Section 4.1 Increasing and Decreasing Functions
|
|
- Owen Chambers
- 5 years ago
- Views:
Transcription
1 Section.1 Increasing and Decreasing Functions The graph of the quadratic function f 1 is a parabola. If we imagine a particle moving along this parabola from left to right, we can see that, while the -coordinates of the ordered pairs steadil increase, the -coordinates of the ordered pairs along the particle s path first decrease and then increase. Determining the intervals in which a function increases and decreases is etremel useful for understanding the behaviour of the function. The following statements give a clear picture: = Intervals of Increase and Decrease We sa that a function f is decreasing on an interval if, for an value of 1 6 on the interval, f f 1. Similarl, we sa that a function f is increasing on an interval if, for an value of on the interval, f f For the parabola with the equation, the change from decreasing -values to increasing -values occurs at 1,, the verte of the parabola. The function f 1 is decreasing on the interval 6 and is increasing on the interval 7. If we eamine tangents to the parabola anwhere on the interval where the -values are decreasing (that is, on 6 ), we see that all of these tangents have negative slopes. Similarl, the slopes of tangents to the graph on the interval where the -values are increasing are all positive. tangent CHAPTER 165
2 For functions that are both continuous and differentiable, we can determine intervals of increasing and decreasing -values using the derivative of the function. In the case of,. For 6, and the slopes of the tangents d d 6, are negative. The interval 6 corresponds to the decreasing portion of the graph of the parabola. For 7, and the slopes of the tangents are positive on d 7, the interval where the graph is increasing. We summarize this as follows: For a continuous and differentiable function, f, the function values (-values) are increasing for all -values where f 1 7, and the function values (-values) are decreasing for all -values where f 1 6. EXAMPLE 1 Using the derivative to reason about intervals of increase and decrease Use our calculator to graph the following functions. Use the graph to estimate the values of for which the function values (-values) are increasing, and the values of for which the -values are decreasing. Verif our estimates with an algebraic solution. a. 3 3 b. 1 Solution a. Using a calculator, we obtain the graph of 3 3. Using the TRACE ke on the calculator, we estimate that the function values are increasing on 6, decreasing on 6 6, and increasing again on 7. To verif these estimates with an algebraic solution, we consider the slopes of the tangents. The slope of a general tangent to the graph of 3 3 is given b We first determine the values of for which. These d 3 6. d values tell us where the function has a local maimum or local minimum value. These are greatest and least values respectivel of a function in relation to its neighbouring values. Setting, d we obtain , These values of locate points on the graph where the slope of the tangent is zero (that is, where the tangent is horizontal) INCREASING AND DECREASING FUNCTIONS
3 Since this is a polnomial function it is continuous so is defined for all values d of. Because onl at and, the derivative must be either d positive or negative for all other values of. We consider the intervals 6, 6 6, and 7. Value of Sign of 3( ) d d 7 d 6 d 7 Slope of Tangents positive negative positive Values of Increasing or Decreasing increasing decreasing increasing So 3 3 is increasing on the intervals 6 and 7 and is decreasing on the interval 6 6. b. Using a calculator, we obtain the graph of Using the TRACE ke 1. on the calculator, we estimate that the function values (-values) are decreasing on 6 1, increasing on , and decreasing again on 7 1. We analze the intervals of increasing/decreasing -values for the function b determining where is positive and where it is negative. d 1 (Epress as a product) d Setting, d we obtain 1 or (Product and chain rules) (Simplif) (Solve) CHAPTER 167
4 These values of locate the points on the graph where the slope of the tangent is. Since the denominator of this rational function can never be, this function is continuous so is defined for all values of. Because at 1and 1, d d we consider the intervals 1 q, 1, 1 1, 1, and 11, q. Value of 1 q, 1 1 1, 1 11, q Sign of d d 6 d 7 d 6 Slope of Tangents negative positive negative Values of Increasing or Decreasing decreasing increasing decreasing Then is increasing on the interval 1 1, 1 and is decreasing on the 1 intervals 1 q, 1 and 11, q. EXAMPLE 8 = f () = f() 6 8 Graphing a function given the graph of the derivative Consider the graph of f 1. Graph f 1. 8 = f () Solution When the derivative, f 1, is positive, the graph of f 1 is rising. When the derivative is negative, the graph is falling. In this eample, the derivative changes sign from positive to negative at.6. This indicates that the graph of f 1 changes from increasing to decreasing, resulting in a local maimum for this value of. The derivative changes sign from negative to positive at.9, indicating the graph of f 1 changes from decreasing to increasing resulting in a local minimum for this value of. One possible graph of f 1 is shown INCREASING AND DECREASING FUNCTIONS
5 IN SUMMARY Ke Ideas A function f is increasing on an interval if, for an value of 1 6 in the interval, f f 1. A function f is decreasing on an interval if, for an value of 1 6 in the interval, f f 1. f( 1 ) f( ) f( 1 ) f( ) a 1 b a 1 b For a function f that is continuous and differentiable on an interval I f1 is increasing on I if f 1 7 for all values of in I f1 is decreasing on I if f 1 6 for all values of in I Need to Know A function increases on an interval if the graph rises from left to right. A function decreases on an interval if the graph falls from left to right. The slope of the tangent at a point on a section of a curve that is increasing is alwas positive. The slope of the tangent at a point on a section of a curve that is decreasing is alwas negative. Eercise.1 K C PART A 1. Determine the points at which f 1 for each of the following functions: a. f c. f b. f 1 d. f Eplain how ou would determine when a function is increasing or decreasing. 3. For each of the following graphs, state i. the intervals where the function is increasing ii. the intervals where the function is decreasing iii. the points where the tangent to the function is horizontal CHAPTER 169
6 a. c. b. d.. Use a calculator to graph each of the following functions. Inspect the graph to estimate where the function is increasing and where it is decreasing. Verif our estimates with algebraic solutions. a. f d. f b. f e. f c. f 1 1 f. f 1 1 A PART B 5. Suppose that f is a differentiable function with the derivative f Determine the values of for which the function f is increasing and the values of for which the function is decreasing. 6. Sketch a graph of a function that is differentiable on the interval 5 and that satisfies the following conditions: The graph of f passes through the points 1 1, and 1, 5. The function f is decreasing on 6 6 1, increasing on 1 6 6, and decreasing again on Find constants a, b, and c such that the graph of f 1 3 a b c will increase to the point 1 3, 18, decrease to the point 11, 1, and then continue increasing. 8. Sketch a graph of a function f that is differentiable and that satisfies the following conditions: f 1 7, when 6 5 f 1 6, when and when 7 1 f 1 5 and f 11 f and f INCREASING AND DECREASING FUNCTIONS
7 9. Each of the following graphs represents the derivative function f 1 of a function f 1. Determine i. the intervals where f 1 is increasing ii. the intervals where f 1 is decreasing iii. the -coordinate for all local etrema of f 1 Assuming that f 1, make a rough sketch of the graph of each function. a. c. =f'() =f'() b. d. =f'() =f'() 1. Use the derivative to show that the graph of the quadratic function f 1 a b c, a 7, is decreasing on the interval 6 b and increasing on the interval 7 b a a. 11. For f 1 3, find where f 1, the intervals on which the function increases and decreases, and all the local etrema. Use graphing technolog to verif our results. 1. Sketch a graph of the function g that is differentiable on the interval 5, decreases on 6 6 3, and increases elsewhere on the domain. The absolute maimum of g is 7, and the absolute minimum is 3. The graph of g has local etrema at 1, and 13, 1. T PART C 13. Let f and g be continuous and differentiable functions on the interval a b. If f and g are both increasing on a b, and if f 1 7 and g1 7 on a b, show that the product fg is also increasing on a b. 1. Let f and g be continuous and differentiable functions on the interval a b. If f and g are both increasing on a b, and if f 1 6 and g1 6 on a b, is the product fg increasing on a b, decreasing, or neither? CHAPTER 171
Math RE - Calculus I Functions Page 1 of 10. Topics of Functions used in Calculus
Math 0-03-RE - Calculus I Functions Page of 0 Definition of a function f() : Topics of Functions used in Calculus A function = f() is a relation between variables and such that for ever value onl one value.
More information4.3 Mean-Value Theorem and Monotonicity
.3 Mean-Value Theorem and Monotonicit 1. Mean Value Theorem Theorem: Suppose that f is continuous on the interval a, b and differentiable on the interval a, b. Then there eists a number c in a, b such
More informationSystems of Linear Equations: Solving by Graphing
8.1 Sstems of Linear Equations: Solving b Graphing 8.1 OBJECTIVE 1. Find the solution(s) for a set of linear equations b graphing NOTE There is no other ordered pair that satisfies both equations. From
More informationDaily Lessons and Assessments for AP* Calculus AB, A Complete Course Page 119 Mark Sparks 2012
Unit # Understanding the Derivative Homework Packet f ( h) f ( Find lim for each of the functions below. Then, find the equation of the tangent line to h 0 h the graph of f( at the given value of. 1. f
More information1 x
Unit 1. Calculus Topic 4: Increasing and decreasing functions: turning points In topic 4 we continue with straightforward derivatives and integrals: Locate turning points where f () = 0. Determine the
More informationMaintaining Mathematical Proficiency
Name Date Chapter 8 Maintaining Mathematical Proficienc Graph the linear equation. 1. = 5. = + 3 3. 1 = + 3. = + Evaluate the epression when =. 5. + 8. + 3 7. 3 8. 5 + 8 9. 8 10. 5 + 3 11. + + 1. 3 + +
More informationMath 3201 UNIT 5: Polynomial Functions NOTES. Characteristics of Graphs and Equations of Polynomials Functions
1 Math 301 UNIT 5: Polnomial Functions NOTES Section 5.1 and 5.: Characteristics of Graphs and Equations of Polnomials Functions What is a polnomial function? Polnomial Function: - A function that contains
More information3 Polynomial and Rational Functions
3 Polnomial and Rational Functions 3.1 Quadratic Functions and Models 3.2 Polnomial Functions and Their Graphs 3.3 Dividing Polnomials 3.4 Real Zeros of Polnomials 3.5 Comple Zeros and the Fundamental
More informationPACKET Unit 4 Honors ICM Functions and Limits 1
PACKET Unit 4 Honors ICM Functions and Limits 1 Day 1 Homework For each of the rational functions find: a. domain b. -intercept(s) c. y-intercept Graph #8 and #10 with at least 5 EXACT points. 1. f 6.
More informationAttributes of Polynomial Functions VOCABULARY
8- Attributes of Polnomial Functions TEKS FCUS Etends TEKS ()(A) Graph the functions f () =, f () =, f () =, f () =, f () = b, f () =, and f () = log b () where b is,, and e, and, when applicable, analze
More informationSection 2.5: Graphs of Functions
Section.5: Graphs of Functions Objectives Upon completion of this lesson, ou will be able to: Sketch the graph of a piecewise function containing an of the librar functions. o Polnomial functions of degree
More informationPolynomial and Rational Functions
Name Date Chapter Polnomial and Rational Functions Section.1 Quadratic Functions Objective: In this lesson ou learned how to sketch and analze graphs of quadratic functions. Important Vocabular Define
More informationQUADRATIC FUNCTION REVIEW
Name: Date: QUADRATIC FUNCTION REVIEW Linear and eponential functions are used throughout mathematics and science due to their simplicit and applicabilit. Quadratic functions comprise another ver important
More informationSTUDY KNOWHOW PROGRAM STUDY AND LEARNING CENTRE. Functions & Graphs
STUDY KNOWHOW PROGRAM STUDY AND LEARNING CENTRE Functions & Graphs Contents Functions and Relations... 1 Interval Notation... 3 Graphs: Linear Functions... 5 Lines and Gradients... 7 Graphs: Quadratic
More informationWriting Quadratic Functions in Standard Form
Chapter Summar Ke Terms standard form (general form) of a quadratic function (.1) parabola (.1) leading coefficient (.) second differences (.) vertical motion model (.3) zeros (.3) interval (.3) open interval
More informationCoordinate geometry. + bx + c. Vertical asymptote. Sketch graphs of hyperbolas (including asymptotic behaviour) from the general
A Sketch graphs of = a m b n c where m = or and n = or B Reciprocal graphs C Graphs of circles and ellipses D Graphs of hperbolas E Partial fractions F Sketch graphs using partial fractions Coordinate
More informationINVESTIGATE the Math
. Graphs of Reciprocal Functions YOU WILL NEED graph paper coloured pencils or pens graphing calculator or graphing software f() = GOAL Sketch the graphs of reciprocals of linear and quadratic functions.
More informationMath 121. Practice Questions Chapters 2 and 3 Fall Find the other endpoint of the line segment that has the given endpoint and midpoint.
Math 11. Practice Questions Chapters and 3 Fall 01 1. Find the other endpoint of the line segment that has the given endpoint and midpoint. Endpoint ( 7, ), Midpoint (, ). Solution: Let (, ) denote the
More information1.2 Functions and Their Properties PreCalculus
1. Functions and Their Properties PreCalculus 1. FUNCTIONS AND THEIR PROPERTIES Learning Targets for 1. 1. Determine whether a set of numbers or a graph is a function. Find the domain of a function given
More informationName Period Date. Practice FINAL EXAM Intro to Calculus (50 points) Show all work on separate sheet of paper for full credit!
Name Period Date Practice FINAL EXAM Intro to Calculus (0 points) Show all work on separate sheet of paper for full credit! ) Evaluate the algebraic epression for the given value or values of the variable(s).
More information4.3 Exercises. local maximum or minimum. The second derivative is. e 1 x 2x 1. f x x 2 e 1 x 1 x 2 e 1 x 2x x 4
SECTION 4.3 HOW DERIVATIVES AFFECT THE SHAPE OF A GRAPH 297 local maimum or minimum. The second derivative is f 2 e 2 e 2 4 e 2 4 Since e and 4, we have f when and when 2 f. So the curve is concave downward
More informationAlgebra 2 Semester Exam Review
Algebra Semester Eam Review 7 Graph the numbers,,,, and 0 on a number line Identif the propert shown rs rs r when r and s Evaluate What is the value of k k when k? Simplif the epression 7 7 Solve the equation
More informationMathematics 309 Conic sections and their applicationsn. Chapter 2. Quadric figures. ai,j x i x j + b i x i + c =0. 1. Coordinate changes
Mathematics 309 Conic sections and their applicationsn Chapter 2. Quadric figures In this chapter want to outline quickl how to decide what figure associated in 2D and 3D to quadratic equations look like.
More informationLESSON #24 - POWER FUNCTIONS COMMON CORE ALGEBRA II
1 LESSON #4 - POWER FUNCTIONS COMMON CORE ALGEBRA II Before we start to analze polnomials of degree higher than two (quadratics), we first will look at ver simple functions known as power functions. The
More informationMathematics 10 Page 1 of 7 The Quadratic Function (Vertex Form): Translations. and axis of symmetry is at x a.
Mathematics 10 Page 1 of 7 Verte form of Quadratic Relations The epression a p q defines a quadratic relation called the verte form with a horizontal translation of p units and vertical translation of
More informationUNCORRECTED SAMPLE PAGES. 3Quadratics. Chapter 3. Objectives
Chapter 3 3Quadratics Objectives To recognise and sketch the graphs of quadratic polnomials. To find the ke features of the graph of a quadratic polnomial: ais intercepts, turning point and ais of smmetr.
More informationPure Core 1. Revision Notes
Pure Core Revision Notes Ma 06 Pure Core Algebra... Indices... Rules of indices... Surds... 4 Simplifing surds... 4 Rationalising the denominator... 4 Quadratic functions... 5 Completing the square....
More informationUNCORRECTED. To recognise the rules of a number of common algebraic relations: y = x 1 y 2 = x
5A galler of graphs Objectives To recognise the rules of a number of common algebraic relations: = = = (rectangular hperbola) + = (circle). To be able to sketch the graphs of these relations. To be able
More informationf(x) = 2x 2 + 2x - 4
4-1 Graphing Quadratic Functions What You ll Learn Scan the tet under the Now heading. List two things ou will learn about in the lesson. 1. Active Vocabular 2. New Vocabular Label each bo with the terms
More informationLESSON #42 - INVERSES OF FUNCTIONS AND FUNCTION NOTATION PART 2 COMMON CORE ALGEBRA II
LESSON #4 - INVERSES OF FUNCTIONS AND FUNCTION NOTATION PART COMMON CORE ALGEBRA II You will recall from unit 1 that in order to find the inverse of a function, ou must switch and and solve for. Also,
More informationAlgebra 2 Honors Summer Packet 2018
Algebra Honors Summer Packet 018 Solving Linear Equations with Fractional Coefficients For these problems, ou should be able to: A) determine the LCD when given two or more fractions B) solve a linear
More informationMath 141 Review for Midterm
Math 141 Review for Midterm There will be two parts to this test. Part 1 will have graph sketching, and no calculator is allowed. Part will have everthing else, and a calculator and/or graphing calculator
More informationCubic and quartic functions
3 Cubic and quartic functions 3A Epanding 3B Long division of polnomials 3C Polnomial values 3D The remainder and factor theorems 3E Factorising polnomials 3F Sum and difference of two cubes 3G Solving
More information1.2 Functions and Their Properties PreCalculus
1. Functions and Their Properties PreCalculus 1. FUNCTIONS AND THEIR PROPERTIES Learning Targets for 1. 1. Determine whether a set of numbers or a graph is a function. Find the domain of a function given
More informationChapter 18 Quadratic Function 2
Chapter 18 Quadratic Function Completed Square Form 1 Consider this special set of numbers - the square numbers or the set of perfect squares. 4 = = 9 = 3 = 16 = 4 = 5 = 5 = Numbers like 5, 11, 15 are
More informationMaintaining Mathematical Proficiency
Chapter Maintaining Mathematical Proficienc Find the -intercept of the graph of the linear equation. 1. = + 3. = 3 + 5 3. = 10 75. = ( 9) 5. 7( 10) = +. 5 + 15 = 0 Find the distance between the two points.
More informationMA123, Chapter 1: Equations, functions and graphs (pp. 1-15)
MA123, Chapter 1: Equations, functions and graphs (pp. 1-15) Date: Chapter Goals: Identif solutions to an equation. Solve an equation for one variable in terms of another. What is a function? Understand
More information5. Zeros. We deduce that the graph crosses the x-axis at the points x = 0, 1, 2 and 4, and nowhere else. And that s exactly what we see in the graph.
. Zeros Eample 1. At the right we have drawn the graph of the polnomial = ( 1) ( 2) ( 4). Argue that the form of the algebraic formula allows ou to see right awa where the graph is above the -ais, where
More informationLesson 4.1 Exercises, pages
Lesson 4.1 Eercises, pages 57 61 When approimating answers, round to the nearest tenth. A 4. Identify the y-intercept of the graph of each quadratic function. a) y = - 1 + 5-1 b) y = 3-14 + 5 Use mental
More informationNorthwest High School s Algebra 2/Honors Algebra 2
Northwest High School s Algebra /Honors Algebra Summer Review Packet 0 DUE Frida, September, 0 Student Name This packet has been designed to help ou review various mathematical topics that will be necessar
More informationLaw of Sines, Law of Cosines, Heron s Formula:
PreAP Math Analsis nd Semester Review Law of Sines, Law of Cosines, Heron s Formula:. Determine how man solutions the triangle has and eplain our reasoning. (FIND YOUR FLOW CHART) a. A = 4, a = 4 ards,
More informationDifferentiation and applications
FS O PA G E PR O U N C O R R EC TE D Differentiation and applications. Kick off with CAS. Limits, continuit and differentiabilit. Derivatives of power functions.4 C oordinate geometr applications of differentiation.5
More informationand y f ( x ). given the graph of y f ( x ).
FUNCTIONS AND RELATIONS CHAPTER OBJECTIVES:. Concept of function f : f ( ) : domain, range; image (value). Odd and even functions Composite functions f g; Identit function. One-to-one and man-to-one functions.
More informationLesson 9.1 Using the Distance Formula
Lesson. Using the Distance Formula. Find the eact distance between each pair of points. a. (0, 0) and (, ) b. (0, 0) and (7, ) c. (, 8) and (, ) d. (, ) and (, 7) e. (, 7) and (8, ) f. (8, ) and (, 0)
More information8 Differential Calculus 1 Introduction
8 Differential Calculus Introduction The ideas that are the basis for calculus have been with us for a ver long time. Between 5 BC and 5 BC, Greek mathematicians were working on problems that would find
More informationLESSON #28 - POWER FUNCTIONS COMMON CORE ALGEBRA II
1 LESSON #8 - POWER FUNCTIONS COMMON CORE ALGEBRA II Before we start to analze polnomials of degree higher than two (quadratics), we first will look at ver simple functions known as power functions. The
More informationCollege Algebra Final, 7/2/10
NAME College Algebra Final, 7//10 1. Factor the polnomial p() = 3 5 13 4 + 13 3 + 9 16 + 4 completel, then sketch a graph of it. Make sure to plot the - and -intercepts. (10 points) Solution: B the rational
More informationReview of Essential Skills and Knowledge
Review of Essential Skills and Knowledge R Eponent Laws...50 R Epanding and Simplifing Polnomial Epressions...5 R 3 Factoring Polnomial Epressions...5 R Working with Rational Epressions...55 R 5 Slope
More informationAPPENDIX D Rotation and the General Second-Degree Equation
APPENDIX D Rotation and the General Second-Degree Equation Rotation of Aes Invariants Under Rotation After rotation of the - and -aes counterclockwise through an angle, the rotated aes are denoted as the
More informationAttributes and Transformations of Quadratic Functions VOCABULARY. Maximum value the greatest. Minimum value the least. Parabola the set of points in a
- Attributes and Transformations of Quadratic Functions TEKS FCUS VCABULARY TEKS ()(B) Write the equation of a parabola using given attributes, including verte, focus, directri, ais of smmetr, and direction
More informationAlgebra I Practice Questions ? 1. Which is equivalent to (A) (B) (C) (D) 2. Which is equivalent to 6 8? (A) 4 3
1. Which is equivalent to 64 100? 10 50 8 10 8 100. Which is equivalent to 6 8? 4 8 1 4. Which is equivalent to 7 6? 4 4 4. Which is equivalent to 4? 8 6 Page 1 of 0 11 Practice Questions 6 1 5. Which
More informationAlgebra 2 CPA Summer Assignment 2018
Algebra CPA Summer Assignment 018 This assignment is designed for ou to practice topics learned in Algebra 1 that will be relevant in the Algebra CPA curriculum. This review is especiall important as ou
More information9.12 Quadratics Review
Algebra Name _ B2g0gD6L jkwudtaaa msvopfwtowiarneq CLOLXCa.I K `Awljla `rtiugohhtfs_ QrIefsfeYrZvtetdf. 9.2 Quadratics Review ) What is the difference between the two mathematical statements below? Then
More informationALGEBRA 1 CP FINAL EXAM REVIEW
ALGEBRA CP FINAL EXAM REVIEW Alg CP Sem Eam Review 0 () Page of 8 Chapter 8: Eponents. Write in rational eponent notation. 7. Write in radical notation. Simplif the epression.. 00.. 6 6. 7 7. 6 6 8. 8
More informationPath of the Horse s Jump y 3. transformation of the graph of the parent quadratic function, y 5 x 2.
- Quadratic Functions and Transformations Content Standards F.BF. Identif the effect on the graph of replacing f() b f() k, k f(), f(k), and f( k) for specific values of k (both positive and negative)
More informationPolynomial and Rational Functions
Polnomial and Rational Functions Figure -mm film, once the standard for capturing photographic images, has been made largel obsolete b digital photograph. (credit film : modification of ork b Horia Varlan;
More information(2.5) 1. Solve the following compound inequality and graph the solution set.
Intermediate Algebra Practice Final Math 0 (7 th ed.) (Ch. -) (.5). Solve the following compound inequalit and graph the solution set. 0 and and > or or (.7). Solve the following absolute value inequalities.
More information5.6 RATIOnAl FUnCTIOnS. Using Arrow notation. learning ObjeCTIveS
CHAPTER PolNomiAl ANd rational functions learning ObjeCTIveS In this section, ou will: Use arrow notation. Solve applied problems involving rational functions. Find the domains of rational functions. Identif
More informationName Please print your name as it appears on the class roster.
Berkele Cit College Practice Problems Math 1 Precalculus - Final Eam Preparation Name Please print our name as it appears on the class roster. SHORT ANSWER. Write the word or phrase that best completes
More informationPolynomial and Rational Functions
Polnomial and Rational Functions Figure -mm film, once the standard for capturing photographic images, has been made largel obsolete b digital photograph. (credit film : modification of ork b Horia Varlan;
More informationCh. 9.3 Vertex to General Form. of a Parabola
Ch. 9.3 Verte to General Form Learning Intentions: of a Parabola Change a quadratic equation from verte to general form. Learn to square a binomial & factor perfectsquare epressions using rectangle diagrams.
More information1.1 Laws of exponents Conversion between exponents and logarithms Logarithm laws Exponential and logarithmic equations 10
CNTENTS Algebra Chapter Chapter Chapter Eponents and logarithms. Laws of eponents. Conversion between eponents and logarithms 6. Logarithm laws 8. Eponential and logarithmic equations 0 Sequences and series.
More informationDerivatives 2: The Derivative at a Point
Derivatives 2: The Derivative at a Point 69 Derivatives 2: The Derivative at a Point Model 1: Review of Velocit In the previous activit we eplored position functions (distance versus time) and learned
More informationGraphing Calculator Computations 2
Graphing Calculator Computations A) Write the graphing calculator notation and B) Evaluate each epression. 4 1) 15 43 8 e) 15 - -4 * 3^ + 8 ^ 4/ - 1) ) 5 ) 8 3 3) 3 4 1 8 3) 7 9 4) 1 3 5 4) 5) 5 5) 6)
More informationMath 111 Lecture Notes
A rational function is of the form R() = p() q() where p and q are polnomial functions. The zeros of a rational function are the values of for which p() = 0, as the function s value is zero where the value
More informationREVIEW KEY VOCABULARY REVIEW EXAMPLES AND EXERCISES
Etra Eample. Graph.. 6. 7. (, ) (, ) REVIEW KEY VOCABULARY quadratic function, p. 6 standard form of a quadratic function, p. 6 parabola, p. 6 verte, p. 6 ais of smmetr, p. 6 minimum, maimum value, p.
More informationCALCULUS APPLICATIONS OF DIFFERENTIATION LESSON PLAN. C3 Topic Overview
CALCULUS C3 Topic Overview C3 APPLICATIONS OF DIFFERENTIATION Differentiation can be used to investigate the behaviour of a function, to find regions where the value of a function is increasing or decreasing
More informationAPPLIED ALGEBRA II SEMESTER 1 EXAM ITEM SPECIFICATION SHEET & KEY
APPLIED ALGEBRA II SEMESTER 1 EXAM ITEM SPECIFICATION SHEET & KEY Constructed Response # Objective Sllabus Objective NV State Standard 1 Graph a polnomial function. 1.1.7.1 Analze graphs of polnomial functions
More informationy intercept Gradient Facts Lines that have the same gradient are PARALLEL
CORE Summar Notes Linear Graphs and Equations = m + c gradient = increase in increase in intercept Gradient Facts Lines that have the same gradient are PARALLEL If lines are PERPENDICULAR then m m = or
More informationWeek #6 - Taylor Series, Derivatives and Graphs Section 4.1
Week #6 - Talor Series, Derivatives and Graphs Section 4.1 From Calculus, Single Variable b Hughes-Hallett, Gleason, McCallum et. al. Copright 2005 b John Wile & Sons, Inc. This material is used b permission
More informationLesson Goals. Unit 2 Functions Analyzing Graphs of Functions (Unit 2.2) Graph of a Function. Lesson Goals
Unit Functions Analzing Graphs of Functions (Unit.) William (Bill) Finch Mathematics Department Denton High School Lesson Goals When ou have completed this lesson ou will: Find the domain and range of
More informationUNIT #8 QUADRATIC FUNCTIONS AND THEIR ALGEBRA REVIEW QUESTIONS
Answer Ke Name: Date: UNIT #8 QUADRATIC FUNCTIONS AND THEIR ALGEBRA REVIEW QUESTIONS Part I Questions. For the quadratic function shown below, the coordinates of its verte are, (), 7 6,, 6 The verte is
More informationMATH section 3.4 Curve Sketching Page 1 of 29
MATH section. Curve Sketching Page of 9 The step by step procedure below is for regular rational and polynomial functions. If a function contains radical or trigonometric term, then proceed carefully because
More information20.2 Connecting Intercepts and Linear Factors
Name Class Date 20.2 Connecting Intercepts and Linear Factors Essential Question: How are -intercepts of a quadratic function and its linear factors related? Resource Locker Eplore Connecting Factors and
More informationConstant no variables, just a number. Linear Note: Same form as f () x mx b. Quadratic Note: Same form as. Cubic x to the third power
Precalculus Notes: Section. Modeling High Degree Polnomial Functions Graphs of Polnomials Polnomial Notation f ( ) a a a... a a a is a polnomial function of degree n. n n 1 n n n1 n 1 0 n is the degree
More informationSolving Polynomial Equations Exponential Growth in Factored Form
7.5 Solving Polnomial Equations Eponential Growth in Factored Form is written in factored form? How can ou solve a polnomial equation that Two polnomial equations are equivalent when the have the same
More informationAlgebra 1 Skills Needed for Success in Math
Algebra 1 Skills Needed for Success in Math A. Simplifing Polnomial Epressions Objectives: The student will be able to: Appl the appropriate arithmetic operations and algebraic properties needed to simplif
More informationA function from a set D to a set R is a rule that assigns a unique element in R to each element in D.
1.2 Functions and Their Properties PreCalculus 1.2 FUNCTIONS AND THEIR PROPERTIES Learning Targets for 1.2 1. Determine whether a set of numbers or a graph is a function 2. Find the domain of a function
More informationIn everyday speech, a continuous. Limits and Continuity. Critical Thinking Exercises
062 Chapter Introduction to Calculus Critical Thinking Eercises Make Sense? In Eercises 74 77, determine whether each statement makes sense or does not make sense, and eplain our reasoning. 74. I evaluated
More informationMathematics. Polynomials and Quadratics. hsn.uk.net. Higher. Contents. Polynomials and Quadratics 52 HSN22100
Higher Mathematics UNIT OUTCOME 1 Polnomials and Quadratics Contents Polnomials and Quadratics 5 1 Quadratics 5 The Discriminant 54 Completing the Square 55 4 Sketching Parabolas 57 5 Determining the Equation
More informationSection 3.3 Graphs of Polynomial Functions
3.3 Graphs of Polynomial Functions 179 Section 3.3 Graphs of Polynomial Functions In the previous section we eplored the short run behavior of quadratics, a special case of polynomials. In this section
More informationMath Honors Calculus I Final Examination, Fall Semester, 2013
Math 2 - Honors Calculus I Final Eamination, Fall Semester, 2 Time Allowed: 2.5 Hours Total Marks:. (2 Marks) Find the following: ( (a) 2 ) sin 2. (b) + (ln 2)/(+ln ). (c) The 2-th Taylor polynomial centered
More information2 variables. is the same value as the solution of. 1 variable. You can use similar reasoning to solve quadratic equations. Work with a partner.
9. b Graphing Essential Question How can ou use a graph to solve a quadratic equation in one variable? Based on what ou learned about the -intercepts of a graph in Section., it follows that the -intercept
More informationUnit #3 Rules of Differentiation Homework Packet
Unit #3 Rules of Differentiation Homework Packet In the table below, a function is given. Show the algebraic analysis that leads to the derivative of the function. Find the derivative by the specified
More informationCHAPTER 3 : QUADRARIC FUNCTIONS MODULE CONCEPT MAP Eercise 1 3. Recognizing the quadratic functions Graphs of quadratic functions 4 Eercis
ADDITIONAL MATHEMATICS MODULE 5 QUADRATIC FUNCTIONS CHAPTER 3 : QUADRARIC FUNCTIONS MODULE 5 3.1 CONCEPT MAP Eercise 1 3. Recognizing the quadratic functions 3 3.3 Graphs of quadratic functions 4 Eercise
More information( ) ( ) ( )( ) Algebra II Semester 2 Practice Exam A DRAFT. = x. Which expression is equivalent to. 1. Find the solution set of. f g x? B.
Algebra II Semester Practice Eam A. Find the solution set of 5 5 +, 9 9 +, 5 5 +, =.. Let f ( ) = and g( ) =. Which epression is equivalent to f g? ( ) ( ) + ( ) 9 9 +, 5. If h( ) = 0 and j( ) h( ) + j
More informationSquare Root Functions as Inverses. Inverse of a Quadratic Function. y f 1 (x) x
6-1 Square Root Functions as Inverses TEKS FOCUS TEKS ()(C) Describe and analze the relationship between a function and its inverse (quadratic and square root, logarithmic and eponential), including the
More informationDerivatives of Multivariable Functions
Chapter 0 Derivatives of Multivariable Functions 0. Limits Motivating Questions In this section, we strive to understand the ideas generated b the following important questions: What do we mean b the limit
More informationMATH 0312 FINAL EXAM REVIEW ITEMS
MATH 012 FINAL EXAM REVIEW ITEMS Name The items on this review are representative of the items that ou might see on our course final eam. No formul sheets are allowed and calculators are not allowed on
More informationKeira Godwin. Time Allotment: 13 days. Unit Objectives: Upon completion of this unit, students will be able to:
Keira Godwin Time Allotment: 3 das Unit Objectives: Upon completion of this unit, students will be able to: o Simplif comple rational fractions. o Solve comple rational fractional equations. o Solve quadratic
More informationTest # 33 QUESTIONS MATH131 091700 COLLEGE ALGEBRA Name atfm131bli www.alvarezmathhelp.com website MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
More information3.1 Graphing Quadratic Functions. Quadratic functions are of the form.
3.1 Graphing Quadratic Functions A. Quadratic Functions Completing the Square Quadratic functions are of the form. 3. It is easiest to graph quadratic functions when the are in the form using transformations.
More informationVertex. March 23, Ch 9 Guided Notes.notebook
March, 07 9 Quadratic Graphs and Their Properties A quadratic function is a function that can be written in the form: Verte Its graph looks like... which we call a parabola. The simplest quadratic function
More informationUnderstanding Part 2 of The Fundamental Theorem of Calculus
Understanding Part of The Fundamental Theorem of Calculus Worksheet 8: The Graph of F () What is an Anti-Derivative? Give an eample that is algebraic: and an eample that is graphical: eample : Below is
More information14.1 Systems of Linear Equations in Two Variables
86 Chapter 1 Sstems of Equations and Matrices 1.1 Sstems of Linear Equations in Two Variables Use the method of substitution to solve sstems of equations in two variables. Use the method of elimination
More information1,3. f x x f x x. Lim. Lim. Lim. Lim Lim. y 13x b b 10 b So the equation of the tangent line is y 13x
1.5 Topics: The Derivative lutions 1. Use the limit definition of derivative (the one with x in it) to find f x given f x 4x 5x 6 4 x x 5 x x 6 4x 5x 6 f x x f x f x x0 x x0 x xx x x x x x 4 5 6 4 5 6
More informationa [A] +Algebra 2/Trig Final Exam Review Fall Semester x [E] None of these [C] 512 [A] [B] 1) Simplify: [D] x z [E] None of these 2) Simplify: [A]
) Simplif: z z z 6 6 z 6 z 6 ) Simplif: 9 9 0 ) Simplif: a a a 0 a a ) Simplif: 0 0 ) Simplif: 9 9 6) Evaluate: / 6 6 6 ) Rationalize: ) Rationalize: 6 6 0 6 9) Which of the following are polnomials? None
More information14.3. Volumes of Revolution. Introduction. Prerequisites. Learning Outcomes
Volumes of Revolution 14.3 Introduction In this Section we show how the concept of integration as the limit of a sum, introduced in Section 14.1, can be used to find volumes of solids formed when curves
More information2 nd Semester Final Exam Review Block Date
Algebra 1B Name nd Semester Final Eam Review Block Date Calculator NOT Allowed Graph each function. Identif the verte and ais of smmetr. 1 (10-1) 1. (10-1). 3 (10-) 3. 4 7 (10-) 4. 3 6 4 (10-1) 5. Predict
More information39. (a) Use trigonometric substitution to verify that. 40. The parabola y 2x divides the disk into two
35. Prove the formula A r for the area of a sector of a circle with radius r and central angle. [Hint: Assume 0 and place the center of the circle at the origin so it has the equation. Then is the sum
More information