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Mindjog. Find the domain of each function. Mindjog. Polynomial and rational functions are differentiable at all points in their domain!. Find the domain of each function. Objective: S.W.B.A.T. find extrema on a given interval in order to solve problems for extreme values.
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Mindjog • Find the domain of each function.
Mindjog Polynomial and rational functions are differentiable at all points in their domain! • Find the domain of each function.
Objective: S.W.B.A.T. • find extrema on a given interval in order to solve problems for extreme values.
Food for thought????? • What are extrema? • What is the difference between relative and absolute extrema? • What is true about the derivative at relative extrema? • What is a critical number?
Finding Extrema • Find critical #s of f in (a, b). • Evaluate f at each critical #. • Evaluate f at each endpoint. • Smallest – Abs. min. Largest – Abs. max.
Min/Max • On an open interval • On a closed Interval • Not at all!
Extreme Value THRM • IF ƒ is continuous on a closed intervalthan it has both a min and a max
Lets take a look! • Y = x2 + 2 (–∞, ∞) • Do you have a max or min?
Lets take a look! • Y = x2 + 2 (–∞, ∞) • Do you have a max or min?
Lets take a look! • Y = x2 + 2 (–∞, ∞) • How about on the interval (–3 , 3)
Lets take a look! • Y = x2 + 2 (–∞, ∞) • How about on the interval (–3 , 3)
Lets take a look! • Y = x2 + 2 (–∞, ∞) • How about on the interval [–3 , 3]
Lets take a look! • Y = x2 + 2 (–∞, ∞) • How about on the interval (–3 , 3)
Let’s Take a look! • ƒ(x) = x3 – 3x (–∞,∞) • Where do the min and max occur?
Let’s Take a look! • ƒ(x) = x3 – 3x (–∞,∞) • What is the slope at those points?
Critical Numbers • Find the derivative and set it equal to zero.
1. What are critical points? • 2. When do absolute max/min and relative max/min occur
Critical Numbers • Find the derivative and set it equal to zero.
Extrema on a closed interval • Find the extrema of each function on the closed interval.
Extrema on a closed interval • Find the extrema of each function on the closed interval.
Extrema on a closed interval • Find the extrema of each function on the closed interval.
Extrema on a closed interval • Find the extrema of each function on the closed interval.
Extrema on a closed interval • Find the extrema of each function on the closed interval.
Summary… • What are the steps for finding the extrema on a closed interval?
Extrema • Absolute Min/Max • Occurs on a closed interval
Extrema • Relative Min/Max • Occurs on a open interval
Objective: S.W.B.A.T. • Understand and apply Rolle’s Theorem and the Mean Value Theorem.
Rolle’s Theorem • Let ƒ be continuous on the closed interval [a , b], and differentiable on the open interval (a, b). If f(a) = f(b) then there is at least one number c in (a, b) such that f’(c) = 0.
Corollary: Rolle’s Theorem • Let ƒ be continuous on the closed interval [a , b]. If f(a) = f(b) then f has a critical number in (a, b).
Corollary: Rolle’s Theorem • Let ƒ be continuous on the closed interval [a , b]. If f(a) = f(b) then f has a critical number in (a, b). Why????????
Using Rolle’s Theorem • Ex: Find all values of c in the interval (-2, 2) such that f’(c) = 0 • 1. Show the function satisfies Rolle’s Theorem. • 2. Set derivative = 0 and solve. • 3. Throw out values not in interval.
Mean Value Theorem • Let ƒ be continuous on the closed interval [a , b], and differentiable on the open interval (a, b) then there exists a number c in (a, b) such that
Mean Value Theorem • Let ƒ be continuous on the closed interval [a , b], and differentiable on the open interval (a, b) then there exists a number c in (a, b) such that
Using the MVT • Ex: For the function f above, find all values of c in (1, 4) such that
Application Speeding Ticket • Two stationary patrol cars equipped with radar are 5 miles apart on a highway. A truck passes the first car at a speed of 55 mph. Four minutes later, the truck passes the second patrol car at 50 mph. Prove that the truck must have exceed the speed limit of 55 mph by more than 10 miles per hour.
Summary….. • What is imperative for the use of Rolle’s or the Mean Value Theorem? • http://www.ies.co.jp/math/java/calc/rolhei/rolhei.html • We now have 3 theorems this chapter. What is the third one? • What is a critical number?
