3.5 - Concavity. a concave up. a concave down

3.5 - Concavity 1. Concave up and concave down

. - Concavit. Concave up and concave down Eample: The graph of f is given below. Determine graphicall the interval on which f is For a function f that is differentiable on an interval I, the graph of f

f x How can we determine algebraically where f is concave up and where f is concave down?

Concavity - 3.5 1. Concave up and concave down Definition For a function f that is differentiable on an interval I, the graph of f is a. If f is concave up on a, b, then the secant line passing through

Section 4.4 Concavity and Points of Inflection

Section 4.4 Concavit and Points of Inflection In Chapter 3, ou saw that the second derivative of a function has applications in problems involving velocit and acceleration or in general rates-of-change

4 Using The Derivative

4 Using The Derivative 4.1 Local Maima and Minima * Local Maima and Minima Suppose p is a point in the domain of f : f has a local minimum at p if f (p) is less than or equal to the values of f for points

MA 220 Lesson 28 Notes Section 3.3 (p. 191, 2 nd half of text)

MA 220 Lesson 28 Notes Section 3.3 (p. 191, 2 nd half of tet) The property of the graph of a function curving upward or downward is defined as the concavity of the graph of a function. Concavity if how

Mth Test 3 Review Stewart 8e Chapter 4. For Test #3 study these problems, the examples in your notes, and the homework.

For Test #3 study these problems, the eamples in your notes, and the homework. I. Absolute Etrema A function, continuous on a closed interval, always has an absolute maimum and absolute minimum. They occur

MA 131 Lecture Notes Chapter 4 Calculus by Stewart

MA 131 Lecture Notes Chapter 4 Calculus by Stewart 4.1) Maimum and Minimum Values 4.3) How Derivatives Affect the Shape of a Graph A function is increasing if its graph moves up as moves to the right and

2.1. Definition: If a < b, then f(a) < f(b) for every a and b in that interval. If a < b, then f(a) > f(b) for every a and b in that interval.

1.1 Concepts: 1. f() is INCREASING on an interval: Definition: If a < b, then f(a) < f(b) for every a and b in that interval. A positive slope for the secant line. A positive slope for the tangent line.

4.4. Concavity and Curve Sketching. Concavity

4.4 Concavit and Curve Sketching 267 4.4 Concavit and Curve Sketching f' decreases CONCAVE DOWN 3 f' increases 0 CONCAVE UP FIGURE 4.25 The graph of ƒsd = 3 is concave down on s - q, 0d and concave up

MA123, Chapter 6: Extreme values, Mean Value Theorem, Curve sketching, and Concavity

MA123, Chapter 6: Etreme values, Mean Value Theorem, Curve sketching, and Concavit Chapter Goals: Appl the Etreme Value Theorem to find the global etrema for continuous function on closed and bounded interval.

ABSOLUTE EXTREMA AND THE MEAN VALUE THEOREM

61 LESSON 4-1 ABSOLUTE EXTREMA AND THE MEAN VALUE THEOREM Definitions (informal) The absolute maimum (global maimum) of a function is the -value that is greater than or equal to all other -values in the

Graphing Functions. 0, < x < 0 1, 0 x < is defined everywhere on R but has a jump discontinuity at x = 0. h(x) =

Graphing Functions Section. of your tetbook is devoted to reviewing a series of steps that you can use to develop a reasonable graph of a function. Here is my version of a list of things to check. You

Section 4.3: Derivatives and the Shapes of Curves

1 Section 4.: Derivatives and the Shapes of Curves Practice HW from Stewart Textbook (not to hand in) p. 86 # 1,, 7, 9, 11, 19, 1,, 5 odd The Mean Value Theorem If f is a continuous function on the closed

Unconstrained and Constrained Optimization

Unconstrained and Constrained Optimization Agenda General Ideas of Optimization Interpreting the First Derivative Interpreting the Second Derivative Unconstrained Optimization Constrained Optimization

Section 4.1: Maximum and Minimum Values

Section 4.: Maimum and Minimum Values In this chapter, we shall consider further applications of the derivative. The main application we shall consider is using derivatives to sketch accurate graphs of

Math RE - Calculus I Application of the derivative (1) Curve Sketching Page 1 of 9

Math 201-103-RE - Calculus I Application of the derivative (1) Curve Sketching Page 1 of 9 Critical numbers - Increasing and decreasing intervals - Relative Etrema Given f(), the derivatives f () and f

This handout will discuss three kinds of asymptotes: vertical, horizontal, and slant.

CURVE SKETCHING This is a handout that will help you systematically sketch functions on a coordinate plane. This handout also contains definitions of relevant terms needed for curve sketching. ASYMPTOTES:

Module 3 Graphing and Optimization

Module 3 Graphing and Optimization One of the most important applications of calculus to real-world problems is in the area of optimization. We will utilize the knowledge gained in the previous chapter,

PRACTICE FINAL - MATH 1210, Spring 2012 CHAPTER 1

PRACTICE FINAL - MATH 2, Spring 22 The Final will have more material from Chapter 4 than other chapters. To study for chapters -3 you should review the old practice eams IN ADDITION TO what appears here.

The Extreme Value Theorem (IVT)

3.1 3.6 old school 1 Extrema If f(c) f(x) (y values) for all x on an interval, then is the (value) of f(x) (the function) on that interval. If f(c) f(x) (y-values) for all x on an interval, then is the

Math 295: Exam 3 Name: Caleb M c Whorter Solutions Fall /16/ Minutes

Math 295: Eam 3 Name: Caleb M c Whorter Solutions Fall 2018 11/16/2018 50 Minutes Write your name on the appropriate line on the eam cover sheet. This eam contains 10 pages (including this cover page)

Section 4.3: How Derivatives Affect the Shape of the Graph

Section 4.3: How Derivatives Affect the Shape of the Graph What does the first derivative of a function tell you about the function? Where on the graph below is f x > 0? Where on the graph below is f x

Chapter 4.1 & 4.2 (Part 1) Practice Problems

Chapter 4. & 4. Part Practice Problems EXPECTED SKILLS: Understand how the signs of the first and second derivatives of a function are related to the behavior of the function. Know how to use the first

MAC 2311 Chapter 3 Review Materials (Part 1) Topics Include Max and Min values, Concavity, Curve Sketching (first and second derivative analysis)

MAC Chapter Review Materials (Part ) Topics Include Ma and Min values, Concavit, Curve Sketching (first and second derivative analsis) MULTIPLE CHOICE. Choose the one alternative that best completes the

Sections 1.3 Computation of Limits

1 Sections 1.3 Computation of Limits We will shortly introduce the it laws. Limit laws allows us to evaluate the it of more complicated functions using the it of simpler ones. Theorem Suppose that c is

The following information is for reviewing the material since Exam 3:

Outcomes List for Math 121 Calculus I Fall 2010-2011 General Information: The purpose of this Outcomes List is to give you a concrete summary of the material you should know, and the skills you should

1. y = f(x) y = f(x + 3) 3. y = f(x) y = f(x 1) 5. y = 3f(x) 6. y = f(3x) 7. y = f(x) 8. y = f( x) 9. y = f(x 3) + 1

.7 Transformations.7. Eercises To see all of the help resources associated with this section, click OSttS Chapter b. Suppose (, ) is on the graph of = f(). In Eercises - 8, use Theorem.7 to find a point

Chapter Seven. Chapter Seven

Chapter Seven Chapter Seven ConcepTests for Section 7. CHAPTER SEVEN 7. Which of the following is an antiderivative of + 7? + 7 + 7 + 7 (e) + C. The most general antiderivative is + 7 + C, so is one possible

Sections 4.3, 4.5 & 4.6: Graphing

Sections 4.3, 4.5 & 4.6: Graphing In this section, we shall see how facts about f () and f () can be used to supply useful information about the graph of f(). Since there are three sections devoted to

AP Calculus AB Mean Value Theorem (MVT) Unit 4 Packet B. 4. on the interval [ ]

WARM-UP: Name For each graph, draw the secant line through the two points on the graph corresponding to the endpoints of the indicated interval. On the indicated interval, draw any tangent lines to the

Math 111 Lecture Notes

A function f is even if for ever in the domain of f it holds that f( ) = f(). Visuall, an even function is smmetric about the -ais. A function f is odd if for ever in the domain of f it holds that f( )

CHAPTER 4 APPLICATIONS OF DERIVATIVES

CHAPTER 4 APPLICATIONS OF DERIVATIVES 4. EXTREME VALUES OF FUNCTIONS. An asolute minimum at c, an asolute maimum at. Theorem guarantees the eistence of such etreme values ecause h is continuous on [aß

Review Sheet Chapter 3

Review Sheet Chapter 3 1. Find the value of the derivative (if it exists) of the function at the extremum point (0,0). A) 0 B) 1 C) -1 D) E) 2. Find the value of the derivative (if it exists) of the function

AP Calculus AB Unit 2 Assessment

Class: Date: 203-204 AP Calculus AB Unit 2 Assessment Multiple Choice Identify the choice that best completes the statement or answers the question. A calculator may NOT be used on this part of the exam.

The term Concavity is used to describe the type of curvature the graph displays at any given point.

4 4 Concavity and the Second Derivative The term Concavity is used to describe the type of curvature the graph displays at any given point. The curve of the graph is called Up at point if the graph is

Optimization Methods: Optimization using Calculus-Stationary Points 1. Module - 2 Lecture Notes 1

Optimization Methods: Optimization using Calculus-Stationary Points 1 Module - Lecture Notes 1 Stationary points: Functions of Single and Two Variables Introduction In this session, stationary points of

CONCAVITY AND INFLECTION POINTS

CONCAVITY AND INFLECTION POINTS Find the Second Derivative of the function, f. Set the Second Derivative equal to zero and solve. Determine whether the Second Derivative is undefined for any x-values.

Downloaded from

1 Class XI: Math Chapter 13: Limits and Derivatives Chapter Notes Key-Concepts 1. The epected value of the function as dictated by the points to the left of a point defines the left hand it of the function

Increasing/Decreasing Behavior

Derivatives and the Shapes of Graphs In this section, we will specifically discuss the information that f (x) and f (x) give us about the graph of f(x); it turns out understanding the first and second

Use Derivatives to Sketch the Graph of a Polynomial Function.

Applications of Derivatives Curve Sketching (using derivatives): A) Polynomial Functions B) Rational Functions Lesson 5.2 Use Derivatives to Sketch the Graph of a Polynomial Function. Idea: 1) Identify

Further Differentiation

Worksheet 39 Further Differentiation Section Discriminant Recall that the epression a + b + c is called a quadratic, or a polnomial of degree The graph of a quadratic is called a parabola, and looks like

MCS 118 Quiz 1. Fall (5pts) Solve the following equations for x. 7x 2 = 4x x 2 5x = 2

MCS 8 Quiz Fall 6. (5pts) Solve the following equations for. 7 = 4 + 3. (5pts) Solve the following equations for. 3 5 = 3. (5pts) Factor 3 + 35 as much as possible. 4. (5pts) Simplify +. 5. (5pts) Solve

Section 4.1 Max and Min Values

Page 1 of 5 Section 4.1 Ma and Min Values Horizontal Tangents: We have looked at graphs and identified horizontal tangents, or places where the slope of the tangent line is zero. Q: For which values does

Graph Sketching. Review: 1) Interval Notation. Set Notation Interval Notation Set Notation Interval Notation. 2) Solving Inequalities

Lesson. Graph Sketching Review: ) Interval Notation Set Notation Interval Notation Set Notation Interval Notation a) { R / < < 5} b) I (, 3) ( 3, ) c){ R} d) I (, ] (0, ) e){ R / > 5} f) I [ 3,5) ) Solving

10. f(x) = 3 2 x f(x) = 3 x 12. f(x) = 1 x 2 + 1

Relations and Functions.6. Eercises To see all of the help resources associated with this section, click OSttS Chapter b. In Eercises -, sketch the graph of the given function. State the domain of the

AP Calculus AB. a.) Midpoint rule with 4 subintervals b.) Trapezoid rule with 4 subintervals

AP Calculus AB Unit 6 Review Name: Date: Block: Section : RAM and TRAP.) Evaluate using Riemann Sums L 4, R 4 for the following on the interval 8 with four subintervals. 4.) Approimate ( )d using a.) Midpoint

Graphs of Other Trig Functions

Graph y = tan. y 0 0 6 3 3 3 5 6 3 3 1 Graphs of Other Trig Functions.58 3 1.7 undefined 3 3 3 1.7-1 0.58 3 CHAT Pre-Calculus 3 The Domain is all real numbers ecept multiples of. (We say the domain is

4.3, Math 1410 Name: And now for something completely different... Well, not really.

4.3, Math 1410 Name: And now for something completely different... Well, not really. How derivatives affect the shape of a graph. Please allow me to offer some explanation as to why the first couple parts

The Fundamental Theorem of Calculus Using the Rule of Three

The Fundamental Theorem of Calculus Using the Rule of Three A. Approimations with Riemann sums. The area under a curve can be approimated through the use of Riemann (or rectangular) sums: n Area f ( k

Increasing/Decreasing Behavior

Derivatives and the Shapes of Graphs In this section, we will specifically discuss the information that f (x) and f (x) give us about the graph of f(x); it turns out understanding the first and second

4.6 Graphs of Other Trigonometric Functions

.6 Graphs of Other Trigonometric Functions Section.6 Graphs of Other Trigonometric Functions 09 Graph of the Tangent Function Recall that the tangent function is odd. That is, tan tan. Consequentl, the

Unit 1: Sections Skill Set

MthSc 106 Fall 2011 Calculus of One Variable I : Calculus by Briggs and Cochran Section 1.1: Review of Functions Unit 1: Sections 1.1 3.3 Skill Set Find the domain and range of a function. 14, 17 13, 15,

AH Properties of Functions.notebook April 19, 2018

Functions Rational functions are of the form where p(x) and q(x) are polynomials. If you can sketch a function without lifting the pencil off the paper, it is continuous. E.g. y = x 2 If there is a break

WebAssign hw2.3 (Homework)

WebAssign hw2.3 (Homework) Current Score : / 98 Due : Wednesday, May 31 2017 07:25 AM PDT Michael Lee Math261(Calculus I), section 1049, Spring 2017 Instructor: Michael Lee 1. /6 pointsscalc8 1.6.001.

Derivatives and Graphs of Functions

Derivatives and Graphs of Functions September 8, 2014 2.2 Second Derivatives, Concavity, and Graphs In the previous section, we discussed how our derivatives can be used to obtain useful information about

Relating Graphs of f and f

Relating Graphs of f and f Do Now: Answer each of the following questions. 1. When the function, f, is increasing, what does that mean about the derivative, f? 2. When the function, f, is decreasing, what

Section 3.1(part), Critical Numbers, Extreme Values, Increasing/Decreasing, Concave Up/Down MATH 1190

Section 3.(part), 3.3-3.4 Critical Numbers, Extreme Values, Increasing/Decreasing, Concave Up/Down MATH 9 9 rel max f (a) = ; slope tangent line = 8 7. slope of tangent line: neg f (a)

Theorem 2(B): Concave DOWNward

Montana State University M161: Survey of Calculus 61 Section 4.2 - Applications of the Second Derivative Honeybees This is a population graph for Cyprian honeybees raised in an apiary. The population is

THS Step By Step Calculus Chapter 3

Name: Class Period: Throughout this packet there will be blanks you are expected to fill in prior to coming to class. This packet follows your Larson Textbook. Do NOT throw away! Keep in 3 ring-binder

Math 205 Test 3 Grading Guidelines Problem 1 Part a: 1 point for figuring out r, 2 points for setting up the equation P = ln 2 P and 1 point for the initial condition. Part b: All or nothing. This is really

Unit 2 Functions Analyzing Graphs of Functions (Unit 2.2)

Unit 2 Functions Analzing Graphs of Functions (Unit 2.2) William (Bill) Finch Mathematics Department Denton High School Introduction Domain/Range Vert Line Zeros Incr/Decr Min/Ma Avg Rate Change Odd/Even

Math 1020 Objectives & Exercises Calculus Concepts Spring 2019

Section of Textbook 1.1 AND Learning Objectives/Testable Skills Identify four representations of a function. Specify input and output variables, input and output descriptions, and input and output units.

dt Acceleration is the derivative of velocity with respect to time. If a body's position at time t is S = f(t), the body's acceleration at time t is

APPLICATIN F DERIVATIVE INTRDUCTIN In this section we eamine some applications in which derivatives are used to represent and interpret the rates at which things change in the world around us. Let S be

4.3 Maximum and Minimum Values of a Function

MA: Prepared b Asst.Prof.Dr. Archara Pacheenburawana 83 4.3 Maimum and Minimum Values of a Function Some of the most important applications of differential calculus are optimization problems, in which

MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.

Eam Name MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Find the intervals on which the function is continuous. 2 1) y = ( + 5)2 + 10 A) (-, ) B)

TABLE OF CONTENTS CHAPTER 1 LIMIT AND CONTINUITY... 26

TABLE OF CONTENTS CHAPTER LIMIT AND CONTINUITY... LECTURE 0- BASIC ALGEBRAIC EXPRESSIONS AND SOLVING EQUATIONS... LECTURE 0- INTRODUCTION TO FUNCTIONS... 9 LECTURE 0- EXPONENTIAL AND LOGARITHMIC FUNCTIONS...

4.3 Finding Local Extreme Values: First and Second Derivatives

Arkansas Tech University MATH 2914: Calculus I Dr. Marcel B. Finan 4.3 Finding Local Extreme Values: First and Second Derivatives Recall that a function f(x) is said to be increasing (respectively decreasing)

Math 1314 Lesson 12 Curve Analysis (Polynomials) This lesson will cover analyzing polynomial functions using GeoGebra.

Math 1314 Lesson 12 Curve Analysis (Polynomials) This lesson will cover analyzing polynomial functions using GeoGebra. Suppose your company embarked on a new marketing campaign and was able to track sales

3 Applications of Di erentiation

9 Applications of Di erentiation This chapter is intended to illustrate some of the applications of di erentiation. The activities of Section. illustrate the relationship between the values of rst and

Secondary Math 3- Honors. 7-4 Inverse Trigonometric Functions

Secondary Math 3- Honors 7-4 Inverse Trigonometric Functions Warm Up Fill in the Unit What You Will Learn How to restrict the domain of trigonometric functions so that the inverse can be constructed. How

Curve Sketching. Calculus U Y2]0x1[5e OK`uJtAaR OSVoufRtIwnaRrre] ]LMLLCr.^ J TAulJlu BrXitgqhItTsq LrfeJsdeKrnv\eydE. Name ID: 1

Calculus U Y]01[5e OK`uJtAaR OSVoufRtIwnaRrre] ]LMLLCr.^ J TAulJlu BrXitgqhItTsq LrfeJsdeKrnv\edE. Curve Sketching Name ID: 1 Date Period For each problem, find the: and intercepts, asmptotes, -coordinates

Polynomial Functions I

Name Student ID Number Group Name Group Members Polnomial Functions I 1. Sketch mm() =, nn() = 3, ss() =, and tt() = 5 on the set of aes below. Label each function on the graph. 15 5 3 1 1 3 5 15 Defn:

2. Suppose we drew many tangent lines for this second curve. How do the slopes of these tangent lines change as we look from left to right?

Do now as a warm up: 1. Suppose we drew many tangent lines for this first curve. How do the slopes of these tangent lines change as we look from left to right? 2. Suppose we drew many tangent lines for

Limits. f(x) and lim. g(x) g(x)

Limits Limit Laws Suppose c is constant, n is a positive integer, and f() and g() both eist. Then,. [f() + g()] = f() + g() 2. [f() g()] = f() g() [ ] 3. [c f()] = c f() [ ] [ ] 4. [f() g()] = f() g()

Chapter III.C Analysis of The Second Derivative Draft Version 4/19/14 Martin Flashman 2005

Chapter III.C Analysis of The Second Derivative Draft Version 4/19/14 Martin Flashman 2005 In this section we will develop applications of the second derivative that explain more graphical features of

Functions Review Packet from November Questions. 1. The diagrams below show the graphs of two functions, y = f(x), and y = g(x). y y

Functions Review Packet from November Questions. The diagrams below show the graphs of two functions, = f(), and = g()..5 = f( ) = g( ).5 6º 8º.5 8º 6º.5 State the domain and range of the function f; the

Review Guide for MAT220 Final Exam Part I. Thursday December 6 th during regular class time.

Review Guide for MAT0 Final Exam Part I. Thursday December 6 th during regular class time. Part is worth 50% of your Final Exam grade. YOUR Syllabus approved calculator can be used on this part of the

(f) Find an interval over which f is concave upwards.

April 4, 2005 Name The total number of points available is 157. work. Throughout this test, show your 1. (24 points) Consider the function f(x) = 2x+9. For this function there are two 6x+3 important intervals:

Sec.4.1 Increasing and Decreasing Functions

U4L1: Sec.4.1 Increasing and Decreasing Functions A function is increasing on a particular interval if for any, then. Ie: As x increases,. A function is decreasing on a particular interval if for any,

Chapter 4. Trigonometric Functions. 4.6 Graphs of Other. Copyright 2014, 2010, 2007 Pearson Education, Inc.

Chapter 4 Trigonometric Functions 4.6 Graphs of Other Trigonometric Functions Copyright 2014, 2010, 2007 Pearson Education, Inc. 1 Objectives: Understand the graph of y = tan x. Graph variations of y =

Answers to Worksheet 5, Math 272

Answers to Worksheet 5, Math 7 1. Calculate the directional derivative of the function f(, y, z) = cos y sin z at the point a = (1, π/, 5π/6) in the direction u = (3, 0, 1). The gradient of f at a is (

INVERSE TRIGONOMETRIC FUNCTIONS

INVERSE TRIGNMETRIC FUNCTINS INTRDUCTIN In chapter we learnt that only one-one and onto functions are invertible. If a function f is one-one and onto then its inverse exists and is denoted by f. We shall

Transformation of curve. a. reflect the portion of the curve that is below the x-axis about the x-axis

Given graph of y f = and sketch:. Linear Transformation cf ( b + a) + d a. translate a along the -ais. f b. scale b along the -ais c. scale c along the y-ais d. translate d along the y-ais Transformation

ReviewUsingDerivatives.nb 1. As we have seen, the connection between derivatives of a function and the function itself is given by the following:

ReviewUsingDerivatives.nb Calculus Review: Using First and Second Derivatives As we have seen, the connection between derivatives of a function and the function itself is given by the following: à If f

NAME: Section # SSN: X X X X

Math 155 FINAL EXAM A May 5, 2003 NAME: Section # SSN: X X X X Question Grade 1 5 (out of 25) 6 10 (out of 25) 11 (out of 20) 12 (out of 20) 13 (out of 10) 14 (out of 10) 15 (out of 16) 16 (out of 24)

Math 1314 Lesson 12 Curve Analysis (Polynomials)

Math 1314 Lesson 12 Curve Analysis (Polynomials) This lesson will cover analyzing polynomial functions using GeoGebra. Suppose your company embarked on a new marketing campaign and was able to track sales

g(x) h(x) f (x) = Examples sin x +1 tan x!

Lecture 4-5A: An Introduction to Rational Functions A Rational Function f () is epressed as a fraction with a functiong() in the numerator and a function h() in the denominator. f () = g() h() Eamples

Math : Differentiation

EP-Program - Strisuksa School - Roi-et Math : Differentiation Dr.Wattana Toutip - Department of Mathematics Khon Kaen University 00 :Wattana Toutip wattou@kku.ac.th http://home.kku.ac.th/wattou. Differentiation

Outputs. Inputs. the point where the graph starts, as we ll see on Example 1.

We ve seen how to work with functions algebraically, by finding domains as well as function values. In this set of notes we ll be working with functions graphically, and we ll see how to find the domain

NOTES: ALGEBRA FUNCTION NOTATION

STARTER: 1. Graph f by completing the table. f, y -1 0 1 4 5 NOTES: ALGEBRA 4.1 FUNCTION NOTATION y. Graph f 4 4 f 4 4, y --5-4 - - -1 0 1 y A Brief Review of Function Notation We will be using function

What is the reasonable domain of this volume function? (c) Can there exist a volume of 0? (d) Estimate a maximum volume for the open box.

MA 15800 Lesson 11 Summer 016 E 1: From a rectangular piece of cardboard having dimensions 0 inches by 0 inches, an open bo is to be made by cutting out identical squares of area from each corner and,

EXERCISE I JEE MAIN. x continuous at x = 0 if a equals (A) 0 (B) 4 (C) 5 (D) 6 Sol. x 1 CONTINUITY & DIFFERENTIABILITY. Page # 20

Page # 0 EXERCISE I 1. A function f() is defined as below cos(sin) cos f() =, 0 and f(0) = a, f() is continuous at = 0 if a equals (A) 0 (B) 4 (C) 5 (D) 6 JEE MAIN CONTINUITY & DIFFERENTIABILITY 4. Let

In this chapter, we define limits of functions and describe some of their properties.

Chapter 2 Limits of Functions In this chapter, we define its of functions and describe some of their properties. 2.. Limits We begin with the ϵ-δ definition of the it of a function. Definition 2.. Let

Lesson #6: Basic Transformations with the Absolute Value Function

Lesson #6: Basic Transformations with the Absolute Value Function Recall: Piecewise Functions Graph:,, What parent function did this piecewise function create? The Absolute Value Function Algebra II with

a) y = x 3 + 3x 2 2 b) = UNIT 4 CURVE SKETCHING 4.1 INCREASING AND DECREASING FUNCTIONS

UNIT 4 CURVE SKETCHING 4.1 INCREASING AND DECREASING FUNCTIONS We read graphs as we read sentences: left to right. Plainly speaking, as we scan the function from left to right, the function is said to

2.2 Limit of a Function and Limit Laws

Limit of a Function and Limit Laws Section Notes Page Let s look at the graph y What is y()? That s right, its undefined, but what if we wanted to find the y value the graph is approaching as we get close

Section 4.4 Rational Functions and Their Graphs. 1, the line x = 0 (y-axis) is its vertical asymptote.

Section 4.4 Rational Functions and Their Graphs p( ) A rational function can be epressed as where p() and q() are q( ) 3 polynomial functions and q() is not equal to 0. For eample, 16 is a rational function.

Example 1: Use the graph of the function f given below to find the following. a. Find the domain of f and list your answer in interval notation

When working with the graph of a function, the inputs (the elements of the domain) are always the values on the horizontal ais (-ais) and the outputs (the elements of the range) are always the values on

Section 4.4 Rational Functions and Their Graphs

Section 4.4 Rational Functions and Their Graphs p( ) A rational function can be epressed as where p() and q() are q( ) 3 polynomial functions and q() is not equal to 0. For eample, is a 16 rational function.

SECTION 6-8 Graphing More General Tangent, Cotangent, Secant, and Cosecant Functions

6-8 Graphing More General Tangent, Cotangent, Secant, and Cosecant Functions 9 duce a scatter plot in the viewing window. Choose 8 for the viewing window. (B) It appears that a sine curve of the form k