Derivatives and Continuity
|
|
- Audrey Benson
- 6 years ago
- Views:
Transcription
1 Derivatives and Continuity As with limits, derivatives do not exist unless the right and left-hand derivatives both exist and are equal. There are three main instances when this happens: One, if the curve is not continuous at a point, the derivative does not exist because there will be no slope on one side of the point. Two, if the curve makes an abrupt turn (a cusp or a corner) the slopes on either side of the point will not be equal. or
2 Three, if there is a vertical tangent at a point the derivative is undefined (does not exist) because the derivative gives the slope of the tangent and that slope will be undefined for a vertical line. Go over them carefully; you will be expected to be able to look at a graph and determine where the derivative exists. These illustrations also emphasize one very important point. Differentiability implies continuity (the function must be continuous in order for a derivative to exist), but continuity does not imply differentiability (as in examples and 3 a continuous function still might not have a derivative if there is a cusp or vertical tangent). Just because the curve is continuous, it does not mean that a derivative must exist. This is a perfect example, by the way, of an AP exam question!
3 Local Linearity The term local linearity refers to the fact that a smooth curve that is differentiable will closely resemble its tangent line IF you zoom in close enough. We will use this later in the course with linear approximations, where we find the tangent line and use it instead of the function to estimate y-values on the curve close to the point of tangency. It is important to be able to visualize this, so do this graphing exercise now. ) Graph the function y = x + x in a normal window. ) Zoom in on x =.5 until the curve looks like a straight line. When you get to the point where the curve looks straight, you essentially have the graph of the tangent line as well. Close to the point of tangency the function and its tangent are the same, and that fact is what lays the groundwork for linear approximations.
4 Evaluating Derivatives with the Calculator One of the things you will be expected to do with your calculator in this course is to evaluate a derivative at a particular point using your calculator. We of course can do this by first finding the derivative the long way and then evaluating it at our x-value, but if all we need is the final value we can do it in one step with the calculator! Don't get too excited though you will get to do this on some long answer free-response questions but for most of time you will be required to show the derivation process. Here is how to evaluate a derivative without first finding the derivative. 3 Suppose I want to find the velocity at t = 6, for the position function st ( ) 4t 5t 45t 30. Instead of first finding the derivative, s '(t) = v (t), and then evaluating that derivative at t = 6, we can do it in one step on the calculator. () Go to the MATH menu and choose nderiv( function. On the TI-8 and 83+ it is #8 in the math menu. () Type in function, variable, value you want to evaluate at. For our example you would type in this: 4t^3+5t^-45t+0, t, 6 and then press enter. The answer will be 46,703. Make sure you can do this, so try it now on your calculator. You will most frequently have to find the derivative in this class, since that is one of the things we need to practice, but there will be times when you can do it the short way with the calculator. The trick is to make sure to use proper notation, because without it you will not get credit for your answer. For the problem we just did, you would need to write s '(6) = v (6) = 46,703. That tells the reader that if you find the derivative of the position function, you will get the velocity function, and that when you evaluate that at t = 6, your answer is 46,703 ft/s (or whatever the units are).
5 Homework Examples #5, 7, 9 #5: For the given graph of a function over the closed interval 3 x, at what domain points does the function appear to be: a) differentiable? b) continuous but not differentiable? c) neither continuous nor differentiable? -3 a) Differentiable for all points on (-3, ) The derivative exists along the entire curve except at the endpoints. At the endpoints there is only a onesided derivative. b) Only at the endpoints, x = -3,, is the The function is everywhere continuous because it is curve continuous but not differentiable. c) There are no points where the curve is neither continuous nor differentiable. everywhere differentiable. The function is everywhere continuous and differentiable.
6 #7: For the given graph of a function over the closed interval 3 x 3, at what domain points does the function appear to be: a) differentiable? b) continuous but not differentiable? c) neither continuous nor differentiable? -3 3 a) Differentiable for all points on (-3, ) except for x = 0. b) The curve is continuous but not differentiable at the endpoints, x = -3, 3. c) At x = 0, the curve is neither continuous nor differentiable. The derivative exists along the entire curve except at the point of discontinuity and the endpoints. There is no point where the function is not differentiable but still continuous. At the break in the curve, the function is neither continuous nor differentiable #9: For the given graph of a function over the closed interval x, at what domain points does the function appear to be: a) differentiable? b) continuous but not differentiable? c) neither continuous nor differentiable? - a) Differentiable for all points on (-, )] except for x = 0 and the endpoints. b) At x = 0, -, the curve is continuous but not differentiable. c) There is no point where the curve is neither continuous nor differentiable. The derivative exists along the entire curve except at the cusp and the endpoints. The function is continuous at the cusp and endpoints even though it is not differentiable. There are no discontinuities on the curve.
7 Homework Examples #, 3, 7 tan x, x 0 #: The function y is not differentiable at x = 0. Tell whether the problem, x 0 is a corner, cusp, vertical tangent, or discontinuity. tan x, x 0 y, x 0 tan 0 0 and f (0) = There is a discontinuity. This is our function. First, check to see if the function is continuous (it is the easiest thing to check). Both pieces are continuous, so the only possible discontinuity would be where the pieces meet. #3: The function y x x corner, cusp, vertical tangent, or discontinuity. is not differentiable at x = 0. Tell whether the problem is a y x x y x x, x 0 y x x 0 There is no discontinuity because to the left of x = 0, y = and at x = 0, y =. 0, x 0 y x 0 There is a corner. This is our function. We can rewrite it since we are looking for the positive square root. Now, we are rewrite it again using a piecewise definition. The x = x when x is positive and x = -x when x is negative. First, check to see if the function is continuous (it is the easiest thing to check). Both pieces are continuous, so the only possible discontinuity would be where the pieces meet. Now look at the slopes on either side of x = 0 to see if they are equal. If they are not, the curve is not smooth. A corner is a sharp turn. A cusp will have infinite slopes on either side, but going in different directions. ** you can also see this on the graph, and for now that is an ok way to answer these questions.
8 #7: Find all values of x for which the function f( x) 3 x 8 x 4x 5 is differentiable. 3 x 8 f( x) x 4x 5 x - 4x - 5 = (x - 5)(x + ) The denominator equals zero when x = 5, - and thus the function will be undefined at those points and there will be discontinuities. The function is differentiable for all x 5, This is our function. Both the numerator and denominator consist of polynomials that are everywhere continuous. Thus, the only points where we might not have a derivative would be values of x where the function is not defined. So, look for values that would make the denominator equal to zero. Those values must be excluded from the domain, and a function is not differentiable at an discontinuity.
Extrema and the Extreme Value Theorem
Extrema and the Extreme Value Theorem Local and Absolute Extrema. Extrema are the points where we will find a maximum or minimum on the curve. If they are local or relative extrema, then they will be the
More informationAB Calculus: Rates of Change and Tangent Lines
AB Calculus: Rates of Change and Tangent Lines Name: The World Record Basketball Shot A group called How Ridiculous became YouTube famous when they successfully made a basket from the top of Tasmania s
More informationThe First Derivative Test
The First Derivative Test We have already looked at this test in the last section even though we did not put a name to the process we were using. We use a y number line to test the sign of the first derivative
More informationMath 1320, Section 10 Quiz IV Solutions 20 Points
Math 1320, Section 10 Quiz IV Solutions 20 Points Please answer each question. To receive full credit you must show all work and give answers in simplest form. Cell phones and graphing calculators are
More informationCH 2: Limits and Derivatives
2 The tangent and velocity problems CH 2: Limits and Derivatives the tangent line to a curve at a point P, is the line that has the same slope as the curve at that point P, ie the slope of the tangent
More informationAP Calculus AB Worksheet - Differentiability
Name AP Calculus AB Worksheet - Differentiability MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. The figure shows the graph of a function. At the
More informationGrade 12 (MCV4UE) AP Calculus Page 1 of 5 Derivative of a Function & Differentiability
Grade 2 (MCV4UE) AP Calculus Page of 5 The Derivative at a Point f ( a h) f ( a) Recall, lim provides the slope of h0 h the tangent to the graph y f ( at the point, f ( a), and the instantaneous rate of
More informationMATH 408N PRACTICE MIDTERM 1
02/0/202 Bormashenko MATH 408N PRACTICE MIDTERM Show your work for all the problems. Good luck! () (a) [5 pts] Solve for x if 2 x+ = 4 x Name: TA session: Writing everything as a power of 2, 2 x+ = (2
More informationLimits Student Study Session
Teacher Notes Limits Student Study Session Solving limits: The vast majority of limits questions can be solved by using one of four techniques: SUBSTITUTING, FACTORING, CONJUGATING, or by INSPECTING A
More informationRolle s Theorem. The theorem states that if f (a) = f (b), then there is at least one number c between a and b at which f ' (c) = 0.
Rolle s Theorem Rolle's Theorem guarantees that there will be at least one extreme value in the interior of a closed interval, given that certain conditions are satisfied. As with most of the theorems
More informationInfinite Limits. Infinite Limits. Infinite Limits. Previously, we discussed the limits of rational functions with the indeterminate form 0/0.
Infinite Limits Return to Table of Contents Infinite Limits Infinite Limits Previously, we discussed the limits of rational functions with the indeterminate form 0/0. Now we will consider rational functions
More informationAnalysis of Functions
Lecture for Week 11 (Secs. 5.1 3) Analysis of Functions (We used to call this topic curve sketching, before students could sketch curves by typing formulas into their calculators. It is still important
More informationRevision notes for Pure 1(9709/12)
Revision notes for Pure 1(9709/12) By WaqasSuleman A-Level Teacher Beaconhouse School System Contents 1. Sequence and Series 2. Functions & Quadratics 3. Binomial theorem 4. Coordinate Geometry 5. Trigonometry
More informationAP Calculus AB Summer Assignment
AP Calculus AB Summer Assignment Name: When you come back to school, it is my epectation that you will have this packet completed. You will be way behind at the beginning of the year if you haven t attempted
More informationContinuity. MATH 161 Calculus I. J. Robert Buchanan. Fall Department of Mathematics
Continuity MATH 161 Calculus I J. Robert Buchanan Department of Mathematics Fall 2017 Intuitive Idea A process or an item can be described as continuous if it exists without interruption. The mathematical
More information3.3 Limits and Infinity
Calculus Maimus. Limits Infinity Infinity is not a concrete number, but an abstract idea. It s not a destination, but a really long, never-ending journey. It s one of those mind-warping ideas that is difficult
More informationAnswers for Calculus Review (Extrema and Concavity)
Answers for Calculus Review 4.1-4.4 (Extrema and Concavity) 1. A critical number is a value of the independent variable (a/k/a x) in the domain of the function at which the derivative is zero or undefined.
More information(A) when x = 0 (B) where the tangent line is horizontal (C) when f '(x) = 0 (D) when there is a sharp corner on the graph (E) None of the above
AP Physics C - Problem Drill 10: Differentiability and Rules of Differentiation Question No. 1 of 10 Question 1. A derivative does not eist Question #01 (A) when 0 (B) where the tangent line is horizontal
More information3.1 Graphs of Polynomials
3.1 Graphs of Polynomials Three of the families of functions studied thus far: constant, linear and quadratic, belong to a much larger group of functions called polynomials. We begin our formal study of
More informationSection 3.2 The Derivative as a Function Graphing the Derivative
Math 80 www.timetodare.com Derivatives Section 3. The Derivative as a Function Graphing the Derivative ( ) In the previous section we defined the slope of the tangent to a curve with equation y= f ( )
More informationMaking Piecewise Functions Continuous and Differentiable by Dave Slomer
Making Piecewise Functions Continuous and Differentiable by Dave Slomer Piecewise-defined functions are applied in areas such as Computer Assisted Drawing (CAD). Many piecewise functions in textbooks are
More information3.1 Day 1: The Derivative of a Function
A P Calculus 3.1 Day 1: The Derivative of a Function I CAN DEFINE A DERIVATIVE AND UNDERSTAND ITS NOTATION. Last chapter we learned to find the slope of a tangent line to a point on a graph by using a
More informationMath 229 Mock Final Exam Solution
Name: Math 229 Mock Final Exam Solution Disclaimer: This mock exam is for practice purposes only. No graphing calulators TI-89 is allowed on this test. Be sure that all of your work is shown and that it
More informationPreliminaries Lectures. Dr. Abdulla Eid. Department of Mathematics MATHS 101: Calculus I
Preliminaries 2 1 2 Lectures Department of Mathematics http://www.abdullaeid.net/maths101 MATHS 101: Calculus I (University of Bahrain) Prelim 1 / 35 Pre Calculus MATHS 101: Calculus MATHS 101 is all about
More informationAB.Q103.NOTES: Chapter 2.4, 3.1, 3.2 LESSON 1. Discovering the derivative at x = a: Slopes of secants and tangents to a curve
AB.Q103.NOTES: Chapter 2.4, 3.1, 3.2 LESSON 1 Discovering the derivative at x = a: Slopes of secants and tangents to a curve 1 1. Instantaneous rate of change versus average rate of change Equation of
More informationChapter 3: The Derivative in Graphing and Applications
Chapter 3: The Derivative in Graphing and Applications Summary: The main purpose of this chapter is to use the derivative as a tool to assist in the graphing of functions and for solving optimization problems.
More informationReview Guideline for Final
Review Guideline for Final Here is the outline of the required skills for the final exam. Please read it carefully and find some corresponding homework problems in the corresponding sections to practice.
More informationCalculus I Practice Test Problems for Chapter 2 Page 1 of 7
Calculus I Practice Test Problems for Chapter Page of 7 This is a set of practice test problems for Chapter This is in no way an inclusive set of problems there can be other types of problems on the actual
More informationMATH 151 Engineering Mathematics I
MATH 151 Engineering Mathematics I Spring 2018, WEEK 3 JoungDong Kim Week 3 Section 2.5, 2.6, 2.7, Continuity, Limits at Infinity; Horizontal Asymptotes, Derivatives and Rates of Change. Section 2.5 Continuity
More informationCh. 3 Equations and Inequalities
Ch. 3 Equations and Inequalities 3.1 Solving Linear Equations Graphically There are 2 methods presented in this section for solving linear equations graphically. Normally I would not cover solving linear
More informationMath 108, Solution of Midterm Exam 3
Math 108, Solution of Midterm Exam 3 1 Find an equation of the tangent line to the curve x 3 +y 3 = xy at the point (1,1). Solution. Differentiating both sides of the given equation with respect to x,
More informationChapter 3 Derivatives
Chapter Derivatives Section 1 Derivative of a Function What you ll learn about The meaning of differentiable Different ways of denoting the derivative of a function Graphing y = f (x) given the graph of
More informationLesson A Limits. Lesson Objectives. Fast Five 9/2/08. Calculus - Mr Santowski
Lesson A.1.3 - Limits Calculus - Mr Santowski 9/2/08 Mr. Santowski - Calculus 1 Lesson Objectives 1. Define its 2. Use algebraic, graphic and numeric (AGN) methods to determine if a it exists 3. Use algebraic,
More informationPTF #AB 07 Average Rate of Change
The average rate of change of f( ) over the interval following: 1. y dy d. f() b f() a b a PTF #AB 07 Average Rate of Change ab, can be written as any of the. Slope of the secant line through the points
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 informationAP Calculus B C Syllabus
AP Calculus B C Syllabus Course Textbook Finney, Ross L., et al. Calculus: Graphical, Numerical, Algebraic. Boston: Addison Wesley, 1999. Additional Texts & Resources Best, George, Stephen Carter, and
More informationUnit 4 Systems of Equations Systems of Two Linear Equations in Two Variables
Unit 4 Systems of Equations Systems of Two Linear Equations in Two Variables Solve Systems of Linear Equations by Graphing Solve Systems of Linear Equations by the Substitution Method Solve Systems of
More informationMath 75B Practice Problems for Midterm II Solutions Ch. 16, 17, 12 (E), , 2.8 (S)
Math 75B Practice Problems for Midterm II Solutions Ch. 6, 7, 2 (E),.-.5, 2.8 (S) DISCLAIMER. This collection of practice problems is not guaranteed to be identical, in length or content, to the actual
More informationMath 115 Second Midterm November 12, 2018
EXAM SOLUTIONS Math 5 Second Midterm November, 08. Do not open this eam until you are told to do so.. Do not write your name anywhere on this eam. 3. This eam has 3 pages including this cover. There are
More informationPre-Calculus Module 4
Pre-Calculus Module 4 4 th Nine Weeks Table of Contents Precalculus Module 4 Unit 9 Rational Functions Rational Functions with Removable Discontinuities (1 5) End Behavior of Rational Functions (6) Rational
More informationMath M111: Lecture Notes For Chapter 3
Section 3.1: Math M111: Lecture Notes For Chapter 3 Note: Make sure you already printed the graphing papers Plotting Points, Quadrant s signs, x-intercepts and y-intercepts Example 1: Plot the following
More informationMath 121: Final Exam Review Sheet
Exam Information Math 11: Final Exam Review Sheet The Final Exam will be given on Thursday, March 1 from 10:30 am 1:30 pm. The exam is cumulative and will cover chapters 1.1-1.3, 1.5, 1.6,.1-.6, 3.1-3.6,
More informationR1: Sets A set is a collection of objects sets are written using set brackets each object in onset is called an element or member
Chapter R Review of basic concepts * R1: Sets A set is a collection of objects sets are written using set brackets each object in onset is called an element or member Ex: Write the set of counting numbers
More information4 3A : Increasing and Decreasing Functions and the First Derivative. Increasing and Decreasing. then
4 3A : Increasing and Decreasing Functions and the First Derivative Increasing and Decreasing! If the following conditions both occur! 1. f (x) is a continuous function on the closed interval [ a,b] and
More informationCalculus with Analytic Geometry I Exam 8 Take Home Part.
Calculus with Analytic Geometry I Exam 8 Take Home Part. INSTRUCTIONS: SHOW ALL WORK. Write clearly, using full sentences. Use equal signs appropriately; don t use them between quantities that are not
More information3.4 Using the First Derivative to Test Critical Numbers (4.3)
118 CHAPTER 3. APPLICATIONS OF THE DERIVATIVE 3.4 Using the First Derivative to Test Critical Numbers (4.3) 3.4.1 Theory: The rst derivative is a very important tool when studying a function. It is important
More informationRadnor High School Course Syllabus Advanced Placement Calculus BC 0460
Radnor High School Modified April 24, 2012 Course Syllabus Advanced Placement Calculus BC 0460 Credits: 1 Grades: 11, 12 Weighted: Yes Prerequisite: Recommended by Department Length: Year Format: Meets
More informationAP Calculus. Derivatives.
1 AP Calculus Derivatives 2015 11 03 www.njctl.org 2 Table of Contents Rate of Change Slope of a Curve (Instantaneous ROC) Derivative Rules: Power, Constant, Sum/Difference Higher Order Derivatives Derivatives
More informationAP Calculus AB Summer Assignment
AP Calculus AB Summer Assignment Name: When you come back to school, you will be epected to have attempted every problem. These skills are all different tools that you will pull out of your toolbo this
More informationWhat is calculus? Consider the way that Archimedes figured out the formula for the area of a circle:
What is calculus? Consider the way that Archimedes figured out the formula for the area of a circle: Calculus involves formally describing what happens as a value becomes infinitesimally small (or large).
More informationMthSc 103 Test 3 Spring 2009 Version A UC , 3.1, 3.2. Student s Printed Name:
Student s Printed Name: Instructor: CUID: Section # : Read each question very carefully. You are NOT permitted to use a calculator on any portion of this test. You are not allowed to use any textbook,
More informationChapter 3 Study Guide
Chapter 3 Study Guide I have listed each section of chapter 3 below and given the main points from each. That being said, there may be information I have missed so it is still a good idea to look at the
More information1 Lecture 25: Extreme values
1 Lecture 25: Extreme values 1.1 Outline Absolute maximum and minimum. Existence on closed, bounded intervals. Local extrema, critical points, Fermat s theorem Extreme values on a closed interval Rolle
More informationMath Lecture 4 Limit Laws
Math 1060 Lecture 4 Limit Laws Outline Summary of last lecture Limit laws Motivation Limits of constants and the identity function Limits of sums and differences Limits of products Limits of polynomials
More informationGUIDED NOTES 2.2 LINEAR EQUATIONS IN ONE VARIABLE
GUIDED NOTES 2.2 LINEAR EQUATIONS IN ONE VARIABLE LEARNING OBJECTIVES In this section, you will: Solve equations in one variable algebraically. Solve a rational equation. Find a linear equation. Given
More informationMA1021 Calculus I B Term, Sign:
MA1021 Calculus I B Term, 2014 Final Exam Print Name: Sign: Write up your solutions neatly and show all your work. 1. (28 pts) Compute each of the following derivatives: You do not have to simplify your
More informationStudent s Printed Name:
Student s Printed Name: Instructor: CUID: Section # : You are not permitted to use a calculator on any portion of this test. You are not allowed to use any textbook, notes, cell phone, laptop, PDA, smart
More information3.2 Differentiability
Section 3 Differentiability 09 3 Differentiability What you will learn about How f (a) Might Fail to Eist Differentiability Implies Local Linearity Numerical Derivatives on a Calculator Differentiability
More informationBE SURE THAT YOU HAVE LOOKED AT, THOUGHT ABOUT AND TRIED THE SUGGESTED PROBLEMS ON THIS REVIEW GUIDE PRIOR TO LOOKING AT THESE COMMENTS!!!
Review Guide for MAT0 Final Eam Part I. Thursday December 7 th during regular class time Part is worth 50% of your Final Eam grade. Syllabus approved calculators can be used on this part of the eam but
More informationEverything Old Is New Again: Connecting Calculus To Algebra Andrew Freda
Everything Old Is New Again: Connecting Calculus To Algebra Andrew Freda (afreda@deerfield.edu) ) Limits a) Newton s Idea of a Limit Perhaps it may be objected, that there is no ultimate proportion of
More informationMath Worksheet 1 SHOW ALL OF YOUR WORK! f(x) = (x a) 2 + b. = x 2 + 6x + ( 6 2 )2 ( 6 2 )2 + 7 = (x 2 + 6x + 9) = (x + 3) 2 2
Names Date. Consider the function Math 0550 Worksheet SHOW ALL OF YOUR WORK! f() = + 6 + 7 (a) Complete the square and write the function in the form f() = ( a) + b. f() = + 6 + 7 = + 6 + ( 6 ) ( 6 ) +
More informationPre-Calculus Mathematics Limit Process Calculus
NOTES : LIMITS AND DERIVATIVES Name: Date: Period: Mrs. Nguyen s Initial: LESSON.1 THE TANGENT AND VELOCITY PROBLEMS Pre-Calculus Mathematics Limit Process Calculus The type of it that is used to find
More informationStudent s Printed Name:
MATH 1060 Test 1 Fall 018 Calculus of One Variable I Version B KEY Sections 1.3 3. Student s Printed Name: Instructor: XID: C Section: No questions will be answered during this eam. If you consider a question
More information106 Chapter 5 Curve Sketching. If f(x) has a local extremum at x = a and. THEOREM Fermat s Theorem f is differentiable at a, then f (a) = 0.
5 Curve Sketching Whether we are interested in a function as a purely mathematical object or in connection with some application to the real world, it is often useful to know what the graph of the function
More informationUnit 4 Day 4 & 5. Piecewise Functions
Unit 4 Day 4 & 5 Piecewise Functions Warm Up 1. Why does the inverse variation have a vertical asymptote? 2. Graph. Find the asymptotes. Write the domain and range using interval notation. a. b. f(x)=
More informationThe degree of the polynomial function is n. We call the term the leading term, and is called the leading coefficient. 0 =
Math 1310 A polynomial function is a function of the form = + + +...+ + where 0,,,, are real numbers and n is a whole number. The degree of the polynomial function is n. We call the term the leading term,
More informationLimits and Their Properties
Chapter 1 Limits and Their Properties Course Number Section 1.1 A Preview of Calculus Objective: In this lesson you learned how calculus compares with precalculus. I. What is Calculus? (Pages 42 44) Calculus
More informationChapter 5: Limits, Continuity, and Differentiability
Chapter 5: Limits, Continuity, and Differentiability 63 Chapter 5 Overview: Limits, Continuity and Differentiability Derivatives and Integrals are the core practical aspects of Calculus. They were the
More informationReplacing the a in the definition of the derivative of the function f at a with a variable x, gives the derivative function f (x).
Definition of The Derivative Function Definition (The Derivative Function) Replacing the a in the definition of the derivative of the function f at a with a variable x, gives the derivative function f
More informationGraphs of polynomials. Sue Gordon and Jackie Nicholas
Mathematics Learning Centre Graphs of polynomials Sue Gordon and Jackie Nicholas c 2004 University of Sydney Mathematics Learning Centre, University of Sydney 1 1 Graphs of Polynomials Polynomials are
More information5.4 Continuity: Preliminary Notions
5.4. CONTINUITY: PRELIMINARY NOTIONS 181 5.4 Continuity: Preliminary Notions 5.4.1 Definitions The American Heritage Dictionary of the English Language defines continuity as an uninterrupted succession,
More informationLecture 3 (Limits and Derivatives)
Lecture 3 (Limits and Derivatives) Continuity In the previous lecture we saw that very often the limit of a function as is just. When this is the case we say that is continuous at a. Definition: A function
More informationFirst Derivative Test TEACHER NOTES
Math Objectives Students will relate the first derivative of a function to its critical s and identify which of these critical s are local extrema. Students will visualize why the first derivative test
More informationNorth Carolina State University
North Carolina State University MA 141 Course Text Calculus I by Brenda Burns-Williams and Elizabeth Dempster August 7, 2014 Section1 Functions Introduction In this section, we will define the mathematical
More informationAP CALCULUS Summer Assignment 2014
Name AP CALCULUS Summer Assignment 014 Welcome to AP Calculus. In order to complete the curriculum before the AP Exam in May, it is necessary to do some preparatory work this summer. The following assignment
More informationStudent s Printed Name: _Key
Student s Printed Name: _Key Instructor: CUID: Section # : You are not permitted to use a calculator on any part of this test. You are not allowed to use any textbook, notes, cell phone, laptop, PDA, or
More informationChapter 2 Overview: Introduction to Limits and Derivatives
Chapter 2 Overview: Introduction to Limits and Derivatives In a later chapter, maximum and minimum points of a curve will be found both by calculator and algebraically. While the algebra of this process
More informationLIMITS AT INFINITY MR. VELAZQUEZ AP CALCULUS
LIMITS AT INFINITY MR. VELAZQUEZ AP CALCULUS RECALL: VERTICAL ASYMPTOTES Remember that for a rational function, vertical asymptotes occur at values of x = a which have infinite its (either positive or
More informationBlue Pelican Calculus First Semester
Blue Pelican Calculus First Semester Student Version 1.01 Copyright 2011-2013 by Charles E. Cook; Refugio, Tx Edited by Jacob Cobb (All rights reserved) Calculus AP Syllabus (First Semester) Unit 1: Function
More informationLast week we looked at limits generally, and at finding limits using substitution.
Math 1314 ONLINE Week 4 Notes Lesson 4 Limits (continued) Last week we looked at limits generally, and at finding limits using substitution. Indeterminate Forms What do you do when substitution gives you
More informationWelcome to Math Video Lessons. Stanley Ocken. Department of Mathematics The City College of New York Fall 2013
Welcome to Math 19500 Video Lessons Prof. Department of Mathematics The City College of New York Fall 013 An important feature of the following Beamer slide presentations is that you, the reader, move
More information(c) Find the equation of the degree 3 polynomial that has the same y-value, slope, curvature, and third derivative as ln(x + 1) at x = 0.
Chapter 7 Challenge problems Example. (a) Find the equation of the tangent line for ln(x + ) at x = 0. (b) Find the equation of the parabola that is tangent to ln(x + ) at x = 0 (i.e. the parabola has
More informationTHE LIMIT PROCESS (AN INTUITIVE INTRODUCTION)
The Limit Process THE LIMIT PROCESS (AN INTUITIVE INTRODUCTION) We could begin by saying that limits are important in calculus, but that would be a major understatement. Without limits, calculus would
More informationChapter 1: January 26 January 30
Chapter : January 26 January 30 Section.7: Inequalities As a diagnostic quiz, I want you to go through the first ten problems of the Chapter Test on page 32. These will test your knowledge of Sections.
More information1.5 Inverse Trigonometric Functions
1.5 Inverse Trigonometric Functions Remember that only one-to-one functions have inverses. So, in order to find the inverse functions for sine, cosine, and tangent, we must restrict their domains to intervals
More informationExtremeValuesandShapeofCurves
ExtremeValuesandShapeofCurves Philippe B. Laval Kennesaw State University March 23, 2005 Abstract This handout is a summary of the material dealing with finding extreme values and determining the shape
More informationDetermining the Intervals on Which a Function is Increasing or Decreasing From the Graph of f
Math 1314 Applications of the First Derivative Determining the Intervals on Which a Function is Increasing or Decreasing From the Graph of f Definition: A function is increasing on an interval (a, b) if,
More informationTest 3 Review. y f(a) = f (a)(x a) y = f (a)(x a) + f(a) L(x) = f (a)(x a) + f(a)
MATH 2250 Calculus I Eric Perkerson Test 3 Review Sections Covered: 3.11, 4.1 4.6. Topics Covered: Linearization, Extreme Values, The Mean Value Theorem, Consequences of the Mean Value Theorem, Concavity
More informationComplex Differentials and the Stokes, Goursat and Cauchy Theorems
Complex Differentials and the Stokes, Goursat and Cauchy Theorems Benjamin McKay June 21, 2001 1 Stokes theorem Theorem 1 (Stokes) f(x, y) dx + g(x, y) dy = U ( g y f ) dx dy x where U is a region of the
More informationDefinition (The carefully thought-out calculus version based on limits).
4.1. Continuity and Graphs Definition 4.1.1 (Intuitive idea used in algebra based on graphing). A function, f, is continuous on the interval (a, b) if the graph of y = f(x) can be drawn over the interval
More information2015 Math Camp Calculus Exam Solution
015 Math Camp Calculus Exam Solution Problem 1: x = x x +5 4+5 = 9 = 3 1. lim We also accepted ±3, even though it is not according to the prevailing convention 1. x x 4 x+4 =. lim 4 4+4 = 4 0 = 4 0 = We
More informationAP Calculus Chapter 9: Infinite Series
AP Calculus Chapter 9: Infinite Series 9. Sequences a, a 2, a 3, a 4, a 5,... Sequence: A function whose domain is the set of positive integers n = 2 3 4 a n = a a 2 a 3 a 4 terms of the sequence Begin
More informationMATH The Derivative as a Function - Section 3.2. The derivative of f is the function. f x h f x. f x lim
MATH 90 - The Derivative as a Function - Section 3.2 The derivative of f is the function f x lim h 0 f x h f x h for all x for which the limit exists. The notation f x is read "f prime of x". Note that
More informationDRAFT - Math 101 Lecture Note - Dr. Said Algarni
2 Limits 2.1 The Tangent Problems The word tangent is derived from the Latin word tangens, which means touching. A tangent line to a curve is a line that touches the curve and a secant line is a line that
More informationUnit 8 - Polynomial and Rational Functions Classwork
Unit 8 - Polynomial and Rational Functions Classwork This unit begins with a study of polynomial functions. Polynomials are in the form: f ( x) = a n x n + a n 1 x n 1 + a n 2 x n 2 +... + a 2 x 2 + a
More informationMath 5a Reading Assignments for Sections
Math 5a Reading Assignments for Sections 4.1 4.5 Due Dates for Reading Assignments Note: There will be a very short online reading quiz (WebWork) on each reading assignment due one hour before class on
More informationSolutions to Math 41 First Exam October 18, 2012
Solutions to Math 4 First Exam October 8, 202. (2 points) Find each of the following its, with justification. If the it does not exist, explain why. If there is an infinite it, then explain whether it
More informationSEE and DISCUSS the pictures on pages in your text. Key picture:
Math 6 Notes 1.1 A PREVIEW OF CALCULUS There are main problems in calculus: 1. Finding a tangent line to a curve though a point on the curve.. Finding the area under a curve on some interval. SEE and DISCUSS
More information(a) The best linear approximation of f at x = 2 is given by the formula. L(x) = f(2) + f (2)(x 2). f(2) = ln(2/2) = ln(1) = 0, f (2) = 1 2.
Math 180 Written Homework Assignment #8 Due Tuesday, November 11th at the beginning of your discussion class. Directions. You are welcome to work on the following problems with other MATH 180 students,
More informationCalculus Example Exam Solutions
Calculus Example Exam Solutions. Limits (8 points, 6 each) Evaluate the following limits: p x 2 (a) lim x 4 We compute as follows: lim p x 2 x 4 p p x 2 x +2 x 4 p x +2 x 4 (x 4)( p x + 2) p x +2 = p 4+2
More information