Mindjog

• Find the domain of each function.

174)(.2

1

3)(.1

34

xxxf

xxf

Mindjog

• Find the domain of each function.

174)(.2

1

3)(.1

34

xxxf

xxf

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

1. Find critical #s of f in (a, b).2. Evaluate f at each critical #.3. Evaluate f at each endpoint.4. 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 interval than 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.

]2,1[,43)(.1 34 xxxf

Extrema on a closed interval

• Find the extrema of each function on the closed interval.

]2,1[,32)(.2 24 xxxf

Extrema on a closed interval

• Find the extrema of each function on the closed interval.

]3,1[,32)(.3 32

xxxf

Extrema on a closed interval

• Find the extrema of each function on the closed interval.

]2,0[,2cossin2)(.4 xxxf

Extrema on a closed interval

• Find the extrema of each function on the closed interval.

]2,1[),25()(.5 3

2

xxxf

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.

24 2)( xxxf

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

ab

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)()(

)('

Using the MVT

• Ex: For the function f above, find all values of c in (1, 4) such that

xxf

45)(

14

)1()4()('

ff

cf

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/rol

hei/rolhei.html

• We now have 3 theorems this chapter. What is the third one?

• What is a critical number?