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

Size: px
Start display at page:

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

Transcription

1 Concavity 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 points x 1, f x 1 and x, f x for any x 1 and x in a, b are above the curve y f x between x 1, f x 1 and x, f x. b. If f is concave down on a, b, then the secant line passing through points x 1, f x 1 and x, f x for any x 1 and x in a, b are below the curve y f x between x 1, f x 1 and x, f x x x a concave up function on 3, 3 a concave down function on 3, 3 Example The graph of f is given below. Determine graphically the interval on which f is a. concave up; b. concave up and decreasing x a. f is concave up on 1.8, 3.3 b. f is concave up and decreasing on 1.8,.5-1 f x How can we determine algebraically where f is concave up and where f is concave down? Theorem Suppose that f is differentiable on an interval I. The graph of f is 1

2 a. concave up on I, iff is increasing on I ; and b. concave down on I,iff is decreasing on I. Or, suppose that f exists on I. The graph of f is a. concave up on I, iff x for all x in I; b. concave down on I,iff x for all x in I.. Inflection points: Definition Suppose that f is continuous on the interval a, b. Letcbein a, b. Then the point c, f c is called an inflection point of f if the graph of f changes concavity at the point c, f c. Note that the graph of f changes concavity at the point c, f c if f changes from increasing to decreasing or from decreasing to increasing at c, f c or f changes from positive to negative or from negative to positive at c, f c. Example Let the graph of f x be given below. Find a. the x coordinate of each inflection point of f; b. where the graph of f is concave up and is concave down x f x a. f x when x, x, x 1 and x. f does not change sign at x. So, the x coordinates of inflection points of f are x, x 1 and x. b. f for x, x 1, x and f for x, 1 x. So, the graph of f is concave up on,, 1, and is concave down on, 1,. Example Let the graph of f x be given below. Find a. the x coordinate of each inflection point of f; b. where the graph of f is concave up and is concave down.

3 x -1 f x f x when x.8, x.5, x. f f is increasing for x.8,.5 x ; f f is decreasing for.8 x.5, x 5. a. So, x.8, x.5, x are the x coordinates of inflection points of f. b. The graph of f is concave up on,.8.5, and is concave down on.8,.5, 5. Example Let f x x 3 9x x 1.Find a. all inflection points of f; b. where the graph of f is concave up and is concave down. Verify your answers by graphing both f and f. a. i. Compute f : f x 6x 18x, f x 1x 18 1 x 3 ii. Solve f x : 1 x 3 x 3. iii. Check signs of f over intervals:, 3, 3, f 1 1 f 1 3, interval, 3 sign of f x 3, Since f changes sign at the point where x 3, 3, 79 is an inflection point of f. b. The graph of f is concave up on 3, and is concave down on, 3 f.. Check the graph of 3

4 x f,... f, -.- f 3. Second Derivative Test: Theorem Suppose that f is continuos on the interval a, b and f c, for some c in a, b. a. If f c, then f c is a local maximum and b. if f c, then f c is a local minimum. Example The graph of f is given below. Suppose that we know f 1,f and f. Determine if f 1, f and f are local maximum, local minimum or neither x -1 f x Example Let f x x 5 x. Find

5 a. the intervals of increase and decrease; b. all local extrema; c. the intervals of concavity; d. all inflection points; and e. sketch the graph of f based on the information in a.-d. The domain of f :,,, Compute f and f : f x 1 5 x x 5, x f x 5 x 3 5 x 3 Find critical numbers of f : f x x 5, x 5; f x is not defined x Determine the sign change of f over, 5, 5,,, 5, 5, f f 1 5, 16 Determine the sign change of f : f x, interval, 5 5,, 5 5, f x f x changes sign as x 3 changes sign. interval,, f x State the results: a. f is increasing on, 5, 5, and is decreasing on 5,,, 5. b. By the first derivative test, f 5 is a local maximum and f 5 is a local minimum. c. f is concave up on, and is concave down on,. d. Though the graph of f changes concavity at x, x is not an inflection point of f since it is not in the domain of f, so there is no inflection point e. Sketch the graph of f based on the information in a.-d x Example Let f x sinx cos x. Find a. the intervals of increase and decrease; b. all local extrema; c. the intervals of concavity; d. all inflection points; and 5

6 e. sketch the graph of f based on the information in a.-d. The domain of f : D f, Compute f and f : f x cos x sinx, f x sinx cos x Find critical numbers of f : f x cos x sinx, sinx cos x, tan x 1, x n, n, 1,,... Determine the sign change of f over... 7, 3, 3,,, 5, 5, 9,... f 1 f 1 1 f 1 1 f 1 1 interval... 7, 3 3,, 5 5, 9... sign of f Determine the sign change of f x sinx cos x : f x, sinx cos x, tan x 1, x 3 n Determine the sign change of f over... 5,,, 3, 3, 7, 7, 11,... f 1 f 1 1 f 1 1 f 1 1 interval... 5,, 3 3, 7 7, sign of f State the results: a. f is increasing on... 3,, 5, 9,... b. f c is a local maximum for c... 7,, 9,... f c is a local minimum for c... 3, 5,... c. f is concave up on... 5,, 3, 7,... and is concave down on..., 3, 7, d. Inflection points of f are: n e. Sketch the graph of f based on the information in a.-d 6

7 x y f x, - - y f x,... y f x 7

8 Example Sketch a graph of a function with the given properties: f f x, for all x; f 1 f x for x, f x for x, f Example Sketch a graph of a function with the given properties: 1. f, f 1 1, f 1 1 f x, for x 1 and x 1, f x for 1 x and x 1; f x for x and x 8

3.5 - Concavity. a concave up. a concave down

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

More information

3.5 - Concavity 1. Concave up and 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

More information

Section 4.3: Derivatives and the Shapes of Curves

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

More information

The Extreme Value Theorem (IVT)

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

More information

Use Derivatives to Sketch the Graph of a Polynomial Function.

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

More information

Section 4.3: How Derivatives Affect the Shape of the Graph

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

More information

Review Sheet Chapter 3

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

More information

AH Properties of Functions.notebook April 19, 2018

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

More information

Chapter 4.1 & 4.2 (Part 1) Practice Problems

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

More information

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

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

More information

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 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

More information

Derivatives and Graphs of Functions

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

More information

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. 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

More information

CONCAVITY AND INFLECTION POINTS

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.

More information

Increasing/Decreasing Behavior

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

More information

4 Using The Derivative

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

More information

Kevin James. MTHSC 102 Section 4.4 Inflection Points and Second Deriva

Kevin James. MTHSC 102 Section 4.4 Inflection Points and Second Deriva MTHSC 102 Section 4.4 Inflection Points and Second Derivatives Example A model for the population of KY from 1980-1993 is p(x) = 0.395x 3 6.67x 2 +30.3x +3661 thousand people where x is the number of years

More information

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. 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

More information

Increasing/Decreasing Behavior

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

More information

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 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

More information

Math 1314 Lesson 12 Curve Analysis (Polynomials)

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

More information

Theorem 2(B): Concave DOWNward

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

More information

4.3 Finding Local Extreme Values: First and Second Derivatives

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)

More information

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

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

More information

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. 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

More information

Relating Graphs of f and f

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

More information

AP Calculus AB Unit 2 Assessment

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.

More information

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

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)

More information

Section 4.4 Concavity and Points of Inflection

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

More information

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

More information

NAME: Section # SSN: X X X X

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)

More information

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

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:

More information

THS Step By Step Calculus Chapter 3

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

More information

Math Lesson 13 Analyzing Other Types of Functions 1

Math Lesson 13 Analyzing Other Types of Functions 1 Math 1314 Lesson 13 Analyzing Other Types of Functions Asymptotes We will need to identify any vertical or horizontal asymptotes of the graph of a function. A vertical asymptote is a vertical line x= a

More information

Sec.4.1 Increasing and Decreasing Functions

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,

More information

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

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

More information

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?

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

More information

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

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.

More information

Unit 1: Sections Skill Set

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,

More information

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.

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.

More information

Math 1314 Lesson 12 Curve Analysis (Polynomials)

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

More information

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

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

More information

ABSOLUTE EXTREMA AND THE MEAN VALUE THEOREM

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

More information

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

(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:

More information

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 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

More information

. The differential of y f (x)

. The differential of y f (x) Calculus I - Prof D Yuen Exam Review version 11/14/01 Please report any typos Derivative Rules Of course you have to remember all your derivative rules Implicit Differentiation Differentiate both sides

More information

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

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

More information

4.4. Concavity and Curve Sketching. Concavity

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

More information

Chapter 5.6: The Other Trig Functions

Chapter 5.6: The Other Trig Functions Chapter 5.6: The Other Trig Functions The other four trig functions, tangent, cotangent, cosecant, and secant are not sinusoids, although they are still periodic functions. Each of the graphs of these

More information

Increasing and Decreasing Functions. MATH 1003 Calculus and Linear Algebra (Lecture 20) Increasing and Decreasing Functions

Increasing and Decreasing Functions. MATH 1003 Calculus and Linear Algebra (Lecture 20) Increasing and Decreasing Functions Increasing and Decreasing Functions MATH 1003 Calculus and Linear Algebra (Lecture 20) Maosheng Xiong Department of Mathematics, HKUST Suppose y = f (x). 1. f (x) is increasing on an interval a < x < b,

More information

To find the intervals on which a given polynomial function is increasing/decreasing using GGB:

To find the intervals on which a given polynomial function is increasing/decreasing using GGB: To find the intervals on which a given polynomial function is increasing/decreasing using GGB: 1. Use GGB to graph the derivative of the function. = ; 2. Find any critical numbers. (Recall that the critical

More information

AB Calculus: Extreme Values of a Function

AB Calculus: Extreme Values of a Function AB Calculus: Extreme Values of a Function Name: Extrema (plural for extremum) are the maximum and minimum values of a function. In the past, you have used your calculator to calculate the maximum and minimum

More information

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

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

More information

This is called the horizontal displacement of also known as the phase shift.

This is called the horizontal displacement of also known as the phase shift. sin (x) GRAPHS OF TRIGONOMETRIC FUNCTIONS Definitions A function f is said to be periodic if there is a positive number p such that f(x + p) = f(x) for all values of x. The smallest positive number p for

More information

Bob Brown Math 251 Calculus 1 Chapter 4, Section 4 Completed 1 CCBC Dundalk

Bob Brown Math 251 Calculus 1 Chapter 4, Section 4 Completed 1 CCBC Dundalk Bob Brown Math 251 Calculus 1 Chapter 4, Section 4 Completed 1 A Function and its Second Derivative Recall page 4 of Handout 3.1 where we encountered the third degree polynomial f(x) x 3 5x 2 4x + 20.

More information

1-1. What you'll Learn About Critical Points/Extreme Values. 1 P a g e

1-1. What you'll Learn About Critical Points/Extreme Values. 1 P a g e CALCULUS: by Rogawski 8) 1 y x 1-1 x Chapter 4.2: Extreme Values What you'll Learn About Critical Points/Extreme Values 12) f(x) 4x - x 1 1 P a g e Determine the extreme values of each function 2 21) f(x)

More information

Math 1525 Excel Lab 9 Fall 2000 This lab is designed to help you discover how to use Excel to identify relative extrema for a given function.

Math 1525 Excel Lab 9 Fall 2000 This lab is designed to help you discover how to use Excel to identify relative extrema for a given function. Math 1525 Excel Lab 9 Fall 2 This lab is designed to help ou discover how to use Excel to identif relative extrema for a given function. Example #1. Stud the data table and graph below for the function

More information

Critical and Inflection Points

Critical and Inflection Points Critical and Inflection Points 1 Finding and Classifying Critical Points A critical point is a point on the graph where the tangent slope is horizontal, (0) or vertical, ( ). or not defined like the minimum

More information

Calculus Course Overview

Calculus Course Overview Description: Walk in the footsteps of Newton and Leibnitz! An interactive text and graphing software combine with the exciting on-line course delivery to make Calculus an adventure. This course includes

More information

AP Calculus. Extreme Values: Graphically. Slide 1 / 163 Slide 2 / 163. Slide 4 / 163. Slide 3 / 163. Slide 5 / 163. Slide 6 / 163

AP Calculus. Extreme Values: Graphically. Slide 1 / 163 Slide 2 / 163. Slide 4 / 163. Slide 3 / 163. Slide 5 / 163. Slide 6 / 163 Slide 1 / 163 Slide 2 / 163 AP Calculus Analyzing Functions Using Derivatives 2015-11-04 www.njctl.org Slide 3 / 163 Table of Contents click on the topic to go to that section Slide 4 / 163 Extreme Values

More information

5.5 Newton s Approximation Method

5.5 Newton s Approximation Method 498CHAPTER 5. USING DERIVATIVES TO ANALYZE FUNCTIONS; FURTHER APPLICATIONS 4 3 y = x 4 3 f(x) = x cosx y = cosx 3 3 x = cosx x cosx = 0 Figure 5.: Figure showing the existence of a solution of x = cos

More information

We can determine this with derivatives: the graph rises where its slope is positive.

We can determine this with derivatives: the graph rises where its slope is positive. Math 1 Derivatives and Graphs Stewart. Increasing and decreasing functions. We will see how to determine the important features of a graph y = f(x) from the derivatives f (x) and f (x), summarizing our

More information

MA 131 Lecture Notes Chapter 4 Calculus by Stewart

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

More information

Math 1020 Objectives & Exercises Calculus Concepts Spring 2019

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.

More information

Mid Term Pre Calc Review

Mid Term Pre Calc Review Mid Term 2015-13 Pre Calc Review I. Quadratic Functions a. Solve by quadratic formula, completing the square, or factoring b. Find the vertex c. Find the axis of symmetry d. Graph the quadratic function

More information

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

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

More information

UNIT 8 STUDY SHEET POLYNOMIAL FUNCTIONS

UNIT 8 STUDY SHEET POLYNOMIAL FUNCTIONS UNIT 8 STUDY SHEET POLYNOMIAL FUNCTIONS KEY FEATURES OF POLYNOMIALS Intercepts of a function o x-intercepts - a point on the graph where y is zero {Also called the zeros of the function.} o y-intercepts

More information

GREENWOOD PUBLIC SCHOOL DISTRICT AP Calculus AB Pacing Guide FIRST NINE WEEKS

GREENWOOD PUBLIC SCHOOL DISTRICT AP Calculus AB Pacing Guide FIRST NINE WEEKS AP Calculus AB FIRST NINE WEEKS 1 Aug. 6-10 5 Introduction to Course Classroom Expectations/Syllabus/Overview Course Requirements/Administer Learning Styles Inventories 1 Limits Estimate limits from graphs

More information

Math Stuart Jones. 4.3 Curve Sketching

Math Stuart Jones. 4.3 Curve Sketching 4.3 Curve Sketching In this section, we combine much of what we have talked about with derivatives thus far to draw the graphs of functions. This is useful in many situations to visualize properties of

More information

The Sine and Cosine Functions

The Sine and Cosine Functions Concepts: Graphs of Tangent, Cotangent, Secant, and Cosecant. We obtain the graphs of the other trig functions by thinking about how they relate to the sin x and cos x. The Sine and Cosine Functions Page

More information

3.1 Maxima/Minima Values

3.1 Maxima/Minima Values 3.1 Maxima/Minima Values Ex 1: Find all critical points for the curve given by f (x)=x 5 25 3 x3 +20x 1 on the interval [-3, 2]. Identify the min and max values. We're guaranteed max and min points if

More information

Graphing. I ll put this information together with some other techniques into a step-by-step graphing procedure. Here it is:

Graphing. I ll put this information together with some other techniques into a step-by-step graphing procedure. Here it is: Graphing 1010005 Calculus provides information which is useful in graphing curves. The first derivative y tells where a curve is increasing and where a curve is decreasing. The second derivative y tells

More information

Worksheet 2.2: Partial Derivatives

Worksheet 2.2: Partial Derivatives Boise State Math 275 (Ultman) Worksheet 2.2: Partial Derivatives From the Toolbox (what you need from previous classes) Be familiar with the definition of a derivative as the slope of a tangent line (the

More information

Comprehensive Practice Handout MATH 1325 entire semester

Comprehensive Practice Handout MATH 1325 entire semester 1 Comprehensive Practice Handout MATH 1325 entire semester Test 1 material Use the graph of f(x) below to answer the following 6 questions. 7 1. Find the value of lim x + f(x) 2. Find the value of lim

More information

1) Find. a) b) c) d) e) 2) The function g is defined by the formula. Find the slope of the tangent line at x = 1. a) b) c) e) 3) Find.

1) Find. a) b) c) d) e) 2) The function g is defined by the formula. Find the slope of the tangent line at x = 1. a) b) c) e) 3) Find. 1 of 7 1) Find 2) The function g is defined by the formula Find the slope of the tangent line at x = 1. 3) Find 5 1 The limit does not exist. 4) The given function f has a removable discontinuity at x

More information

Math Exam 2a. 1) Take the derivatives of the following. DO NOT SIMPLIFY! 2 c) y = tan(sec2 x) ) b) y= , for x 2.

Math Exam 2a. 1) Take the derivatives of the following. DO NOT SIMPLIFY! 2 c) y = tan(sec2 x) ) b) y= , for x 2. Math 111 - Exam 2a 1) Take the derivatives of the following. DO NOT SIMPLIFY! a) y = ( + 1 2 x ) (sin(2x) - x- x 1 ) b) y= 2 x + 1 c) y = tan(sec2 x) 2) Find the following derivatives a) Find dy given

More information

Math 1330 Test 3 Review Sections , 5.1a, ; Know all formulas, properties, graphs, etc!

Math 1330 Test 3 Review Sections , 5.1a, ; Know all formulas, properties, graphs, etc! Math 1330 Test 3 Review Sections 4.1 4.3, 5.1a, 5. 5.4; Know all formulas, properties, graphs, etc! 1. Similar to a Free Response! Triangle ABC has right angle C, with AB = 9 and AC = 4. a. Draw and label

More information

2 Unit Bridging Course Day 10

2 Unit Bridging Course Day 10 1 / 31 Unit Bridging Course Day 10 Circular Functions III The cosine function, identities and derivatives Clinton Boys / 31 The cosine function The cosine function, abbreviated to cos, is very similar

More information

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

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

More information

Concavity. Notice the location of the tangents to each type of curve.

Concavity. Notice the location of the tangents to each type of curve. Concavity We ve seen how knowing where a unction is increasing and decreasing gives a us a good sense o the shape o its graph We can reine that sense o shape by determining which way the unction bends

More information

Graphs of Polynomial Functions. Use the information that you get from the Maple output to do the following:

Graphs of Polynomial Functions. Use the information that you get from the Maple output to do the following: 1 Calculus 1 Maple Lab #4 Revised 10/29/13 Graphs of Polynomial Functions Name: Phone Number: In this lab you will utilize the first and second derivative of a polynomial, and use that information to help

More information

Math 1120, Section 4 Calculus Test 2. November 5, 2008 Name. work. 1. (15 points) Consider the function f(x) = (2x + 3) 2 (x 1) 2.

Math 1120, Section 4 Calculus Test 2. November 5, 2008 Name. work. 1. (15 points) Consider the function f(x) = (2x + 3) 2 (x 1) 2. November 5, 2008 Name The total number of points available is 139 work Throughout this test, show your 1 (15 points) Consider the function f(x) = (2x + 3) 2 (x 1) 2 (a) Use the product rule to find f (x)

More information

Translation of graphs (2) The exponential function and trigonometric function

Translation of graphs (2) The exponential function and trigonometric function Lesson 35 Translation of graphs (2) The exponential function and trigonometric function Learning Outcomes and Assessment Standards Learning Outcome 2: Functions and Algebra Assessment Standard Generate

More information

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

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)

More information

Math 1314 Lesson 13 Analyzing Other Types of Functions

Math 1314 Lesson 13 Analyzing Other Types of Functions Math 1314 Lesson 13 Analyzing Other Types of Functions Asymptotes We will need to identify any vertical or horizontal asymptotes of the graph of a function. A vertical asymptote is a vertical line x =

More information

2.3 Graph Sketching: Asymptotes and Rational Functions Math 125

2.3 Graph Sketching: Asymptotes and Rational Functions Math 125 .3 Graph Sketching: Asymptotes and Rational Functions Math 15.3 GRAPH SKETCHING: ASYMPTOTES AND RATIONAL FUNCTIONS All the functions from the previous section were continuous. In this section we will concern

More information

STEP Support Programme. Assignment 13

STEP Support Programme. Assignment 13 STEP Support Programme Assignment 13 Warm-up You probably already know that for a graph with gradient dy : if dy > 0 then the graph is increasing ; if dy < 0 then the graph is decreasing. The sign of the

More information

Math 104, Spring 2010 Course Log

Math 104, Spring 2010 Course Log Math 104, Spring 2010 Course Log Date: 1/11 Sections: 1.3, 1.4 Log: Lines in the plane. The point-slope and slope-intercept formulas. Functions. Domain and range. Compositions of functions. Inverse functions.

More information

0.6 Graphing Transcendental Functions

0.6 Graphing Transcendental Functions 0.6 Graphing Transcendental Functions There are some special functions we need to be able to graph easily. Directions follow for exponential functions (see page 68), logarithmic functions (see page 71),

More information

End Behavior and Symmetry

End Behavior and Symmetry Algebra 2 Interval Notation Name: Date: Block: X Characteristics of Polynomial Functions Lesson Opener: Graph the function using transformations then identify key characteristics listed below. 1. y x 2

More information

Sections 4.3, 4.5 & 4.6: Graphing

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

More information

Module 4 Graphs of the Circular Functions

Module 4 Graphs of the Circular Functions MAC 1114 Module 4 Graphs of the Circular Functions Learning Objectives Upon completing this module, you should be able to: 1. Recognize periodic functions. 2. Determine the amplitude and period, when given

More information

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. 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 =

More information

MEI Desmos Tasks for AS Pure

MEI Desmos Tasks for AS Pure Task 1: Coordinate Geometry Intersection of a line and a curve 1. Add a quadratic curve, e.g. y = x² 4x + 1 2. Add a line, e.g. y = x 3 3. Select the points of intersection of the line and the curve. What

More information

Why are soda cans shaped the way they are?

Why are soda cans shaped the way they are? Why are soda cans shaped the way they are? ... but other cans vary in shape? Our second Midterm Exam will be on Tuesday 11/4 Two times (check your time following these instructions) 6-8 PM at AUST 108,

More information

c x y f() f (x) Determine the Determine the Approximate c : Replacin on the AP exam: under-approximation

c x y f() f (x) Determine the Determine the Approximate c : Replacin on the AP exam: under-approximation Tangent Lines and Linear Approximations Students should be able to: Determine the slope of tangent line to a curve at a point Determine the equations of tangent lines and normal lines Approximate a value

More information

AP Calculus BC Course Description

AP Calculus BC Course Description AP Calculus BC Course Description COURSE OUTLINE: The following topics define the AP Calculus BC course as it is taught over three trimesters, each consisting of twelve week grading periods. Limits and

More information

Math 1314 Lesson 13 Analyzing Other Types of Functions

Math 1314 Lesson 13 Analyzing Other Types of Functions Math 1314 Lesson 13 Analyzing Other Types of Functions Asymptotes We will need to identify any vertical or horizontal asymptotes of the graph of a function. A vertical asymptote is a vertical line x a

More information

Further Differentiation

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

More information

MEI GeoGebra Tasks for AS Pure

MEI GeoGebra Tasks for AS Pure Task 1: Coordinate Geometry Intersection of a line and a curve 1. Add a quadratic curve, e.g. y = x 2 4x + 1 2. Add a line, e.g. y = x 3 3. Use the Intersect tool to find the points of intersection of

More information