Lesson 2-4 Finding Maximums and Minimums of Polynomial Functions.

Slides:

Advertisements

Similar presentations

Factor and Solve Quadratic Equations

Advertisements

Section 5.4 I can use calculus to solve optimization problems.

Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley.

Pre – CalcLesson 2.4 Finding Maximums and Minimums of Polynomial Functions For quadratic functions: f(x) = ax 2 + bx + c To fin d the max. or min. 1 st.

Applications of Differentiation

3.4 Quadratic Modeling.

A rectangular dog pen is constructed using a barn wall as one side and 60m of fencing for the other three sides. Find the dimensions of the pen that.

4.4 Optimization Finding Optimum Values. A Classic Problem You have 40 feet of fence to enclose a rectangular garden. What is the maximum area that you.

2.8 Analyzing Graphs of Polynomial Functions p. 373

Pg. 116 Homework Pg. 117#44 – 51 Pg. 139#78 – 86 #20 (-2, 1)#21[0, 3] #22 (-∞, 2]U[3, ∞)#24(-∞, -3]U[½, ∞) #25 (0, 1)#26(-∞, -3]U(1, ∞) #27 [-2, 0]U[4,

Objectives Use properties of end behavior to analyze, describe, and graph polynomial functions. Identify and use maxima and minima of polynomial functions.

Using Calculus to Solve Optimization Problems

AIM: APPLICATIONS OF FUNCTIONS? HW P. 27 #74, 76, 77, Functions Worksheet #1-3 1 Expressing a quantity as a function of another quantity. Do Now: Express.

Rev.S08 MAC 1105 Module 4 Quadratic Functions and Equations.

Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley.

4.7 Applied Optimization Wed Jan 14

6.4 - The Quadratic Formula

© 2008 Pearson Addison-Wesley. All rights reserved Chapter 1 Section 8-5 Quadratic Functions, Graphs, and Models.

Copyright © 2007 Pearson Education, Inc. Slide 3-1.

Copyright © 2014, 2010, 2006 Pearson Education, Inc. 1 Chapter 3 Quadratic Functions and Equations.

Quiz Show. Polynomial Operations 100ABCDE Evaluating Polynomials 200ABCDE Factoring & Solving Polynomials 300ABCDE App. Problems & Theorems 400ABCDE Polynomial.

Warm-ups Find each product. 1. (x + 2)(x + 7)2. (x – 11)(x + 5) 3. (x – 10) 2 Factor each polynomial. 4. x x x 2 + 2x – x 2.

6.8 Analyzing Graphs of Polynomial Functions

Optimization. Objective  To solve applications of optimization problems  TS: Making decisions after reflection and review.

C2: Maxima and Minima Problems

Notes Over 6.8 Using x-Intercepts to Graph a Polynomial Function Graph the function. x-inter: 1, -2 End behavior: degree 3 L C: positive Bounces off of.

Theorems I LOVE Parametric Equations Factors, Remainders, And Roots (oh my!) Max-MinInequalitiesGraph.

Factoring to Solve Quadratic Equations ALGEBRA 1 LESSON 10-5 (For help, go to Lessons 2-2 and 9-6.) Solve and check each equation n = 22. – 9 =

Section 4.5 Optimization and Modeling. Steps in Solving Optimization Problems 1.Understand the problem: The first step is to read the problem carefully.

Precalculus Section 2.4 Use polynomials to find maximum and minimum values Example 1 page 69 Area = length x width A(x) = (60 – 2x)(x) A(x) = 60x - 2x².

Make a Model A box company makes boxes to hold popcorn. Each box is made by cutting the square corners out of a rectangular sheet of cardboard. The rectangle.

Optimization Problems Section 4-4. Example  What is the maximum area of a rectangle with a fixed perimeter of 880 cm? In this instance we want to optimize.

A25 & 26-Optimization (max & min problems). Guidelines for Solving Applied Minimum and Maximum Problems 1.Identify all given quantities and quantities.

Chapter 11 Maximum and minimum points and optimisation problems Learning objectives:  Understand what is meant by stationary point  Find maximum and.

Building Boxes What is the largest volume open top box that you can build from an 8 ½ by 11 inch sheet of paper?

2.6 Extreme Values of Functions

EQ: How are extreme values useful in problem solving situations?

Sect. 3-7 Optimization.

2.1 Quadratic Functions Standard form Applications.

Ch. 5 – Applications of Derivatives

Copyright © 2006 Pearson Education, Inc

Chapter 3: Polynomial Functions

Chapter 4: Rational Power, and Root Functions

Analyzing Graphs of Polynomial Functions

MAXIMIZING AREA AND VOLUME

Module 4 Quadratic Functions and Equations

Calculus I (MAT 145) Dr. Day Wednesday Nov 8, 2017

Calculus I (MAT 145) Dr. Day Friday Nov 10, 2017

Dilrajkumar 06 X;a.

Optimization Chapter 4.4.

6.8 Analyzing Graphs of Polynomial Functions

Factoring to Solve Quadratic Equations

Polynomial Functions.

Chapters 1 & 2 Review Day.

A farmer has 100m of fencing to attach to a long wall, making a rectangular pen. What is the optimal width of this rectangle to give the pen the largest.

Polynomials: Application

Sketch the graph of each function. 1. −

Using Calculus to Solve Optimization Problems

Optimization (Max/Min)

Objectives Use properties of end behavior to analyze, describe, and graph polynomial functions. Identify and use maxima and minima of polynomial functions.

Day 168 – Cubical and cuboidal structures

Chapter 3: Polynomial Functions

Objectives Use properties of end behavior to analyze, describe, and graph polynomial functions. Identify and use maxima and minima of polynomial functions.

Tell me everything you can about this relationship:

6.7 Using the Fundamental Theorem of Algebra

Differentiation and Optimisation

Optimization (Max/Min)

Finding Maximums & Minimums of Polynomial Functions fguilbert.

Presentation transcript:

Lesson 2-4 Finding Maximums and Minimums of Polynomial Functions

Objective:

Objective: To write a polynomial function for a given situation and to find the maximum or minimum value of the function.

: For quadratic functions:

To find the minimum or maximum:

: For quadratic functions: To find the minimum or maximum: 1 st : Find x.

: For quadratic functions: To find the minimum or maximum: 1 st : Find x. 2 nd : Substitute x back into the equation to find y.

: For quadratic functions: To find the minimum or maximum: This y value is the minimum or maximum.

A rectangular dog pen is constructed using a barn wall as one side and 60m of fence for the other three sides. Find the dimensions of the pen that give the greatest possible area.

1. Draw a picture.

A rectangular dog pen is constructed using a barn wall as one side and 60m of fence for the other three sides. Find the dimensions of the pen that give the greatest possible area. 1. Draw a picture. 2. Let x = length (in meters) of one on the sides perpendicular to the barn.

A rectangular dog pen is constructed using a barn wall as one side and 60m of fence for the other three sides. Find the dimensions of the pen that give the greatest possible area. 1. Draw a picture. 2. Let x = length (in meters) of one on the sides perpendicular to the barn. 3. Determine the length of the other sides in terms of x. length = x width = ??

A rectangular dog pen is constructed using a barn wall as one side and 60m of fence for the other three sides. Find the dimensions of the pen that give the greatest possible area. 4. Recall that the area of a rectangle : A = ??

A rectangular dog pen is constructed using a barn wall as one side and 60m of fence for the other three sides. Find the dimensions of the pen that give the greatest possible area. 4. Recall that the area of a rectangle : A = ?? 5. Therefore, A(x) = x( ? ) A(x) = ?

A rectangular dog pen is constructed using a barn wall as one side and 60m of fence for the other three sides. Find the dimensions of the pen that give the greatest possible area. 4. Recall that the area of a rectangle : A = ?? 5. Therefore, A(x) = x( ? ) A(x) = ?

A rectangular dog pen is constructed using a barn wall as one side and 60m of fence for the other three sides. Find the dimensions of the pen that give the greatest possible area. 4. Recall that the area of a rectangle : A = ?? 5. Therefore, A(x) = x( ? ) A(x) = ? Therefore, the maximum occurs when x = ?? (Note: this is not the maximum, just where it occurs)

A rectangular dog pen is constructed using a barn wall as one side and 60m of fence for the other three sides. Find the dimensions of the pen that give the greatest possible area. So, the dimensions would have to be ??

A rectangular dog pen is constructed using a barn wall as one side and 60m of fence for the other three sides. Find the dimensions of the pen that give the greatest possible area. So, the dimensions would have to be ?? So, the maximum area to enclose would be ??

Squares with sides of length x are cut from the corners of a rectangular piece of sheet metal with dimensions 6”x10”. The metal is folded to make an open-top box. What is the maximum volume of such a box?

1. Draw a picture.

Squares with sides of length x are cut from the corners of a rectangular piece of sheet metal with dimensions 6”x10”. The metal is folded to make an open-top box. What is the maximum volume of such a box? 1. Draw a picture. 2. Identify the dimensions in terms of x.

Squares with sides of length x are cut from the corners of a rectangular piece of sheet metal with dimensions 6”x10”. The metal is folded to make an open-top box. What is the maximum volume of such a box? 1. Draw a picture. 2. Identify the dimensions in terms of x. Height = x Length = 10 – 2x Width = 6 – 2x

Squares with sides of length x are cut from the corners of a rectangular piece of sheet metal with dimensions 6”x10”. The metal is folded to make an open-top box. What is the maximum volume of such a box? 3. Come up with a formula.

Squares with sides of length x are cut from the corners of a rectangular piece of sheet metal with dimensions 6”x10”. The metal is folded to make an open-top box. What is the maximum volume of such a box? 3. Come up with a formula. Recall the volume of a rectangular solid: V = l x w x h

Squares with sides of length x are cut from the corners of a rectangular piece of sheet metal with dimensions 6”x10”. The metal is folded to make an open-top box. What is the maximum volume of such a box? 3. Come up with a formula. Recall the volume of a rectangular solid: V = l x w x h Therefore: V(x) = (10-2x)(6-2x)(x)

Squares with sides of length x are cut from the corners of a rectangular piece of sheet metal with dimensions 6”x10”. The metal is folded to make an open-top box. What is the maximum volume of such a box? 3. Come up with a formula. Recall the volume of a rectangular solid: V = l x w x h Therefore: V(x) = (10-2x)(6-2x)(x) So, 10 – 2x = 0 and 6 – 2x = 0 and x = 0.

Squares with sides of length x are cut from the corners of a rectangular piece of sheet metal with dimensions 6”x10”. The metal is folded to make an open-top box. What is the maximum volume of such a box? 3. Come up with a formula. Recall the volume of a rectangular solid: V = l x w x h Therefore: V(x) = (10-2x)(6-2x)(x) So, 10 – 2x = 0 and 6 – 2x = 0 and x = 0. When solved; x = 5, x = 3, and x = 0.

Squares with sides of length x are cut from the corners of a rectangular piece of sheet metal with dimensions 6”x10”. The metal is folded to make an open-top box. What is the maximum volume of such a box? Plot these points on a graph, to find: Leading term: Constant term:

Squares with sides of length x are cut from the corners of a rectangular piece of sheet metal with dimensions 6”x10”. The metal is folded to make an open-top box. What is the maximum volume of such a box? Plot these points on a graph, to find: Leading term: 4x 3  a = 4 Constant term:

Squares with sides of length x are cut from the corners of a rectangular piece of sheet metal with dimensions 6”x10”. The metal is folded to make an open-top box. What is the maximum volume of such a box? Plot these points on a graph, to find: Leading term: 4x 3  a = 4  (graph rises) Constant term:

Squares with sides of length x are cut from the corners of a rectangular piece of sheet metal with dimensions 6”x10”. The metal is folded to make an open-top box. What is the maximum volume of such a box? Plot these points on a graph, to find: Leading term: 4x 3  a = 4  (graph rises) Constant term: 0  y-intercept

Now, make a sketch and rationalize the local max. This occurs between the roots of 0 and 3. Now, use your grapher to find the local max. It is when x = 1.21 and y = Therefore, the maximum volume is in 3. Answer problems from class exercises #1-5 in class.

Assignment: Pgs. 70 – 71 C.E.  1-4 all W.E.  1, 3, 9, 10, 11