x5 - 2x3 - 8 is a continuous function on the given interval [-1,3] and f(-1)<0 and f(3)>0, by the Intermediate Value Theorem x5 - 2x3 - 8 has a zero between the interval [-1,3] .

*You must state the function is continous

Real-Life Application:When the temperature outside changes into the night from 85 degrees to 70 degrees, by the Intermediate Value Theorem , if the function p(t) describes the decrease in temperature where t is time, then at some time the temperature will be 76 degrees.

Additional Problems

Quiz

Please pull out a sheet of paper for a five question assessment. At the end of this exam, we will verify your answers and reward those who have paid attention throughout our presentation.

1.What are the six types of continuous functions?

2.Is there a solution to 2x5 - 4x3 - 8x+3 = 0 on the contained interval [-2,5] ?

3.What is the inverse of g(x) = x²+3?

4.Solve log5 25

5.What type of discontinuity is

show at x=-3

Answers: 1. rational, polynomial, radical, exponential, trigonometric, logarithmic 2. Since 2x5 - 4x3 - 8x+3 is a continuous function on the given interval [-2,5] and f(-2)<0 and f(5)>0, by the Intermediate Value Theorem 2x5 - 4x3 - 8x+3 has a zero between the interval [-2,5] . 3. No inverse 4. 2 5. Jump

Final RemarksThrough this project, we refreshed our minds on how to find the slope of parallel

and perpendicular lines, questions with absolute value, and the distance formula. We learned the equation for a circle, a logarithmic function, and exponential function. In addition, we revisited properties of logarithms and learned more about determining whether or not a function is continuous. Finally, we learned about the Intermediate Value Theorem for continuous functions. Ultimately, this project exemplifies the ability technology has to enhance learning, but the most interesting viewpoint was applying what we learned in real-life examples.

Calculus Song