Section 4 1 Local Maxima and Minima Section

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Section 4. 1 Local Maxima and Minima Section 4. 1: Local Maxima and Minima

Section 4. 1 Local Maxima and Minima Section 4. 1: Local Maxima and Minima Applied Calculus, 3/E by Deborah Hughes-Hallet Copyright 2006 by John Wiley & Sons. All rights reserved.

Figure 4. 2 and the first box on page 177. Applied Calculus, 3/E by

Figure 4. 2 and the first box on page 177. Applied Calculus, 3/E by Deborah Hughes-Hallet Copyright 2006 by John Wiley & Sons. All rights reserved.

Figure 4. 6 Slide 2: 2 boxes on pg 176 and Figure 4. 6

Figure 4. 6 Slide 2: 2 boxes on pg 176 and Figure 4. 6 f’’(x)>0 concave up (min) f’’(x)<0 concave down (max) f(x) = 3 x 3 – x + 5 Applied Calculus, 3/E by Deborah Hughes-Hallet Copyright 2006 by John Wiley & Sons. All rights reserved.

f(x) = 3 x 3 – x + 5 f(x) = 2 x 4+8

f(x) = 3 x 3 – x + 5 f(x) = 2 x 4+8 x 3+3 Applied Calculus, 3/E by Deborah Hughes-Hallet Copyright 2006 by John Wiley & Sons. All rights reserved.

f(x) = 2 x 4+8 x 3+3 Applied Calculus, 3/E by Deborah Hughes-Hallet Copyright

f(x) = 2 x 4+8 x 3+3 Applied Calculus, 3/E by Deborah Hughes-Hallet Copyright 2006 by John Wiley & Sons. All rights reserved.

2 x Find a so that f(x) = + ax has a critical point

2 x Find a so that f(x) = + ax has a critical point at x = 3 Applied Calculus, 3/E by Deborah Hughes-Hallet Copyright 2006 by John Wiley & Sons. All rights reserved.

f(x) = x 3 – 6 x + 1 Applied Calculus, 3/E by Deborah

f(x) = x 3 – 6 x + 1 Applied Calculus, 3/E by Deborah Hughes-Hallet Copyright 2006 by John Wiley & Sons. All rights reserved.

f(x) = x 3 – 6 x + 1 f(x) = 3 x 5

f(x) = x 3 – 6 x + 1 f(x) = 3 x 5 – 5 x 3 Applied Calculus, 3/E by Deborah Hughes-Hallet Copyright 2006 by John Wiley & Sons. All rights reserved.

f(x) = 3 x 5 – 5 x 3 Applied Calculus, 3/E by Deborah

f(x) = 3 x 5 – 5 x 3 Applied Calculus, 3/E by Deborah Hughes-Hallet Copyright 2006 by John Wiley & Sons. All rights reserved.

Test question 15. f(x) = (x+1)2(x-3)3 Applied Calculus, 3/E by Deborah Hughes-Hallet Copyright 2006

Test question 15. f(x) = (x+1)2(x-3)3 Applied Calculus, 3/E by Deborah Hughes-Hallet Copyright 2006 by John Wiley & Sons. All rights reserved.

Section 4. 2 Inflection Points Section 4. 2: Inflection Points Applied Calculus, 3/E by

Section 4. 2 Inflection Points Section 4. 2: Inflection Points Applied Calculus, 3/E by Deborah Hughes-Hallet Copyright 2006 by John Wiley & Sons. All rights reserved.

Slide 1: Blue box on pg. 182 and Figure 4. 16 . . Figure

Slide 1: Blue box on pg. 182 and Figure 4. 16 . . Figure 4. 16 Applied Calculus, 3/E by Deborah Hughes-Hallet Copyright 2006 by John Wiley & Sons. All rights reserved.

Chapter 4 Key Points f(x) = 0 Zero (graph passes thru xaxis) f’(x) =

Chapter 4 Key Points f(x) = 0 Zero (graph passes thru xaxis) f’(x) = 0 Critical Point (max/min? ) f’’ (x) = 0 Inflection Point (change in concavity? ) Applied Calculus, 3/E by Deborah Hughes-Hallet Copyright 2006 by John Wiley & Sons. All rights reserved.

f(x) = x 3 – x 2 + x - 1 Applied Calculus, 3/E

f(x) = x 3 – x 2 + x - 1 Applied Calculus, 3/E by Deborah Hughes-Hallet Copyright 2006 by John Wiley & Sons. All rights reserved.

There is one minimum and one inflection point. Find them mathematically and then graph

There is one minimum and one inflection point. Find them mathematically and then graph to confirm. f(x) = 3 x 5 + 5 x 3 Applied Calculus, 3/E by Deborah Hughes-Hallet Copyright 2006 by John Wiley & Sons. All rights reserved.

f(x) = 3 x 5 + 5 x 3 y=3 x 5+5 x 3

f(x) = 3 x 5 + 5 x 3 y=3 x 5+5 x 3 -4 x 2 -x Applied Calculus, 3/E by Deborah Hughes-Hallet Copyright 2006 by John Wiley & Sons. All rights reserved.

y=3 x 5+5 x 3 -4 x 2 -x f(x) = x 3 -3

y=3 x 5+5 x 3 -4 x 2 -x f(x) = x 3 -3 x+10 Applied Calculus, 3/E by Deborah Hughes-Hallet Copyright 2006 by John Wiley & Sons. All rights reserved.

f(x) = x 3 -3 x+10 f(x) = x 3/6 -x 2/4 -x+2 Applied

f(x) = x 3 -3 x+10 f(x) = x 3/6 -x 2/4 -x+2 Applied Calculus, 3/E by Deborah Hughes-Hallet Copyright 2006 by John Wiley & Sons. All rights reserved.

f(x) = x 3/6 -x 2/4 -x+2 Applied Calculus, 3/E by Deborah Hughes-Hallet Copyright

f(x) = x 3/6 -x 2/4 -x+2 Applied Calculus, 3/E by Deborah Hughes-Hallet Copyright 2006 by John Wiley & Sons. All rights reserved.

Section 4. 3 Global Maxima and Minima Section 4. 3: Global Maxima and Minima

Section 4. 3 Global Maxima and Minima Section 4. 3: Global Maxima and Minima Applied Calculus, 3/E by Deborah Hughes-Hallet Copyright 2006 by John Wiley & Sons. All rights reserved.

Slide 1: Blue box on pg. 188 and Figure 4. 30. . . Figure

Slide 1: Blue box on pg. 188 and Figure 4. 30. . . Figure 4. 30 Applied Calculus, 3/E by Deborah Hughes-Hallet Copyright 2006 by John Wiley & Sons. All rights reserved.

. . . . Figure 4. 31 Applied Calculus, 3/E by Deborah Hughes-Hallet Copyright

. . . . Figure 4. 31 Applied Calculus, 3/E by Deborah Hughes-Hallet Copyright 2006 by John Wiley & Sons. All rights reserved.

Graph f(x)=x 3 - ex for -2< x < 5. Approximate the global max.

Graph f(x)=x 3 - ex for -2< x < 5. Approximate the global max. f(x) = 3/4 x 4 – 3/2 x 2 Applied Calculus, 3/E by Deborah Hughes-Hallet Copyright 2006 by John Wiley & Sons. All rights reserved.

Find all max and min of f(x) = 3/4 x 4 – 3/2 x

Find all max and min of f(x) = 3/4 x 4 – 3/2 x 2 f(x) = (1 -x)/(1+x 2) Applied Calculus, 3/E by Deborah Hughes-Hallet Copyright 2006 by John Wiley & Sons. All rights reserved.

Find the global max of f(x) = (1 -x)/(1+x 2) f(x) = 2 x

Find the global max of f(x) = (1 -x)/(1+x 2) f(x) = 2 x 3 – 9 x 2 + 12 x + 1 Applied Calculus, 3/E by Deborah Hughes-Hallet Copyright 2006 by John Wiley & Sons. All rights reserved.

f(x) = 2 x 3 – 9 x 2 + 12 x + 1

f(x) = 2 x 3 – 9 x 2 + 12 x + 1 f(x) = 4 x – x 2 - 5 Applied Calculus, 3/E by Deborah Hughes-Hallet Copyright 2006 by John Wiley & Sons. All rights reserved.

f(x) = 4 x – x 2 - 5 f(x) = 2 x 3

f(x) = 4 x – x 2 - 5 f(x) = 2 x 3 + 3 x 2 – 36 x +100 Applied Calculus, 3/E by Deborah Hughes-Hallet Copyright 2006 by John Wiley & Sons. All rights reserved.

f(x) = 2 x 3 + 3 x 2 – 36 x +100 For

f(x) = 2 x 3 + 3 x 2 – 36 x +100 For x between -6 and + 5 f(x) = x 4 + 3 x 3 - 5 Applied Calculus, 3/E by Deborah Hughes-Hallet Copyright 2006 by John Wiley & Sons. All rights reserved.

f(x) = x 4 + 3 x 3 - 5 For x = -3

f(x) = x 4 + 3 x 3 - 5 For x = -3 to + 1 f(x) = x – ln x for x>0 Applied Calculus, 3/E by Deborah Hughes-Hallet Copyright 2006 by John Wiley & Sons. All rights reserved.

f(x) = x – ln x for x>0 Applied Calculus, 3/E by Deborah Hughes-Hallet

f(x) = x – ln x for x>0 Applied Calculus, 3/E by Deborah Hughes-Hallet Copyright 2006 by John Wiley & Sons. All rights reserved.

Section 4. 4 Profit, Cost, and Revenue Section 4. 4: Profit, Cost, and Revenue

Section 4. 4 Profit, Cost, and Revenue Section 4. 4: Profit, Cost, and Revenue Applied Calculus, 3/E by Deborah Hughes-Hallet Copyright 2006 by John Wiley & Sons. All rights reserved.

Slide 1: Figure 4. 43 Applied Calculus, 3/E by Deborah Hughes-Hallet Copyright 2006 by

Slide 1: Figure 4. 43 Applied Calculus, 3/E by Deborah Hughes-Hallet Copyright 2006 by John Wiley & Sons. All rights reserved.

Slide 2: Problem 1 Applied Calculus, 3/E by Deborah Hughes-Hallet Copyright 2006 by John

Slide 2: Problem 1 Applied Calculus, 3/E by Deborah Hughes-Hallet Copyright 2006 by John Wiley & Sons. All rights reserved.

Slide 3: Problem 2 Applied Calculus, 3/E by Deborah Hughes-Hallet Copyright 2006 by John

Slide 3: Problem 2 Applied Calculus, 3/E by Deborah Hughes-Hallet Copyright 2006 by John Wiley & Sons. All rights reserved.

Slide 4: Problem 28 Applied Calculus, 3/E by Deborah Hughes-Hallet Copyright 2006 by John

Slide 4: Problem 28 Applied Calculus, 3/E by Deborah Hughes-Hallet Copyright 2006 by John Wiley & Sons. All rights reserved.

. . Figure 4. 57 Slide 1: Box on pg. 203 and Figure 4.

. . Figure 4. 57 Slide 1: Box on pg. 203 and Figure 4. 57 – first without the line, then add line and labels Applied Calculus, 3/E by Deborah Hughes-Hallet Copyright 2006 by John Wiley & Sons. All rights reserved.

. . Figure 4. 57 Slide 1: Box on pg. 203 and Figure 4.

. . Figure 4. 57 Slide 1: Box on pg. 203 and Figure 4. 57 – first without the line, then add line and labels Applied Calculus, 3/E by Deborah Hughes-Hallet Copyright 2006 by John Wiley & Sons. All rights reserved.

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