Algebra II: Strand 2. Linear Functions; Topic 2. Slope and Rate of Change; Task 2.2.1
|
|
- Edith Carpenter
- 6 years ago
- Views:
Transcription
1 1 TASK 2.2.1: AVERAGE RATES OF CHANGE Solutions One of the ways in which we describe functions is by whether they are increasing, decreasing, or constant on an interval in their domain. If the graph of f consistently rises on the interval then we conclude that the function f is increasing on the interval. If the graph of f consistently falls on the interval then we conclude that the function f is decreasing on the interval. If the graph of f remains constant (horizontal) then we conclude that the function f is constant on the interval. This can be expressed symbolically as: A function f is increasing on an interval if for any x 1 and x 2 in the interval, where x 2 > x 1, then f (x 2 ) > f (x 1 ). A function f is decreasing on an interval if for any x 1 and x 2 in the interval, where x 2 > x 1, then f (x 2 ) < f (x 1 ). A function f is constant on an interval if for any x 1 and x 2 in the interval, where x 2 > x 1, then f (x 2 ) = f (x 1 ). A single function can be increasing on some intervals, decreasing on others, and constant on yet others. Consider the function shown below: f is increasing on the interval (-1,3) because for any x 1 and x 2 in the interval (-1,3), where x 2 > x 1, then f (x 2 ) > f (x 1 ). f is constant on an interval (3,6) because for any x 1 and x 2 in the interval (3,6), where x 2 > x 1, then f (x 2 ) = f (x 1 ). f is decreasing on the interval (6,10) because for any x 1 and x 2 in the interval (6,10), where x 2 > x 1, then f (x 2 ) < f (x 1 ).
2 2 It is useful to not only determine whether or not a function is increasing or decreasing over an interval but also how fast the function is either increasing or decreasing. To investigate how fast a function is increasing or decreasing over an interval, we calculate the change in f(x) relative to the change in x over the interval. This yields the average rate of change over the interval. To make this notion more mathematically precise we define the average rate of change of f from c to x as follows: If c is in the domain of a function y=f(x), the average rate of change of f from c to x is defined as: Average rate of change =!y!x f (x) " f (c) =, x # c x " c This expression is also called the difference quotient of f at c. Graphically, we can see that the average rate of change of f from c to x corresponds to the slope of the secant line passing through (c, f(c)) and (x, f(x)) as in the figure below: For a given value of c, the average rate of change of f from c to x is a function of x. Thus, we can say that for a given value of c, the function r c given by: r c (x) =!y f (x) " f (c) = is a function of f and represents the average rate of change of f!x x " c from c to x for any x not equal to c. If a function is increasing on the interval (c,x) then the sign of r c is positive. Why? When f is increasing x>c implies f(x)>f(c) so both (x-c) and (f(x)-f(c)) are positive.
3 3 If a function is decreasing on the interval (c,x) then the sign of r c is negative. Why? When f is decreasing x>c implies f(x)<f(c) so (x-c) is positive and (f(x)- f(c)) is negative and then their ratio is negative. If a function is constant on the interval (c,x) then r c (x)=0. Why? When f is constant x>c implies f(x)=f(c) so (x-c) is positive and (f(x)-f(c)) is zero and then their ratio is zero. Note that r c (x)>0 does not imply that f is increasing on (c,x). However, if we find that r c (x)>0 for any x in the interval (c,d) then we can conclude that f is increasing on the interval (c,d). The following illustrates this point: Consider the function f with graph f (8)! f (2) If we let c=2 and x = 8, then r 2 (8) = = 5! 4 = 1 is positive, but f is not 8! increasing on the interval (2,8). However, for any x in the interval (2,3), r 2 (x) is positive which would imply that f is increasing on the interval (2,3). Let s Try This! Complete the function tables for each of the following functions by finding f x i ( ) = f for the given values of x. You can use the graphing i calculator to make your calculations by following these steps: ( ) and
4 4 Enter the x values for the given function in L1. Press STAT, then ENTER to access the list editor. Enter the x i values in L1. Cursor to list L2 and move the cursor up to highlight the L2 at the top of the column. Type the function, replacing the x in the function with L1 (2 nd 1). For example, for x + 2, enter L The screen shots are for Problem 1, f (x) = 2x 1: Press ENTER. This will calculate the function values and store them in L2.
5 5 Move the cursor to list L3 and cursor up to highlight the L3 at the top of the column. Press 2 nd STAT (LIST), OPS, ΔList (7) Type L2 (2 nd 2), and close the parenthesis. Press ENTER. This will calculate Δf (x) for each pair of successive function values in list L2. This is also r(x) since the x values given are such that Δx = 1.
6 6 Graph each function on the graphing calculator and sketch the graph on the grid provided. Press Y= and enter Y1= 2x -1. Press ZOOM 6.
7 7 1. f (x) = 2x! 1 i x i f ( x i ) ( ) = f
8 8 2. f (x) =!3x + 2 i x i f ( x i ) ( ) = f
9 9 3. f (x) = 1 2 x! 2 i x i f ( x i ) ( ) = f
10 10 4. f (x) = x 2 i x i f ( x i ) ( ) = f
11 11 5. f (x) = x 3 i x i f ( x i ) ( ) = f
12 12 6. f (x) = 2 x i x i f ( x i ) ( ) = f
13 Reflect and Summarize What observations can you make about the average rate of change of linear functions compared to the average rate of change of non-linear functions? Linear functions have a constant average rate of change. Non-linear functions do not have a constant average rate of change. 2. Using the average rate of change of a function of f from c to x, r c (x) = f (x)! f (c), x! c and the slope-intercept form of the equation of a line, y = mx + b, prove that the slope, m, is the average rate of change of the function on any interval x 1,x 2 ( ). The rate of change of f from x 1 to x 2 is given by ( ) = f ( x 2 )! f ( x 1 ) r x1 x 2 x 2! x 1 for x 1 " x 2 ( ) = mx + b, we can express Since f x f (x 1 ) = mx 1 + b and f (x 2 ) = mx 2 + b, and substitute into r x1 ( x 2 ), giving ( ( ) = mx + b 2 )! ( mx 1 + b) r x1 x 2 x 2! x 1 Simplifying the numerator, we have r x1 ( x 2 ) = mx + b! mx! b 2 1 = mx! mx + b! b 2 1 = mx! mx 2 1 x 2! x 1 x 2! x 1 x 2! x 1 factoring out the m, we have ( ) = m ( x! x 2 1) r x1 x 2 x 2! x 1 = m for x 1 " x 2 Since x 1 and x 2 had no other restrictions other than x 1 " x 2, we have shown that the average rate of change of f from x 1 to x 2, for any x 1 and x 2 is m.
14 14 Math notes This task builds toward the understanding that the average rate of change of a linear function over any interval is constant and is equal to the slope of the linear function. We learn later in calculus that the slope of a linear function also corresponds to the instantaneous rate of change at any given point. Teaching notes It is helpful to introduce this task by sketching several graphs of functions and asking participants to note that parts of the graphs are rising, parts are falling, and parts are horizontal and that we have a mathematical way of describing this. Remind participants that we are examining the change in y over the change in x when we calculate values f (x)! f (c) for r c (x) =. Have participants work in individually initially and then move x! c them into groups to compare findings. Assign each group one of the questions 1-6 to present on chart paper. After group presentations of questions 1-6, have the participants work in groups on Reflect and Summarize task. After hearing participant responses to question 2 of the Reflect and Summarize task, the facilitator should have participants think about how they might describe the instantaneous rate of change of a linear function by building from their work that for an arbitrary interval (c,d) ( c! d ) the average rate of change (slope of the secant line through (c,f(c)) and (d, f(d)) is always constant and equal to the slope of the linear function. Have the participants graphically determine the instantaneous rate of change as d c. Extensions For each function in exercises 1-5, have participants derive a general formula for the f (x)! f (c) function r c (x) =. The facilitator may foreshadow calculus concepts of x! c instantaneous rates of change (derivatives) and limits by asking participants what the limiting function r c (x) would be as we consider smaller and smaller intervals (c,x).the facilitator may encourage participants to rewrite the function r c (x) in terms of the change f (c +!x) " f (c) in x:!x = x " c. They would then examine r c ( c +!x) =.!x
15 15 TASK 2.2.1: AVERAGE RATES OF CHANGE TASK 2.2.1: AVERAGE RATES OF CHANGE One of the ways in which we describe functions is by whether they are increasing, decreasing, or constant on an interval in their domain. If the graph of f consistently rises on the interval then we conclude that the function f is increasing on the interval. If the graph of f consistently falls on the interval then we conclude that the function f is decreasing on the interval. If the graph of f remains constant (horizontal) then we conclude that the function f is constant on the interval. This can be expressed symbolically as: A function f is increasing on an interval if for any x 1 and x 2 in the interval, where x 2 > x 1, then f (x 2 ) > f (x 1 ). A function f is decreasing on an interval if for any x 1 and x 2 in the interval, where x 2 > x 1, then f (x 2 ) < f (x 1 ). A function f is constant on an interval if for any x 1 and x 2 in the interval, where x 2 > x 1, then f (x 2 ) = f (x 1 ). A single function can be increasing on some intervals, decreasing on others, and constant on yet others. Consider the function shown below: f is increasing on the interval (-1,3) because for any x 1 and x 2 in the interval (-1,3), where x 2 > x 1, then f (x 2 ) > f (x 1 ). f is constant on an interval (3,6) because for any x 1 and x 2 in the interval (3,6), where x 2 > x 1, then f (x 2 ) = f (x 1 ). f is decreasing on the interval (6,10) because for any x 1 and x 2 in the interval (6,10), where x 2 > x 1, then f (x 2 ) < f (x 1 ).
16 16 It is useful to not only determine whether or not a function is increasing or decreasing over an interval but also how fast the function is either increasing or decreasing. To investigate how fast a function is increasing or decreasing over an interval, we calculate the change in f(x) relative to the change in x over the interval. This yields the average rate of change over the interval. To make this notion more mathematically precise we define the average rate of change of f from c to x as follows: If c is in the domain of a function y=f(x), the average rate of change of f from c to x is defined as: Average rate of change =!y!x f (x) " f (c) =, x # c x " c This expression is also called the difference quotient of f at c. Graphically, we can see that the average rate of change of f from c to x corresponds to the slope of the secant line passing through (c, f(c)) and (x, f(x)) as in the figure below: For a given value of c, the average rate of change of f from c to x is a function of x. Thus, we can say that for a given value of c, the function r c given by: r c (x) =!y f (x) " f (c) = is a function of f and represents the average rate of change of f!x x " c from c to x for any x not equal to c. If a function is increasing on the interval (c,x) then the sign of r c is positive. Why? If a function is decreasing on the interval (c,x) then the sign of r c is negative. Why?
17 17 If a function is constant on the interval (c,x) then r c (x)=0. Why? Note that r c (x)>0 does not imply that f is increasing on (c,x). However, if we find that r c (x)>0 for any x in the interval (c,d) then we can conclude that f is increasing on the interval (c,d). The following illustrates this point: Consider the function f with graph f (8)! f (2) If we let c=2, then r 2 (8) = = 5! 4 = 1 is positive, but f is not increasing on 8! the interval (2,8). However, for any x in the interval (2,3), r 2 (x) is positive which would imply that f is increasing on the interval (2,3). Let s Try This! Complete the function tables for each of the following functions by finding f x i ( ) = f for the given values of x. You can use the graphing i calculator to make your calculations by following these steps: ( ) and Enter the x values for the given function in L1. Cursor to list L2 and move the cursor up to highlight the L2 at the top of the column. Type the function, replacing the x in the function with L1 (2 nd 1). For example, for x + 2, enter L Press ENTER. This will calculate the function values and store them in L2.
18 Move the cursor to list L3 and cursor up to highlight the L3 at the top of the column. Press 2 nd STAT (LIST), OPS, ΔList (7) Type L2 (2 nd 2), and close the parenthesis. Press ENTER. This will calculate Δy for each pair of successive function values in list L2. This is also x 2 ( ) since the x values given are such that Δx = 1. Graph each function on the graphing calculator and sketch the graph on the grid provided. 1. f (x) = 2x! 1 18 i x i f ( x i ) ( ) = f
19 19 2. f (x) =!3x + 2 i x i f ( x i ) ( ) = f
20 20 3. f (x) = 1 2 x! 2 i x i f ( x i ) ( ) = f
21 21 4. f (x) = x 2 i x i f ( x i ) ( ) = f
22 22 5. f (x) = x 3 i x i f ( x i ) ( ) = f
23 23 6. f (x) = 2 x i x i f ( x i ) ( ) = f
24 Reflect and Summarize What observations can you make about the average rate of change of linear functions compared to the average rate of change of non-linear functions? 2. Using the average rate of change of a function of f from c to x, r c (x) = f (x)! f (c), x! c and the slope-intercept form of the equation of a line, y = mx + b, prove that the slope, m, is the average rate of change of the function on any interval x 1,x 2 ( ).
Math 131. The Derivative and the Tangent Line Problem Larson Section 2.1
Math 131. The Derivative and the Tangent Line Problem Larson Section.1 From precalculus, the secant line through the two points (c, f(c)) and (c +, f(c + )) is given by m sec = rise f(c + ) f(c) f(c +
More informationRewriting Absolute Value Functions as Piece-wise Defined Functions
Rewriting Absolute Value Functions as Piece-wise Defined Functions Consider the absolute value function f ( x) = 2x+ 4-3. Sketch the graph of f(x) using the strategies learned in Algebra II finding the
More informationAlgebra I Calculator Activities
First Nine Weeks SOL Objectives Calculating Measures of Central Tendency SOL A.17 Organize a set of data Calculate the mean, median, mode, and range of a set of data Describe the relationships between
More informationLesson 31 - Average and Instantaneous Rates of Change
Lesson 31 - Average and Instantaneous Rates of Change IBHL Math & Calculus - Santowski 1 Lesson Objectives! 1. Calculate an average rate of change! 2. Estimate instantaneous rates of change using a variety
More informationTangent Lines and Derivatives
The Derivative and the Slope of a Graph Tangent Lines and Derivatives Recall that the slope of a line is sometimes referred to as a rate of change. In particular, we are referencing the rate at which the
More information3.1 Day 1: The Derivative of a Function
A P Calculus 3.1 Day 1: The Derivative of a Function I CAN DEFINE A DERIVATIVE AND UNDERSTAND ITS NOTATION. Last chapter we learned to find the slope of a tangent line to a point on a graph by using a
More informationChapter 1. Functions and Graphs. 1.5 More on Slope
Chapter 1 Functions and Graphs 1.5 More on Slope 1/21 Chapter 1 Homework 1.5 p200 2, 4, 6, 8, 12, 14, 16, 18, 22, 24, 26, 29, 30, 32, 46, 48 2/21 Chapter 1 Objectives Find slopes and equations of parallel
More informationGUIDED NOTES 4.1 LINEAR FUNCTIONS
GUIDED NOTES 4.1 LINEAR FUNCTIONS LEARNING OBJECTIVES In this section, you will: Represent a linear function. Determine whether a linear function is increasing, decreasing, or constant. Interpret slope
More informationAP Calculus Summer Prep
AP Calculus Summer Prep Topics from Algebra and Pre-Calculus (Solutions are on the Answer Key on the Last Pages) The purpose of this packet is to give you a review of basic skills. You are asked to have
More informationReteach 2-3. Graphing Linear Functions. 22 Holt Algebra 2. Name Date Class
-3 Graphing Linear Functions Use intercepts to sketch the graph of the function 3x 6y 1. The x-intercept is where the graph crosses the x-axis. To find the x-intercept, set y 0 and solve for x. 3x 6y 1
More informationElement x in D is called the input or the independent variable of the function.
P a g e 1 Chapter 1. Functions and Mathematical Models Definition: Function A function f defined on a collection D of numbers is a rule that assigns to each number x in D a specific number f(x) or y. We
More information10/22/16. 1 Math HL - Santowski SKILLS REVIEW. Lesson 15 Graphs of Rational Functions. Lesson Objectives. (A) Rational Functions
Lesson 15 Graphs of Rational Functions SKILLS REVIEW! Use function composition to prove that the following two funtions are inverses of each other. 2x 3 f(x) = g(x) = 5 2 x 1 1 2 Lesson Objectives! The
More informationThis lesson examines the average and
NATIONAL MATH + SCIENCE INITIATIVE Mathematics 5 4 1 5 4 1 1 4 5 1 4 5 LEVEL Algebra or Math in a unit on quadratic functions MODULE/CONNECTION TO AP* Rate of Change: Average and Instantaneous *Advanced
More informationSection 3.1 Extreme Values
Math 132 Extreme Values Section 3.1 Section 3.1 Extreme Values Example 1: Given the following is the graph of f(x) Where is the maximum (x-value)? What is the maximum (y-value)? Where is the minimum (x-value)?
More informationSect 2.4 Linear Functions
36 Sect 2.4 Linear Functions Objective 1: Graphing Linear Functions Definition A linear function is a function in the form y = f(x) = mx + b where m and b are real numbers. If m 0, then the domain and
More informationAccel Alg E. L. E. Notes Solving Quadratic Equations. Warm-up
Accel Alg E. L. E. Notes Solving Quadratic Equations Warm-up Solve for x. Factor. 1. 12x 36 = 0 2. x 2 8x Factor. Factor. 3. 2x 2 + 5x 7 4. x 2 121 Solving Quadratic Equations Methods: (1. By Inspection)
More informationCalculus. Weijiu Liu. Department of Mathematics University of Central Arkansas 201 Donaghey Avenue, Conway, AR 72035, USA
Calculus Weijiu Liu Department of Mathematics University of Central Arkansas 201 Donaghey Avenue, Conway, AR 72035, USA 1 Opening Welcome to your Calculus I class! My name is Weijiu Liu. I will guide you
More information2015 SUMMER MATH PACKET
Name: Date: 05 SUMMER MATH PACKET College Algebra Trig. - I understand that the purpose of the summer packet is for my child to review the topics they have already mastered in previous math classes and
More informationExample: f(x) = 2x² + 1 Solution: Math 2 VM Part 5 Quadratic Functions April 25, 2017
Math 2 Variable Manipulation Part 5 Quadratic Functions MATH 1 REVIEW THE CONCEPT OF FUNCTIONS The concept of a function is both a different way of thinking about equations and a different way of notating
More informationPeriod: Date: Lesson 3B: Properties of Dilations and Equations of lines
Name: Period: Date: : Properties of Dilations and Equations of lines Learning Targets I can identify the properties of dilation mentioned as followed: dilation takes a line not passing through the center
More informationMINI LESSON. Lesson 2a Linear Functions and Applications
MINI LESSON Lesson 2a Linear Functions and Applications Lesson Objectives: 1. Compute AVERAGE RATE OF CHANGE 2. Explain the meaning of AVERAGE RATE OF CHANGE as it relates to a given situation 3. Interpret
More informationAP Calculus Worksheet: Chapter 2 Review Part I
AP Calculus Worksheet: Chapter 2 Review Part I 1. Given y = f(x), what is the average rate of change of f on the interval [a, b]? What is the graphical interpretation of your answer? 2. The derivative
More informationInstructor Notes for Chapters 3 & 4
Algebra for Calculus Fall 0 Section 3. Complex Numbers Goal for students: Instructor Notes for Chapters 3 & 4 perform computations involving complex numbers You might want to review the quadratic formula
More informationMATH 113: ELEMENTARY CALCULUS
MATH 3: ELEMENTARY CALCULUS Please check www.ualberta.ca/ zhiyongz for notes updation! 6. Rates of Change and Limits A fundamental philosophical truth is that everything changes. In physics, the change
More informationFoundations for Functions
Activity: TEKS: Overview: Materials: Regression Exploration (A.2) Foundations for functions. The student uses the properties and attributes of functions. The student is expected to: (D) collect and organize
More information1 Functions, Graphs and Limits
1 Functions, Graphs and Limits 1.1 The Cartesian Plane In this course we will be dealing a lot with the Cartesian plane (also called the xy-plane), so this section should serve as a review of it and its
More informationSection 2.1: The Derivative and the Tangent Line Problem Goals for this Section:
Section 2.1: The Derivative and the Tangent Line Problem Goals for this Section: Find the slope of the tangent line to a curve at a point. Day 1 Use the limit definition to find the derivative of a function.
More informationAB Calculus: Rates of Change and Tangent Lines
AB Calculus: Rates of Change and Tangent Lines Name: The World Record Basketball Shot A group called How Ridiculous became YouTube famous when they successfully made a basket from the top of Tasmania s
More informationMath 131. Rolle s and Mean Value Theorems Larson Section 3.2
Math 3. Rolle s and Mean Value Theorems Larson Section 3. Many mathematicians refer to the Mean Value theorem as one of the if not the most important theorems in mathematics. Rolle s Theorem. Suppose f
More informationLecture 5 - Logarithms, Slope of a Function, Derivatives
Lecture 5 - Logarithms, Slope of a Function, Derivatives 5. Logarithms Note the graph of e x This graph passes the horizontal line test, so f(x) = e x is one-to-one and therefore has an inverse function.
More informationSolving a Linear-Quadratic System
CC-18 Solving LinearQuadratic Systems Objective Content Standards A.REI.7 Solve a simple system consisting of a linear equation and a quadratic equation in two variables... A.REI.11 Explain why the x-coordinates
More informationTo get horizontal and slant asymptotes algebraically we need to know about end behaviour for rational functions.
Concepts: Horizontal Asymptotes, Vertical Asymptotes, Slant (Oblique) Asymptotes, Transforming Reciprocal Function, Sketching Rational Functions, Solving Inequalities using Sign Charts. Rational Function
More informationThe Coordinate Plane; Graphs of Equations of Two Variables. A graph of an equation is the set of all points which are solutions to the equation.
Hartfield College Algebra (Version 2015b - Thomas Hartfield) Unit TWO Page 1 of 30 Topic 9: The Coordinate Plane; Graphs of Equations of Two Variables A graph of an equation is the set of all points which
More informationFor those of you who are taking Calculus AB concurrently with AP Physics, I have developed a
AP Physics C: Mechanics Greetings, For those of you who are taking Calculus AB concurrently with AP Physics, I have developed a brief introduction to Calculus that gives you an operational knowledge of
More informationPolynomial Functions and Models
1 CA-Fall 2011-Jordan College Algebra, 4 th edition, Beecher/Penna/Bittinger, Pearson/Addison Wesley, 2012 Chapter 4: Polynomial Functions and Rational Functions Section 4.1 Polynomial Functions and Models
More informationSlope Fields and Differential Equations. Copyright Cengage Learning. All rights reserved.
Slope Fields and Differential Equations Copyright Cengage Learning. All rights reserved. Objectives Review verifying solutions to differential equations. Review solving differential equations. Review using
More informationAbsolute and Local Extrema
Extrema of Functions We can use the tools of calculus to help us understand and describe the shapes of curves. Here is some of the data that derivatives f (x) and f (x) can provide about the shape of the
More informationMath 122 Fall Handout 11: Summary of Euler s Method, Slope Fields and Symbolic Solutions of Differential Equations
1 Math 122 Fall 2008 Handout 11: Summary of Euler s Method, Slope Fields and Symbolic Solutions of Differential Equations The purpose of this handout is to review the techniques that you will learn for
More information1.1 : (The Slope of a straight Line)
1.1 : (The Slope of a straight Line) Equations of Nonvertical Lines: A nonvertical line L has an equation of the form y mx b. The number m is called the slope of L and the point (0, b) is called the y-intercept.
More informationLesson 28: Another Computational Method of Solving a Linear System
Lesson 28: Another Computational Method of Solving a Linear System Student Outcomes Students learn the elimination method for solving a system of linear equations. Students use properties of rational numbers
More informationconverges to a root, it may not always be the root you have in mind.
Math 1206 Calculus Sec. 4.9: Newton s Method I. Introduction For linear and quadratic equations there are simple formulas for solving for the roots. For third- and fourth-degree equations there are also
More informationCharacteristics of Linear Functions (pp. 1 of 8)
Characteristics of Linear Functions (pp. 1 of 8) Algebra 2 Parent Function Table Linear Parent Function: x y y = Domain: Range: What patterns do you observe in the table and graph of the linear parent
More informationThe Derivative of a Function Measuring Rates of Change of a function. Secant line. f(x) f(x 0 ) Average rate of change of with respect to over,
The Derivative of a Function Measuring Rates of Change of a function y f(x) f(x 0 ) P Q Secant line x 0 x x Average rate of change of with respect to over, " " " " - Slope of secant line through, and,
More informationDRAFT - Math 101 Lecture Note - Dr. Said Algarni
2 Limits 2.1 The Tangent Problems The word tangent is derived from the Latin word tangens, which means touching. A tangent line to a curve is a line that touches the curve and a secant line is a line that
More informationMA 123 September 8, 2016
Instantaneous velocity and its Today we first revisit the notion of instantaneous velocity, and then we discuss how we use its to compute it. Learning Catalytics session: We start with a question about
More informationRational Functions. Elementary Functions. Algebra with mixed fractions. Algebra with mixed fractions
Rational Functions A rational function f (x) is a function which is the ratio of two polynomials, that is, Part 2, Polynomials Lecture 26a, Rational Functions f (x) = where and are polynomials Dr Ken W
More informationGUIDED NOTES 5.6 RATIONAL FUNCTIONS
GUIDED NOTES 5.6 RATIONAL FUNCTIONS LEARNING OBJECTIVES In this section, you will: Use arrow notation. Solve applied problems involving rational functions. Find the domains of rational functions. Identify
More informationAlgebra II: Strand 5. Power, Polynomial, and Rational Functions; Topic 3. Rational Functions; Task 5.3.1
TASK 5.3.: RATIONAL FUNCTIONS AND THEIR RECIPROCALS Solutions Rational functions appear frequently in business, science, engineering, and medical applications. This activity explores some aspects of rational
More informationChapter 3A -- Rectangular Coordinate System
Fry Texas A&M University! Fall 2016! Math 150 Notes! Section 3A! Page61 Chapter 3A -- Rectangular Coordinate System A is any set of ordered pairs of real numbers. A relation can be finite: {(-3, 1), (-3,
More informationExponential Functions Dr. Laura J. Pyzdrowski
1 Names: (4 communication points) About this Laboratory An exponential function is an example of a function that is not an algebraic combination of polynomials. Such functions are called trancendental
More informationLesson 4 Linear Functions and Applications
In this lesson, we take a close look at Linear Functions and how real world situations can be modeled using Linear Functions. We study the relationship between Average Rate of Change and Slope and how
More informationMath 261 Calculus I. Test 1 Study Guide. Name. Decide whether the limit exists. If it exists, find its value. 1) lim x 1. f(x) 2) lim x -1/2 f(x)
Math 261 Calculus I Test 1 Study Guide Name Decide whether the it exists. If it exists, find its value. 1) x 1 f(x) 2) x -1/2 f(x) Complete the table and use the result to find the indicated it. 3) If
More informationSummer Packet A Math Refresher For Students Entering IB Mathematics SL
Summer Packet A Math Refresher For Students Entering IB Mathematics SL Name: PRECALCULUS SUMMER PACKET Directions: This packet is required if you are registered for Precalculus for the upcoming school
More informationWEEK 7 NOTES AND EXERCISES
WEEK 7 NOTES AND EXERCISES RATES OF CHANGE (STRAIGHT LINES) Rates of change are very important in mathematics. Take for example the speed of a car. It is a measure of how far the car travels over a certain
More informationSection 1.6. Functions
Section 1.6 Functions Definitions Relation, Domain, Range, and Function The table describes a relationship between the variables x and y. This relationship is also described graphically. x y 3 2 4 1 5
More informationCalculus 221 worksheet
Calculus 221 worksheet Graphing A function has a global maximum at some a in its domain if f(x) f(a) for all other x in the domain of f. Global maxima are sometimes also called absolute maxima. A function
More information1.2 Supplement: Mathematical Models: A Catalog of Essential Functions
Math 131 -copyright Angela Allen, Fall 2011 1 1.2 Supplement: Mathematical Models: A Catalog of Essential Functions Note: Some of these examples and figures come from your textbook Single Variable Calculus:
More information3.Applications of Differentiation
3.Applications of Differentiation 3.1. Maximum and Minimum values Absolute Maximum and Absolute Minimum Values Absolute Maximum Values( Global maximum values ): Largest y-value for the given function Absolute
More informationLet's look at some higher order equations (cubic and quartic) that can also be solved by factoring.
GSE Advanced Algebra Polynomial Functions Polynomial Functions Zeros of Polynomial Function Let's look at some higher order equations (cubic and quartic) that can also be solved by factoring. In the video,
More informationSection 1.4. Meaning of Slope for Equations, Graphs, and Tables
Section 1.4 Meaning of Slope for Equations, Graphs, and Tables Finding Slope from a Linear Equation Finding Slope from a Linear Equation Example Find the slope of the line Solution Create a table using
More informationCHAPTER 3 DIFFERENTIATION
CHAPTER 3 DIFFERENTIATION 3.1 THE DERIVATIVE AND THE TANGENT LINE PROBLEM You will be able to: - Find the slope of the tangent line to a curve at a point - Use the limit definition to find the derivative
More informationFormulas that must be memorized:
Formulas that must be memorized: Position, Velocity, Acceleration Speed is increasing when v(t) and a(t) have the same signs. Speed is decreasing when v(t) and a(t) have different signs. Section I: Limits
More informationYou are looking at a textile fabric pattern. Please answer the following questions.
Introductory Activity: You are looking at a textile fabric pattern. Please answer the following questions. 1.) What different patterns do you see inside the fabric? 2.) How are the shapes in the pattern
More informationMath 211 Business Calculus TEST 3. Question 1. Section 2.2. Second Derivative Test.
Math 211 Business Calculus TEST 3 Question 1. Section 2.2. Second Derivative Test. p. 1/?? Math 211 Business Calculus TEST 3 Question 1. Section 2.2. Second Derivative Test. Question 2. Section 2.3. Graph
More informationFinal Exam Review. MATH Intuitive Calculus Fall 2013 Circle lab day: Mon / Fri. Name:. Show all your work.
MATH 11012 Intuitive Calculus Fall 2013 Circle lab day: Mon / Fri Dr. Kracht Name:. 1. Consider the function f depicted below. Final Exam Review Show all your work. y 1 1 x (a) Find each of the following
More informationMATH 60 Course Notebook Chapter #1
MATH 60 Course Notebook Chapter #1 Integers and Real Numbers Before we start the journey into Algebra, we need to understand more about the numbers and number concepts, which form the foundation of Algebra.
More informationLesson 23: The Defining Equation of a Line
Classwork Exploratory Challenge/Exercises 1 3 1. Sketch the graph of the equation 9xx +3yy = 18 using intercepts. Then, answer parts (a) (f) that follow. a. Sketch the graph of the equation yy = 3xx +6
More informationUnit 6 Quadratic Relations of the Form y = ax 2 + bx + c
Unit 6 Quadratic Relations of the Form y = ax 2 + bx + c Lesson Outline BIG PICTURE Students will: manipulate algebraic expressions, as needed to understand quadratic relations; identify characteristics
More informationMidterm Review. Name: Class: Date: ID: A. Short Answer. 1. For each graph, write the equation of a radical function of the form y = a b(x h) + k.
Name: Class: Date: ID: A Midterm Review Short Answer 1. For each graph, write the equation of a radical function of the form y = a b(x h) + k. a) b) c) 2. Determine the domain and range of each function.
More informationReview Assignment II
MATH 11012 Intuitive Calculus KSU Name:. Review Assignment II 1. Let C(x) be the cost, in dollars, of manufacturing x widgets. Fill in the table with a mathematical expression and appropriate units corresponding
More informationPoint of intersection
Name: Date: Period: Exploring Systems of Linear Equations, Part 1 Learning Goals Define a system of linear equations and a solution to a system of linear equations. Identify whether a system of linear
More information1) The line has a slope of ) The line passes through (2, 11) and. 6) r(x) = x + 4. From memory match each equation with its graph.
Review Test 2 Math 1314 Name Write an equation of the line satisfying the given conditions. Write the answer in standard form. 1) The line has a slope of - 2 7 and contains the point (3, 1). Use the point-slope
More informationLesson 4 - Limits & Instantaneous Rates of Change
Lesson Objectives Lesson 4 - Limits & Instantaneous Rates of Cange SL Topic 6 Calculus - Santowski 1. Calculate an instantaneous rate of cange using difference quotients and limits. Calculate instantaneous
More informationTangent Planes, Linear Approximations and Differentiability
Jim Lambers MAT 80 Spring Semester 009-10 Lecture 5 Notes These notes correspond to Section 114 in Stewart and Section 3 in Marsden and Tromba Tangent Planes, Linear Approximations and Differentiability
More informationMaximum and Minimum Values (4.2)
Math 111.01 July 17, 2003 Summer 2003 Maximum and Minimum Values (4.2) Example. Determine the points at which f(x) = sin x attains its maximum and minimum. Solution: sin x attains the value 1 whenever
More informationEDEXCEL NATIONAL CERTIFICATE UNIT 4 MATHEMATICS FOR TECHNICIANS OUTCOME 1
Learning outcomes EDEXCEL NATIONAL CERTIFICATE UNIT 4 MATHEMATICS FOR TECHNICIANS OUTCOME 1 TUTORIAL 2 - LINEAR EQUATIONS AND GRAPHS On completion of this unit a learner should: 1 Know how to use algebraic
More informationChapter 5B - Rational Functions
Fry Texas A&M University Math 150 Chapter 5B Fall 2015 143 Chapter 5B - Rational Functions Definition: A rational function is The domain of a rational function is all real numbers, except those values
More informationSection 3.1. Best Affine Approximations. Difference Equations to Differential Equations
Difference Equations to Differential Equations Section 3.1 Best Affine Approximations We are now in a position to discuss the two central problems of calculus as mentioned in Section 1.1. In this chapter
More information1.2 Functions & Function Notation
1.2 Functions & Function Notation A relation is any set of ordered pairs. A function is a relation for which every value of the independent variable (the values that can be inputted; the t s; used to call
More informationPreCalculus Summer Assignment (2018/2019)
PreCalculus Summer Assignment (2018/2019) We are thrilled to have you join the Pre-Calculus family next year, and we want you to get a jump-start over the summer! You have learned so much valuable information
More informationChapter 2 Analysis of Graphs of Functions
Chapter Analysis of Graphs of Functions Chapter Analysis of Graphs of Functions Covered in this Chapter:.1 Graphs of Basic Functions and their Domain and Range. Odd, Even Functions, and their Symmetry..
More informationDIFFERENTIAL EQUATIONS
DIFFERENTIAL EQUATIONS Chapter 1 Introduction and Basic Terminology Most of the phenomena studied in the sciences and engineering involve processes that change with time. For example, it is well known
More informationGraphical Solutions of Linear Systems
Graphical Solutions of Linear Systems Consistent System (At least one solution) Inconsistent System (No Solution) Independent (One solution) Dependent (Infinite many solutions) Parallel Lines Equations
More informationSolving Polynomial and Rational Inequalities Algebraically. Approximating Solutions to Inequalities Graphically
10 Inequalities Concepts: Equivalent Inequalities Solving Polynomial and Rational Inequalities Algebraically Approximating Solutions to Inequalities Graphically (Section 4.6) 10.1 Equivalent Inequalities
More information2.2 The Derivative Function
2.2 The Derivative Function Arkansas Tech University MATH 2914: Calculus I Dr. Marcel B. Finan Recall that a function f is differentiable at x if the following it exists f f(x + h) f(x) (x) =. (2.2.1)
More informationAP Calculus AB. Introduction. Slide 1 / 233 Slide 2 / 233. Slide 4 / 233. Slide 3 / 233. Slide 6 / 233. Slide 5 / 233. Limits & Continuity
Slide 1 / 233 Slide 2 / 233 AP Calculus AB Limits & Continuity 2015-10-20 www.njctl.org Slide 3 / 233 Slide 4 / 233 Table of Contents click on the topic to go to that section Introduction The Tangent Line
More informationCalculus I Homework: The Derivatives of Polynomials and Exponential Functions Page 1
Calculus I Homework: The Derivatives of Polynomials and Exponential Functions Page 1 Questions Example Differentiate the function y = ae v + b v + c v 2. Example Differentiate the function y = A + B x
More informationLesson 5: The Graph of the Equation y = f(x)
Lesson 5: The Graph of the Equation y = f(x) Learning targets: I can identify when a function is increasing, decreasing, positive and negative and use interval notation to describe intervals where the
More informationNorth Carolina State University
North Carolina State University MA 141 Course Text Calculus I by Brenda Burns-Williams and Elizabeth Dempster August 7, 2014 Section1 Functions Introduction In this section, we will define the mathematical
More informationSections 8.1 & 8.2 Systems of Linear Equations in Two Variables
Sections 8.1 & 8.2 Systems of Linear Equations in Two Variables Department of Mathematics Porterville College September 7, 2014 Systems of Linear Equations in Two Variables Learning Objectives: Solve Systems
More informationAP Calculus AB. Slide 1 / 233. Slide 2 / 233. Slide 3 / 233. Limits & Continuity. Table of Contents
Slide 1 / 233 Slide 2 / 233 AP Calculus AB Limits & Continuity 2015-10-20 www.njctl.org Table of Contents click on the topic to go to that section Slide 3 / 233 Introduction The Tangent Line Problem Definition
More informationCartesian Plane. Analytic Geometry. Unit Name
3.1cartesian Unit Name Analytic Geometry Unit Goals 1. Create table of values in order to graph &/or determine if a relation is linear. Determine slope 3. Calculate missing information for linearelationships.
More information2 the maximum/minimum value is ( ).
Math 60 Ch3 practice Test The graph of f(x) = 3(x 5) + 3 is with its vertex at ( maximum/minimum value is ( ). ) and the The graph of a quadratic function f(x) = x + x 1 is with its vertex at ( the maximum/minimum
More informationMATH 1040 Objectives List
MATH 1040 Objectives List Textbook: Calculus, Early Transcendentals, 7th edition, James Stewart Students should expect test questions that require synthesis of these objectives. Unit 1 WebAssign problems
More informationChapter 2.1 Relations and Functions
Analyze and graph relations. Find functional values. Chapter 2.1 Relations and Functions We are familiar with a number line. A number line enables us to locate points, denoted by numbers, and find distances
More informationAP Calculus AB. Limits & Continuity.
1 AP Calculus AB Limits & Continuity 2015 10 20 www.njctl.org 2 Table of Contents click on the topic to go to that section Introduction The Tangent Line Problem Definition of a Limit and Graphical Approach
More informationSkill 6 Exponential and Logarithmic Functions
Skill 6 Exponential and Logarithmic Functions Skill 6a: Graphs of Exponential Functions Skill 6b: Solving Exponential Equations (not requiring logarithms) Skill 6c: Definition of Logarithms Skill 6d: Graphs
More informationLimit. Chapter Introduction
Chapter 9 Limit Limit is the foundation of calculus that it is so useful to understand more complicating chapters of calculus. Besides, Mathematics has black hole scenarios (dividing by zero, going to
More informationSuccessful completion of the core function transformations unit. Algebra manipulation skills with squares and square roots.
Extension A: Circles and Ellipses Algebra ; Pre-Calculus Time required: 35 50 min. Learning Objectives Math Objectives Students will write the general forms of Cartesian equations for circles and ellipses,
More informationMath M111: Lecture Notes For Chapter 3
Section 3.1: Math M111: Lecture Notes For Chapter 3 Note: Make sure you already printed the graphing papers Plotting Points, Quadrant s signs, x-intercepts and y-intercepts Example 1: Plot the following
More information