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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
Mindjog
  • Find the domain of each function.
mindjog1
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
Objective: S.W.B.A.T.
  • find extrema on a given interval in order to solve problems for extreme values.
food for thought
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
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
Min/Max
  • On an open interval
  • On a closed Interval
  • Not at all!
extreme value thrm
Extreme Value THRM
  • IF ƒ is continuous on a closed intervalthan it has both a min and a max
lets take a look
Lets take a look!
  • Y = x2 + 2 (–∞, ∞)
  • Do you have a max or min?
lets take a look1
Lets take a look!
  • Y = x2 + 2 (–∞, ∞)
  • Do you have a max or min?
lets take a look2
Lets take a look!
  • Y = x2 + 2 (–∞, ∞)
  • How about on the interval (–3 , 3)
lets take a look3
Lets take a look!
  • Y = x2 + 2 (–∞, ∞)
  • How about on the interval (–3 , 3)
lets take a look4
Lets take a look!
  • Y = x2 + 2 (–∞, ∞)
  • How about on the interval [–3 , 3]
lets take a look5
Lets take a look!
  • Y = x2 + 2 (–∞, ∞)
  • How about on the interval (–3 , 3)
let s take a look
Let’s Take a look!
  • ƒ(x) = x3 – 3x (–∞,∞)
  • Where do the min and max occur?
let s take a look1
Let’s Take a look!
  • ƒ(x) = x3 – 3x (–∞,∞)
  • What is the slope at those points?
critical numbers
Critical Numbers
  • Find the derivative and set it equal to zero.
slide17
1. What are critical points?
  • 2. When do absolute max/min and relative max/min occur
critical numbers1
Critical Numbers
  • Find the derivative and set it equal to zero.
extrema on a closed interval
Extrema on a closed interval
  • Find the extrema of each function on the closed interval.
extrema on a closed interval1
Extrema on a closed interval
  • Find the extrema of each function on the closed interval.
extrema on a closed interval2
Extrema on a closed interval
  • Find the extrema of each function on the closed interval.
extrema on a closed interval3
Extrema on a closed interval
  • Find the extrema of each function on the closed interval.
extrema on a closed interval4
Extrema on a closed interval
  • Find the extrema of each function on the closed interval.
summary
Summary…
  • What are the steps for finding the extrema on a closed interval?
extrema
Extrema
  • Absolute Min/Max
    • Occurs on a closed interval
extrema1
Extrema
  • Relative Min/Max
    • Occurs on a open interval
objective s w b a t1
Objective: S.W.B.A.T.
  • Understand and apply Rolle’s Theorem and the Mean Value Theorem.
rolle s 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
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 theorem1
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
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
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 theorem1
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
Using the MVT
  • Ex: For the function f above, find all values of c in (1, 4) such that
application speeding ticket
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.
summary1
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?