Cornell Notes Section 1.3 Day 2 Section 1.4 Day 1 Write a summary of the lesson Section 1.3 Day 3 Write 2 questions (L1-L3) Section 1.4 Day 1 Read through highlight, underline, add “?”

Exit Ticket for Feedback 1. Is the following function continuous? Prove your answer. 𝑓 𝑥 = 𝑥 2 +3, 𝑥≤2 10−𝑥, 𝑥>2

Section 1.4 Day 2 Continuity and One-Sided Limits AP Calculus AB

Learning Targets Define and evaluate one-sided limits Define and determine continuity at a point and on an open/closed interval Define and determine a continuous function Define and determine forms of discontinuity (removable/non- removable) Define and operate with the greatest integer function Describe the continuity of composite functions Evaluate a point to make a function continuous Apply properties of continuity Define and apply Intermediate Value Theorem

Discontinuities Example 1 Find the points of discontinuity and determine if they are removable or non-removable Example 1: 𝑥−1 𝑥 2 −4𝑥+3 1. We are looking for holes and vertical asymptotes 2. Factor: 𝑥−1 𝑥−1 𝑥−3 3. Notice (𝑥−1) cancels and produces a hole at 𝑥=1. Removable Discontinuity 4. (𝑥−3) is in the denominator and produces a vertical asymptote at 𝑥=3. Non-Removable Discontinuity

Discontinuities Example 2 Find the points of discontinuity, determine the type, and if it’s removable, determine what value would extend the function to be continuous. Example 2: 𝑓 𝑥 = 3−𝑥, 𝑥<2 𝑥 2 +1, 𝑥>2 1. 𝑓(𝑥) is continuous on its domain. However, it is not defined at 𝑥=2. It is discontinuous at 𝑥=2. 2. To check if it is removable or non-removable, let’s look at the limits from the right and left

Discontinuities Example 2 𝑓 𝑥 = 3−𝑥, 𝑥<2 𝑥 2 +1, 𝑥>2 3. lim 𝑥→ 2 − 𝑓(𝑥) =3 −2=1 lim 𝑥→ 2 + 𝑓(𝑥) = 2 2 +1=2 4. The limits do not match. Thus, we have a non-removable (jump) discontinuity.

Properties of Continuity If b is a real number and f and g are continuous at x = c, then the following functions are also continuous at c. 1. Scalar Multiple: 𝑏∙𝑓 2. Sum and Difference: 𝑓±𝑔 3. Product: 𝑓∙g 4. Quotient: 𝑓 𝑔 , if 𝑔 𝑐 ≠0

Continuous Functions Polynomial Functions Rational Functions Radical Functions Trigonometric Functions 1. Keep in mind that these are continuous on their domains and not necessarily at every interval 2. Knowing these values and the properties of continuity shows that there are many continuous functions. Example 1: 𝑓 𝑥 =𝑥+ sin 𝑥 is continuous on its domain Example 2: 𝑓 𝑥 = 𝑥 2 +1 cos 𝑥 is continuous on its domain

Continuity of a Composite Function If 𝑓 is continuous at 𝑐 and 𝑔 is continuous at 𝑓(𝑐), then the composite function 𝑔(𝑓 𝑥 ) is continuous at c. Continuous at c Continuous at c Continuous at f(c)

Puzzle! In your groups, draw a graph to represent this situation. Put it on a notecard and be prepared to defend your result. I have drawn a function 𝑓(𝑥) on [1,5] with the value of 5 between 𝑓(1) and 𝑓(5). However, there is no value in the interval [1,5] for which the function will actually be 5. I challenge you to do the same!

Puzzle Possible Solution 1. Notice that the value of 5 is a possible value between 𝑓 1 =2 and 𝑓 5 =6. 2. Notice that there is no value between [1,5] in which the function actually is 5. There is a hole at that value. What is a commonality amongst all our graphs?

Observation

Observation Take 2 minutes to answer the following questions on your own. 1. What is a speed you are 100% sure you traveled within the few seconds you were day dreaming? Why? 2. What is a speed that you could have gone within the time you were day dreaming, but aren’t 100% sure? Why?

Observation In 2007, the iPhone-8GB cost about $599 Later in 2007, the iPhone-8GB cost about $399 In 2009, the iPhone3G-8GB cost about $99 Answer the following, on your own 1. What is a price between 2007-2009 that you are 100% sure the iPhone was sold for? Why? 2. What is price between 2007-2009 that the iPhone could have been sold for, but you aren’t 100% sure of? Why?

Intermediate Value Theorem If 𝑓 is continuous on the closed interval [𝑎,𝑏] and 𝑘 is any number between 𝑓 𝑎 and 𝑓(𝑏), then there is at least one number 𝑐 in [𝑎,𝑏] such that 𝑓 𝑐 =𝑘.

Example 1 Let 𝑓 𝑥 = 𝑥 3 +2𝑥 −1. Explain why there must be a value 𝑐 for 0<𝑐<1 such that 𝑓 𝑐 =0. 1. Since 𝑓(𝑥) is a polynomial, it is continuous on the given interval. 2. We need to find one value below 0 and one value above 0 to use IVT. Let’s check the end points of the interval. 3. 𝑓 0 =−1 and 𝑓 1 =2. Thus, 𝑓 0 <0<𝑓 1 . 4. By IVT, there must be a 𝑐 in 0<𝑐<1 such that 𝑓 𝑐 =0.

Example 2 Let 𝑓(𝑥) and 𝑔(𝑥) be strictly increasing continuous functions and ℎ 𝑥 =𝑓 𝑔 𝑥 −4. Explain why there must be a value 𝑟 for 2<𝑟<6 such that ℎ 𝑟 =−1 1. ℎ(𝑥) is continuous 2. ℎ 2 =𝑓 𝑔 2 −4=𝑓 1 −4 =−2−4=−6 3. ℎ 6 =𝑓 𝑔 6 −4=𝑓 5 −4 =7−4=3 4. Since ℎ 2 <−1<ℎ(6), by IVT, there is some 𝑟 in 2<𝑟<6 such that ℎ 𝑟 =−1. X f(x) g(x) 1 -2 2 3 4 5 7 6 10

Exit Ticket for Feedback 1. Let 𝑓 𝑥 = 𝑥 3 −3 𝑥 2 +1. Explain why there must be a value 𝑐 for 0<𝑐<2 such that 𝑓 𝑐 =0. 2. Find the points of discontinuity and determine the type. If it’s removable, determine what value would extend the function to be continuous. 𝑓 𝑥 = 𝑥 2 , 𝑥<1 2−𝑥, 𝑥>1 2, 𝑥=1