Calculus Chapter 2: Limits Section 2.8: The Intermediate Value Theorem (IVT) SWBAT: use the intermediate value theorem to prove that continuous functions have values at a given point on a given interval. Standard: F.IF. Ø Understand the concept of a function and the use of function notation. “Do Now” Sketch the graph 𝑓(𝑥). At each point of discontinuity, determine the type of discontinuity and state whether it is left or right continuous. 𝑥 𝑓𝑜𝑟 𝑥 < 1
𝑓 𝑥 = 3 𝑓𝑜𝑟 1 ≤ 𝑥 ≤ 3 𝑥 𝑓𝑜𝑟 𝑥 > 3
Graph the following: 𝑥 ! + 1,
−3 ≤ 𝑥 ≤ 0
ℎ 𝑥 =
!
!
−1 − 𝑥 − 𝑥 , 0 < 𝑥 ≤ 2
Calculus Chapter 2: Limits Draw a continuous function, 𝑓(𝑥) that meets the following conditions: 𝑓 2 = −3 𝑎𝑛𝑑 𝑓 5 = 4 Intermediate Value Theorem: If 𝑓(𝑥) is continuous on a closed interval 𝑎, 𝑏 𝑎𝑛𝑑 𝑓(𝑎) ≠ 𝑓(𝑏), then for every value M between 𝑓 𝑎 𝑎𝑛𝑑 𝑓(𝑏), there exists at least one value 𝑐 ∈ (𝑎, 𝑏) such that 𝑓 𝑐 = 𝑀. Ø Speeding up in a car Ø Altitude of a plane taking off Example: Prove that 𝑔 𝑥 = 2𝑥 ! − 1 has at least one solution in the interval [0, 2]. Step 1: Check to see that the function is continuous on the given interval. Step 2: Evaluate the function at each endpoint. Step 3: Make a statement using the IVT. Calculus Chapter 2: Limits Example: Prove that 𝑓 𝑥 = 𝑥 ! takes on the value 0.5 in the interval [0, 1]. Step 1: Check to see that the function is continuous on the given interval. Step 2: Evaluate the function at each endpoint. Step 3: Make a statement using the IVT. Example: Why can’t we use the intermediate value theorem to say that 𝑓(𝑥) takes on a particular value in the interval [0, 2]. 𝑥! − 1
𝑓 𝑥 =
𝑥−1
Calculus Chapter 2: Limits !
Example: Show that 𝑔 𝑥 = !!! takes on the value 0.599 for some 𝑥 ∈ [1, 2]. Closure: What characteristic must the function possess in order to use the intermediate value theorem? Homework: Pg. 109 #’s 1 – 5