Mathematical Modeling

Slides:

Advertisements

Similar presentations

Inverse, Joint, and Combined Variation

Advertisements

Lesson 12.1 Inverse Variation pg. 642

What You Will Learn Recognize and solve direct and joint variation problems Recognize and solve inverse variation problems.

Direct Variation Certain formulas occur so frequently in applied situations that they are given special names. Variation formulas show how one quantity.

Modeling Using Variation

Date: 2.2 Power Functions with Modeling Definition Power Function Any function that can be written in the form: is a power function. The constant a is.

3.4-1 Variation. Many natural (physical) phenomena exhibit variation = one quantity (quantities) changing on account of another (or several) Principle.

Copyright © 2014, 2010, 2007 Pearson Education, Inc.

1 PRECALCULUS I Dr. Claude S. Moore Danville Community College Graphs and Lines Intercepts, symmetry, circles Slope, equations, parallel, perpendicular.

Lesson 8-4: Direct, Joint, and Inverse Variation.

Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. Chapter 10 Graphing Equations and Inequalities.

Section 3.6 Variation. Direct Variation If a situation gives rise to a linear function f(x) = kx, or y = kx, where k is a positive constant, we say.

Variation Variation describes the relationship between two or more variables. Two variables, x and y, can vary directly or inversely. Three or more variables.

EXAMPLE 5 Write a joint variation equation The variable z varies jointly with x and y. Also, z = –75 when x = 3 and y = –5. Write an equation that relates.

Direct Variation: y varies directly as x (y is directly proportional to x), if there is a nonzero constant k such th at 3.7 – Variation The number k is.

Variation and Proportion Indirect Proportion. The formula for indirect variation can be written as y=k/x where k is called the constant of variation.

Variation. Direct Variation if there is some nonzero constant k such that k is called the constant of variation.

Chapter 1 Section 4. Direct Variation and Proportion Direct Variation: The variable y varies directly as x if there is a nonzero constant k such that.

Direct and Inverse Variation

PRECALCULUS I EXPONENTIAL & LOG EQUATIONS Dr. Claude S. Moore Danville Community College.

ObjectivesExamples DefinitionActivity Prepared by: Michael Lacsina Val de Guzman.

Direct and Inverse Variation

Variation Chapter 9.1. Direct Variation As x increases/decreases, y increases/decreases too. y = kx k is called the Constant of Variation k ≠ 0 “y varies.

The domain of the composite function f(g(x)) is the set of all x in the domain of g such that g(x) is in the domain of f. The composition of the function.

Mathematical Modeling & Variation MATH Precalculus S. Rook.

1 PRECALCULUS I Dr. Claude S. Moore Danville Community College Mathematical Modeling Direct, inverse, joint variations; Least squares regression.

1 © 2010 Pearson Education, Inc. All rights reserved © 2010 Pearson Education, Inc. All rights reserved Chapter 3 Polynomial and Rational Functions.

Lesson 2.8, page 357 Modeling using Variation Objectives: To find equations of direct, inverse, and joint variation, and to solve applied problems involving.

2.8 Modeling Using Variation Pg. 364 #2-10 (evens), (evens) Objectives –Solve direct variation problems. –Solve inverse variation problems. –Solve.

Copyright © 2009 Pearson Education, Inc. CHAPTER 2: More on Functions 2.1 Increasing, Decreasing, and Piecewise Functions; Applications 2.2 The Algebra.

Finding and Using Inverse Functions: Lesson 50. LESSON OBJECTIVE: 1)Recognize and solve direct and joint variation problems. 2)Recognize and solve inverse.

2-3 D IRECT V ARIATION Objective: To write and interpret direct variation equations.

Variation and Proportion Direct Proportion. The formula for direct variation can be written as y=kx where k is called the constant of variation.   The.

Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall.

Lesson 6 & 7 Unit 5. Objectives Be able to find equations for direct variation: y = kx Be able to find equations for inverse variation: y = k/x Be able.

2.7 Variation. Direct Variation Let x and y denote 2 quantities. Then y varies directly with x, or y is directly proportional to x, if there is a nonzero.

Direct Variation We say that two positive quantities x and y vary directly if an increase in one causes a proportional increase in the other. In this case,

W RITING E QUATIONS OF L INES Lesson 2.4. V OCABULARY Direct Variation: two variables x and y show direct variation provided y = kx where k is a nonzero.

Section 2.6 Variation and Applications Copyright ©2013, 2009, 2006, 2001 Pearson Education, Inc.

U SING D IRECT AND I NVERSE V ARIATION D IRECT V ARIATION The variables x and y vary directly if, for a constant k, k  0. y = kx, Objective Lesson 11.3.

1.11 Modeling Variation.

Direct Variation 3.6. Direct Variation  Direct Variation is when two variables can be expressed as y=kx where k is a constant and k is not 0.  k is.

2.4 and10.2 Direct/Inverse/Joint Variation ©2001 by R. Villar All Rights Reserved.

X = Y. Direct variation X 1 X = Y 1 Y 2.

U SING D IRECT AND I NVERSE V ARIATION D IRECT V ARIATION The variables x and y vary directly if, for a constant k, y x =k,k, k  0. or y = kx,

Section 1.6 Mathematical Modeling

3.8 Direct, Inverse, and Joint Variation

Chapter 3.1 Variation.

Wed 2/24 Lesson 8 – 1 Learning Objective: To find direct, inverse, & joint variations Hw: Lesson 8 – 1 & 2 – 2 WS.

NOTES 2.3 & 9.1 Direct and Inverse Variation. Direct Variation A function in the form y = kx, where k is not 0 Constant of variation (k) is the coefficient.

3.8 – Direct, Inverse, and Joint Variation. Direct Variation When two variables are related in such a way that the ratio of their values remains constant.

Inverse Variation Lesson 11-1 SOL A.8. Inverse Variation.

9.1: Inverse and Joint Variation Objectives: Students will be able to… Write and use inverse variation models Write and use joint variation models.

Section 3.5 Mathematical Modeling Objective(s): To learn direct, inverse and joint variations. To learn how to apply the variation equations to find the.

6/22/2016Section 6.41 Section 6.4 Ratio, Proportion, and Variation Objectives 1.Solve proportions. 2.Solve problems using proportions. 3.Solve direct variation.

6/22/ Ratio, Proportion and Variation 1 Starter An HMO pamphlet contains the following recommended weight for women: Give yourself 100 pounds for.

Copyright 2013, 2009, 2005, 2002 Pearson, Education, Inc.

Copyright © 2014, 2010, 2007 Pearson Education, Inc.

3.8 Direct, Inverse, and Joint Variation

Copyright © 2014, 2010, 2007 Pearson Education, Inc.

Advanced Math Topics Mrs. Mongold

Variation Objectives: Construct a Model using Direct Variation

Inverse & Joint Variation

5-2 Direct Variation What is Direct Variation?

Inverse Variation Chapter 8 Section 8.10.

The variables x and y vary directly if, for a constant k,

9-2 Direct, Inverse, and Joint Variation

Chapter 1: Lesson 1.10 Mathematical Modeling & Relations

Mathematical Modeling and Variation

Presentation transcript:

Mathematical Modeling PRECALCULUS I Mathematical Modeling Direct, inverse, joint variations; Least squares regression Dr. Claude S. Moore Danville Community College

Direct Variation Statements 1. y varies directly as x. 2. y is directly proportional to x. 3. y = kx for some nonzero constant m. NOTE: k is the constant of variation or the constant of proportionality. Example: If y = 3 when x = 2, find k. y = kx yields 3 = m(2) or m = 1.5. Thus, y = 1.5x.

Direct Variation as nth Power 1. y varies directly as the nth power of x. 2. y is directly proportional to the nth power of x. 3. y = kxn for some nonzero constant k. NOTE: k is the constant of variation or constant of proportionality.

Inverse Variation Statements 1. y varies inversely as x. 2. y is inversely proportional to x. 3. y = k / x for some nonzero constant k. NOTE: k is the constant of variation or the constant of proportionality. Example: If y = 3 when x = 2, find k. y = k / x yields 3 = k / 2 or k = 6. Thus, y = 6 / x.

Joint Variation Statements 1. z varies jointly as x and y. 2. z is jointly proportional to x and y. 3. z = kxy for some nonzero constant k. NOTE: k is the constant of variation. Example: If z = 15 when x = 2 and y = 3, find k. z = kxy yields 15 = k(2)(3) or k = 15/6 = 2.5. Thus, z = 2.5xy.

USING DIRECT AND INVERSE VARIATION IN REAL LIFE Writing and Using a Model BICYCLING A bicyclist tips the bicycle when making turn. The angle B of the bicycle from the vertical direction is called the banking angle. banking angle, B

Writing and Using a Model Turning Radius Banking angle (degrees) Find an inverse variation model that relates B and r. SOLUTION From the graph, you can see that B = 32° when r = 3.5 feet. B = k r Write direct variation model. 32 = k 3.5 Substitute 32 for B and 3.5 for r. 112 = k Solve for k. The model is B = , where B is in degrees and r is in feet. 112 r

Writing and Using a Model Turning Radius Banking angle (degrees) Use the model to find the banking angle for a turning radius of 5 feet. SOLUTION Substitute 5 for r in the model you just found. B = 112 5 = 22.4 When the turning radius is 5 feet, the banking angle is about 22°.