C-N M151 Lecture Notes (part 1) Based on Stewart s Calculus (2013) B. A. Starnes
|
|
- Edward West
- 5 years ago
- Views:
Transcription
1 Lecture Calculus is the study of infinite Mathematics. In essence, it is the extension of algebraic concepts to their limfinity(l). What does that even mean? Well, let's begin with the notion of functions and proceed from there. As on tb. 2, a function (f) is a rule that assigns to each element x in a set D exactly one element y (= f(x)) in set E. The set D is called the domain of f, and E is called the codomain of f. The range of f may be any set containing E. We usually call x the independent variable and y the dependent variable. Why? There are generally 4 ways to represent a function. 1. verbal description 2. tabular 3. via formula 4. Graphically Usually we can detect whether or not a relation is a function by graphically using the vertical line test (VLT). If a vertical line can pass through the Cartesian graph of the relation and intersect it at more than one point, then the relation is not a function. A function for which to every y there corresponds one and only one x is called a 1 1 funtion. Here are some examples. Ex. Consider the following relations and list their descriptives. y = x This is a function, but not 1 1. The domain is R, and the codomain is [1, ). x 2 + y 2 = 1 This not a function since the graph is a circle. Both domain and codomain are [-1, 1]. x 3 3 = y This is a 1 1 function. Both domain and codomain are R. [] 1S
2 Lecture Some functions may be defined in a piecewise fashion. For instance the function may be defined one way on one part of the domain and another way on a second part. Here's another example. Ex. Consider the following function. f(x) = -x 2 I x (-,0) + x 2 I x [0, ). Graph this bad boy. You'll see that this is, in fact, a 1 1 function. [] Genesis 1 24Then God said, Let the earth bring forth the living creature according to its kind: cattle and creeping thing and beast of the earth, each according to its kind ; and it was so. 2S
3 Lecture If a function f(x) satisfies f(-x) = f(x) x D, then f is said to be an even function. If it satisfies f(-x) = -f(x) x D, then f is said to be an odd function. Ex. Are there any even or odd functions in the previous example problems? Yes, in fact, f(x) = x is an even function. And x 2 + y 2 = 1 is an even relation. [] Lecture On p. 7 we see that a function is said to increasing on an interval I if f(x 1 ) < f(x 2 ) whenever x 1 < x 2 in I, and decreasing on I if f(x 1 ) > f(x 2 ) whenever x 1 < x 2 in I. Ex. Of the previous functions mentioned which are increasing on [0, 1]? f(x) = x f(x) = -x 2 I x (-,0) + x 2 I x [0, ) f(x) = x 3 3 = y. [] Student problem set A (Section 1.1) 4, 6, 8, 10, 11, 13, 22, 24, 28, 36, 42, 56 Student Solution Set A A6. Yes, it's a function where D = [-2, 2] and E = [-1, 2]. A22. [(a + h) 3 a 3 ]/h = (a 3 + 3a 2 h + 3ah 2 + h 3 a 3 )/h = 3a 2 + 3ah + h 2. [] Lecture As part of the idea of the mathematics of infinity, we'll begin with the definition of a limit (tb. 31). Let f(x) be defined on an open interval containing the number a (except possibly at a itself). Then lim x-> a f(x) = L means that for every ε > 0, there exists a δ > 0 such that if x a < δ => f(x) L < ε. Similarly lim x-> a + f(x) = L means that for x > a and for every ε > 0, there exists a δ > 0 such that if x a < δ => f(x) L < ε. The expression lim x-> a - f(x) = L is defined similarly. The idea is that we want to find out what the function does as x approaches a certain number. Here's an old Eagles song to demonstrate the concept. 3S
4 Ex. If f(x) = 5x/4, find lim x->4 f(x). In viewing the graph to the right it is clear that the function is a line going right through the point (4, 5). Therefore we have no problem asserting that lim x->4 5x/4 = 5. [] Ex. Show that lim x->4 5x/4 = 5. Let ε > 0 and choose δ =.8ε. If x 4 < δ => So then (5/4) x 4 = 5x/4 5 < 5δ/4 = ε (def. of length). lim x->4 5x/4 = 5 (p. 3). //. [] Ex. Show lim x->3 x 2 = 9. This is actually a C-N M405 type problem, so don't get too worked up. We'll begin by choosing/letting > 0. Does there exist > 0 so that if 0 < x 3 < => x 2 9 <? 4S
5 Ex. (cont.) Let's choose = ( /6). Now using properties of inequalities, for x 3, x + 3 < 3 + δ, and 0 < x 2 9 < x 3 (x + 3) 2 + 3δ = ε 2 /36 + ε/2 < ε.(cauchy-schwarz Inequality). So x 2 9 < ε. Then lim x->3 x 2 = 9 (p. 3). //. [] Student problem set B (Section 1.3) 4, 6, 12, 16, 20, 24, 28, 30 Student Solution Set B B4. a) 3, c) DNE. B30. Choose δ = ε/2. [] Lecture Here are some rules for limits from tb 35. Of course, note that for the time being all limits exist, all quantities are finite and no denominators can = 0. In the common vernacular (class repeat): 1. The limit of a sum is the sum of the limits. 2. The limit of a product is the product of the limits. 3. The limit of an inverse is the inverse of the limits. 4. The limit of a quotient is the quotient of the limits. And then, stemming from these, here are some common (and useful) results. 1. lim x->a [f(x)] n = [lim x->a f(x)] n, n N. 2. lim x->a [f(x)] 1/n = [lim x->a f(x)] 1/n, n N. Further, we see that if f(x) is a polynomial, trigonometric or rational function and a D, then lim x->a f(x) = f(a). Ex. Here is a function with some interesting features. We see that lim x->a2 f(x) = L2, lim x->a1 + f(x) = L2, lim x->a1 - f(x) = L1, and L f(x) = L2. [] 5S
6 Lecture On tb 38, 39 we see that for two functions f(x) = g(x), x a, => lim x->a f(x) = lim x->a g(x) provided the limits exist. Further, lim x->a f(x) = L iff lim x->a- f(x) = L = lim x->a+ f(x). On tb 41 we have two theorems, the second of which is actually called the Sandwich theorem. First if f(x) g(x) in an open interval containing a (except possibly at a) => lim x->a f(x) lim x->a g(x) provided the limits exist. This leads to the Sandwich theorem. If f(x) g(x) h(x) in an open interval containing a (except possibly at a) and lim x->a f(x) = L = lim x->a h(x) => lim x->a g(x) = L. As a result of the Sandwich theorem we have the following sign of the times. lim x->0 sin(x)/x = 1 (why?). Now here are some examples. Ex. Find lim x->0 f(x) where f(x) = -xi x (-, 0) + xi x (0, ) (the original V). Well, we know that lim x->0- f(x) = lim x->0- -x = 0 = lim x->0- x = lim x->0+ f(x). So by p. 6, lim x->0 f(x) = 0. [] Ex. Find lim x->0 f(x) where f(x) = (x 2 + 6x + 9)/(2x 3). No worries. We know that lim x->0 (x 2 + 6x + 9) = 9, and that lim x->0 (2x - 3) = -3. So by p. 5, lim x->0 (x 2 + 6x + 9)/(2x 3) = 9/(-3) = -3. [] Ex. Suppose f(x) = (x 2 9)/(x + 3). Find lim x->-3 f(x). Even though we may be tempted to utilize the results from p. 5, we cannot. Why? Because we have a singularity at x = -3. What to do? Here's the graph from Geogebra. 6S
7 Ex. (cont.) But wait!?! Isn't there supposed to be an asymptote at x = 3??? What just happened? Actually, a little cancellation just happened (how?). As a result the new f(x) = (x 3)I x (R \ -3). So by p. 6, lim x->-3 f(x) = -6. [] Student problem set C (Section 1.4) 1, 2, 6, 18, 19, 28, 30, 40, 44, 48, 62 Student Solution Set C C C40. L = 0. The limit must be taken from the left of c since the speed of light ( 186K mi/sec) is as fast as we can conceive of sans going back in time. [] Lecture On tb 46 we see that, simply put, a function, f, is said to be continuous at real number a if lim x->a f(x) = f(a). So what does this mean? Actually 3 things. First, the function is defined at a. Next, that the limit of f exists at a. And third, that the first two must be equal. Ex. If we revisit the graph of f(x) = (x 2 9)/(x + 3) it appears that everything is hunky dory! Of course, that is not the case. The function is discontinuous at x = -3 since it is not defined there. The limit parts are fine, but the non-existence of the function makes it discontinuous. []. Lecture Practically speaking, a function is continuous over some interval if the graph of the function can be drawn in one fell swoop, sans lifting the pen off of the paper. Recall from p 5 that the precise definition of continuity is true for all polynomial, trigonometric and rational functions. The discontinuity in the function above is called a removable discontinuity. In an earlier example we saw this graph. 7S
8 Lecture Here lim x->0 +f(x) is an example of an infinite discontinuity, and lim x->a1 f(x) is an example of a jump discontinuity. Further if for a real number a lim x->a +f(x) = f(a) then f is said to be right continuous at a. Left continuous is defined similarly. Now let's extend this concept. A function f is said to be continuous on an interval of R if it is continuous at every number in the interval (meaning f is right continuous at the leftmost number in the interval and vice versa). Tb 51 tells us that all polynomials are continuous on R, and all rational, root and trigonometric functions are continuous on R except at those values not in their domains. As the reader might imagine from the global result on limits on p 5, if f and g are both continuous at a real number a, then 1. The sum, f + g, is continuous. 2. The product, fg, is continuous. 3. The quotient, f/g, is continuous. Practice saying this, and let's do some examples. Ex. Consider the function f(x) given below. Note that f has an infinite discontinuity at x = -3, a jump discontinuities at x = 1, and no discontinuity at x = 0. So what the heck is going on as x??? [] Ex. Now try to draw the graph of a function f that is a) right continuous at -2, but not left continuous there, b) has a removable continuity at x = 3, c) has a jump continuity at x = 6. [] Ex. Consider f(x) = (1 x 2 )I x (-, 1) + x -1 I x [1, ). Why is this function discontinuous at a = 1? Notice that the first part is a polynomial and is defined only up to x = 1. Notwithstanding, by p 5 lim x->1- f(x) = 0. On the right of x = 1, x-1 is a rational function, so again by p 5 lim x->1+ f(x) = 1. By the definition on p 7, then, the limit DNE, therefore f is discontinuous at a = 1. [] 8S
9 Lecture Let's take a look now at some features of continuous functions. A function f is said to be continuous on an interval (a, b) if it is continuous at every number in (a, b). We can also say that f is continuous on [a, b] if f is right continuous at a, and left continuous at b. f is continuous on R if it is continuous on (-, ).Here a, b R ext. What does this mean? We must be definitive on the concept of. Ex. Let's revisit the beautiful graph of a function we saw earlier. Where is the function f continuous. We can assume that f continues indefinitely in both directions. So we see that lim x->- + f(x) = (maybe)? and that Also lim x-> + f(x) = 0. lim x->-3- f(x) = - (maybe) = lim x->-3+ f(x)? f also has discontinuities at x = 1, although it is right continuous at both numbers. Finally we can write that f is continuous on Ex. (-, -3) (-3, -1] and on (-1, 1] and (1, ). [] Okay what about this bad boy we saw earlier? f(x) = (1 x 2 )I x (-, 1) + x -1 I x [1, ). Since we also saw that f was discontinuous at x = 1, but is also right continuous there, then we can write that f is continuous on (-, 1) and [1, ). [] Student problem set D (Section 1.5) 4, 6, 8, 10, 16, 18, 20, 30, 32 Student Solution Set D D4. [-4, 2), (-2, 2), [2, 4), (4, 6), (6, 8). D30. lim x-> /4- sinx = (2)/2 = lim x-> /4+ cosx. => lim x-> /4 f(x) = f( /4). Since the function is continuous elsewhere (p 8), then it is continuous on R (p 7). //. [] 9S
10 Lecture 10S
Advanced Mathematics Unit 2 Limits and Continuity
Advanced Mathematics 3208 Unit 2 Limits and Continuity NEED TO KNOW Expanding Expanding Expand the following: A) (a + b) 2 B) (a + b) 3 C) (a + b)4 Pascals Triangle: D) (x + 2) 4 E) (2x -3) 5 Random Factoring
More informationAdvanced Mathematics Unit 2 Limits and Continuity
Advanced Mathematics 3208 Unit 2 Limits and Continuity NEED TO KNOW Expanding Expanding Expand the following: A) (a + b) 2 B) (a + b) 3 C) (a + b)4 Pascals Triangle: D) (x + 2) 4 E) (2x -3) 5 Random Factoring
More information1.10 Continuity Brian E. Veitch
1.10 Continuity Definition 1.5. A function is continuous at x = a if 1. f(a) exists 2. lim x a f(x) exists 3. lim x a f(x) = f(a) If any of these conditions fail, f is discontinuous. Note: From algebra
More informationEQ: What are limits, and how do we find them? Finite limits as x ± Horizontal Asymptote. Example Horizontal Asymptote
Finite limits as x ± The symbol for infinity ( ) does not represent a real number. We use to describe the behavior of a function when the values in its domain or range outgrow all finite bounds. For example,
More informationContinuity, Intermediate Value Theorem (2.4)
Continuity, Intermediate Value Theorem (2.4) Xiannan Li Kansas State University January 29th, 2017 Intuitive Definition: A function f(x) is continuous at a if you can draw the graph of y = f(x) without
More informationInfinite Limits. By Tuesday J. Johnson
Infinite Limits By Tuesday J. Johnson Suggested Review Topics Algebra skills reviews suggested: Evaluating functions Graphing functions Working with inequalities Working with absolute values Trigonometric
More informationMath 115 Spring 11 Written Homework 10 Solutions
Math 5 Spring Written Homework 0 Solutions. For following its, state what indeterminate form the its are in and evaluate the its. (a) 3x 4x 4 x x 8 Solution: This is in indeterminate form 0. Algebraically,
More information1.2 Functions and Their Properties Name:
1.2 Functions and Their Properties Name: Objectives: Students will be able to represent functions numerically, algebraically, and graphically, determine the domain and range for functions, and analyze
More informationDetermine whether the formula determines y as a function of x. If not, explain. Is there a way to look at a graph and determine if it's a function?
1.2 Functions and Their Properties Name: Objectives: Students will be able to represent functions numerically, algebraically, and graphically, determine the domain and range for functions, and analyze
More informationMTAEA Continuity and Limits of Functions
School of Economics, Australian National University February 1, 2010 Continuous Functions. A continuous function is one we can draw without taking our pen off the paper Definition. Let f be a real-valued
More information1 + lim. n n+1. f(x) = x + 1, x 1. and we check that f is increasing, instead. Using the quotient rule, we easily find that. 1 (x + 1) 1 x (x + 1) 2 =
Chapter 5 Sequences and series 5. Sequences Definition 5. (Sequence). A sequence is a function which is defined on the set N of natural numbers. Since such a function is uniquely determined by its values
More informationThe main way we switch from pre-calc. to calc. is the use of a limit process. Calculus is a "limit machine".
A Preview of Calculus Limits and Their Properties Objectives: Understand what calculus is and how it compares with precalculus. Understand that the tangent line problem is basic to calculus. Understand
More informationCalculus I Exam 1 Review Fall 2016
Problem 1: Decide whether the following statements are true or false: (a) If f, g are differentiable, then d d x (f g) = f g. (b) If a function is continuous, then it is differentiable. (c) If a function
More informationMATH CALCULUS I 1.5: Continuity
MATH 12002 - CALCULUS I 1.5: Continuity Professor Donald L. White Department of Mathematical Sciences Kent State University D.L. White (Kent State University) 1 / 12 Definition of Continuity Intuitively,
More informationChapter 2: Functions, Limits and Continuity
Chapter 2: Functions, Limits and Continuity Functions Limits Continuity Chapter 2: Functions, Limits and Continuity 1 Functions Functions are the major tools for describing the real world in mathematical
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 informationCalculus I. 1. Limits and Continuity
2301107 Calculus I 1. Limits and Continuity Outline 1.1. Limits 1.1.1 Motivation:Tangent 1.1.2 Limit of a function 1.1.3 Limit laws 1.1.4 Mathematical definition of a it 1.1.5 Infinite it 1.1. Continuity
More informationMATH 151 Engineering Mathematics I
MATH 151 Engineering Mathematics I Fall 2018, WEEK 3 JoungDong Kim Week 3 Section 2.3, 2.5, 2.6, Calculating Limits Using the Limit Laws, Continuity, Limits at Infinity; Horizontal Asymptotes. Section
More informationA function is actually a simple concept; if it were not, history would have replaced it with a simpler one by now! Here is the definition:
1.2 Functions and Their Properties A function is actually a simple concept; if it were not, history would have replaced it with a simpler one by now! Here is the definition: Definition: Function, Domain,
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 information56 CHAPTER 3. POLYNOMIAL FUNCTIONS
56 CHAPTER 3. POLYNOMIAL FUNCTIONS Chapter 4 Rational functions and inequalities 4.1 Rational functions Textbook section 4.7 4.1.1 Basic rational functions and asymptotes As a first step towards understanding
More informationSection 2.5. Evaluating Limits Algebraically
Section 2.5 Evaluating Limits Algebraically (1) Determinate and Indeterminate Forms (2) Limit Calculation Techniques (A) Direct Substitution (B) Simplification (C) Conjugation (D) The Squeeze Theorem (3)
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 informationThe function graphed below is continuous everywhere. The function graphed below is NOT continuous everywhere, it is discontinuous at x 2 and
Section 1.4 Continuity A function is a continuous at a point if its graph has no gaps, holes, breaks or jumps at that point. If a function is not continuous at a point, then we say it is discontinuous
More informationter. on Can we get a still better result? Yes, by making the rectangles still smaller. As we make the rectangles smaller and smaller, the
Area and Tangent Problem Calculus is motivated by two main problems. The first is the area problem. It is a well known result that the area of a rectangle with length l and width w is given by A = wl.
More informationChapter 1 Functions and Limits
Contents Chapter 1 Functions and Limits Motivation to Chapter 1 2 4 Tangent and Velocity Problems 3 4.1 VIDEO - Secant Lines, Average Rate of Change, and Applications......................... 3 4.2 VIDEO
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 informationTo get horizontal and slant asymptotes algebraically we need to know about end behaviour for rational functions.
Concepts: Horizontal Asymptotes, Vertical Asymptotes, Slant (Oblique) Asymptotes, Transforming Reciprocal Function, Sketching Rational Functions, Solving Inequalities using Sign Charts. Rational Function
More informationAn Intro to Limits Sketch to graph of 3
Limits and Their Properties A Preview of Calculus Objectives: Understand what calculus is and how it compares with precalculus.understand that the tangent line problem is basic to calculus. Understand
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 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 informationWe can see that f(2) is undefined. (Plugging x = 2 into the function results in a 0 in the denominator)
In order to be successful in AP Calculus, you are expected to KNOW everything that came before. All topics from Algebra I, II, Geometry and of course Precalculus are expected to be mastered before you
More informationWarm-Up: Sketch a graph of the function and use the table to find the lim 3x 2. lim 3x. x 5. Level 3 Finding Limits Algebraically.
Warm-Up: 1. Sketch a graph of the function and use the table to find the lim 3x 2 x 5 lim 3x 2. 2 x 1 finding limits algibraically Ch.15.1 Level 3 2 Target Agenda Purpose Evaluation TSWBAT: Find the limit
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 informationMATH 114 Calculus Notes on Chapter 2 (Limits) (pages 60-? in Stewart)
Still under construction. MATH 114 Calculus Notes on Chapter 2 (Limits) (pages 60-? in Stewart) As seen in A Preview of Calculus, the concept of it underlies the various branches of calculus. Hence we
More informationCalculus I. George Voutsadakis 1. LSSU Math 151. Lake Superior State University. 1 Mathematics and Computer Science
Calculus I George Voutsadakis 1 1 Mathematics and Computer Science Lake Superior State University LSSU Math 151 George Voutsadakis (LSSU) Calculus I November 2014 1 / 67 Outline 1 Limits Limits, Rates
More informationand lim lim 6. The Squeeze Theorem
Limits (day 3) Things we ll go over today 1. Limits of the form 0 0 (continued) 2. Limits of piecewise functions 3. Limits involving absolute values 4. Limits of compositions of functions 5. Limits similar
More information3. LIMITS AND CONTINUITY
3. LIMITS AND CONTINUITY Algebra reveals much about many functions. However, there are places where the algebra breaks down thanks to division by zero. We have sometimes stated that there is division by
More informationLimits, Continuity, and the Derivative
Unit #2 : Limits, Continuity, and the Derivative Goals: Study and define continuity Review limits Introduce the derivative as the limit of a difference quotient Discuss the derivative as a rate of change
More information2.2 The Limit of a Function
2.2 The Limit of a Function Introductory Example: Consider the function f(x) = x is near 0. x f(x) x f(x) 1 3.7320508 1 4.236068 0.5 3.8708287 0.5 4.1213203 0.1 3.9748418 0.1 4.0248457 0.05 3.9874607 0.05
More informationAP Calculus Summer Prep
AP Calculus Summer Prep Topics from Algebra and Pre-Calculus (Solutions are on the Answer Key on the Last Pages) The purpose of this packet is to give you a review of basic skills. You are asked to have
More informationCalculus (Math 1A) Lecture 6
Calculus (Math 1A) Lecture 6 Vivek Shende September 5, 2017 Hello and welcome to class! Hello and welcome to class! Last time Hello and welcome to class! Last time We introduced limits, and discussed slopes
More information2.4 The Precise Definition of a Limit
2.4 The Precise Definition of a Limit Reminders/Remarks: x 4 < 3 means that the distance between x and 4 is less than 3. In other words, x lies strictly between 1 and 7. So, x a < δ means that the distance
More information10/22/16. 1 Math HL - Santowski SKILLS REVIEW. Lesson 15 Graphs of Rational Functions. Lesson Objectives. (A) Rational Functions
Lesson 15 Graphs of Rational Functions SKILLS REVIEW! Use function composition to prove that the following two funtions are inverses of each other. 2x 3 f(x) = g(x) = 5 2 x 1 1 2 Lesson Objectives! The
More informationChapter 1. Functions 1.1. Functions and Their Graphs
1.1 Functions and Their Graphs 1 Chapter 1. Functions 1.1. Functions and Their Graphs Note. We start by assuming that you are familiar with the idea of a set and the set theoretic symbol ( an element of
More informationMATH 1910 Limits Numerically and Graphically Introduction to Limits does not exist DNE DOES does not Finding Limits Numerically
MATH 90 - Limits Numerically and Graphically Introduction to Limits The concept of a limit is our doorway to calculus. This lecture will explain what the limit of a function is and how we can find such
More informationTopics from Algebra and Pre-Calculus. (Key contains solved problems)
Topics from Algebra and Pre-Calculus (Key contains solved problems) Note: The purpose of this packet is to give you a review of basic skills. You are asked not to use the calculator, except on p. (8) and
More informationKEMATH1 Calculus for Chemistry and Biochemistry Students. Francis Joseph H. Campeña, De La Salle University Manila
KEMATH1 Calculus for Chemistry and Biochemistry Students Francis Joseph H Campeña, De La Salle University Manila February 9, 2015 Contents 1 Conic Sections 2 11 A review of the coordinate system 2 12 Conic
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 informationCalculus. Central role in much of modern science Physics, especially kinematics and electrodynamics Economics, engineering, medicine, chemistry, etc.
Calculus Calculus - the study of change, as related to functions Formally co-developed around the 1660 s by Newton and Leibniz Two main branches - differential and integral Central role in much of modern
More informationFunctional Limits and Continuity
Chapter 4 Functional Limits and Continuity 4.1 Discussion: Examples of Dirichlet and Thomae Although it is common practice in calculus courses to discuss continuity before differentiation, historically
More informationMATH 1902: Mathematics for the Physical Sciences I
MATH 1902: Mathematics for the Physical Sciences I Dr Dana Mackey School of Mathematical Sciences Room A305 A Email: Dana.Mackey@dit.ie Dana Mackey (DIT) MATH 1902 1 / 46 Module content/assessment Functions
More informationMATH 1130 Exam 1 Review Sheet
MATH 1130 Exam 1 Review Sheet The Cartesian Coordinate Plane The Cartesian Coordinate Plane is a visual representation of the collection of all ordered pairs (x, y) where x and y are real numbers. This
More informationRational Functions. Elementary Functions. Algebra with mixed fractions. Algebra with mixed fractions
Rational Functions A rational function f (x) is a function which is the ratio of two polynomials, that is, Part 2, Polynomials Lecture 26a, Rational Functions f (x) = where and are polynomials Dr Ken W
More 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 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 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 informationMULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. A) 6 B) 14 C) 10 D) Does not exist
Assn 3.1-3.3 Name MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Find the limit, if it exists. 1) Find: lim x -1 6x + 5 5x - 6 A) -11 B) - 1 11 C)
More information1.3 Limits and Continuity
.3 Limits and Continuity.3. Limits Problem 8. What will happen to the functional values of as x gets closer and closer to 2? f(x) = Solution. We can evaluate f(x) using x values nearer and nearer to 2
More informationMathematic 108, Fall 2015: Solutions to assignment #7
Mathematic 08, Fall 05: Solutions to assignment #7 Problem # Suppose f is a function with f continuous on the open interval I and so that f has a local maximum at both x = a and x = b for a, b I with a
More informationContinuity and One-Sided Limits. By Tuesday J. Johnson
Continuity and One-Sided Limits By Tuesday J. Johnson Suggested Review Topics Algebra skills reviews suggested: Evaluating functions Rationalizing numerators and/or denominators Trigonometric skills reviews
More informationLimits and Continuity
Limits and Continuity Philippe B. Laval Kennesaw State University January 2, 2005 Contents Abstract Notes and practice problems on its and continuity. Limits 2. Introduction... 2.2 Theory:... 2.2. GraphicalMethod...
More informationChapter 2. Limits and Continuity. 2.1 Rates of change and Tangents to Curves. The average Rate of change of y = f(x) with respect to x over the
Chapter 2 Limits and Continuity 2.1 Rates of change and Tangents to Curves Definition 2.1.1 : interval [x 1, x 2 ] is The average Rate of change of y = f(x) with respect to x over the y x = f(x 2) f(x
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 informationMath 12 Final Exam Review 1
Math 12 Final Exam Review 1 Part One Calculators are NOT PERMITTED for this part of the exam. 1. a) The sine of angle θ is 1 What are the 2 possible values of θ in the domain 0 θ 2π? 2 b) Draw these angles
More informationMath 106 Calculus 1 Topics for first exam
Chapter 2: Limits and Continuity Rates of change and its: Math 06 Calculus Topics for first exam Limit of a function f at a point a = the value the function should take at the point = the value that the
More informationPre-Calculus Midterm Practice Test (Units 1 through 3)
Name: Date: Period: Pre-Calculus Midterm Practice Test (Units 1 through 3) Learning Target 1A I can describe a set of numbers in a variety of ways. 1. Write the following inequalities in interval notation.
More informationChapter 10 Introduction to the Derivative
Chapter 0 Introduction to the Derivative The concept of a derivative takes up half the study of Calculus. A derivative, basically, represents rates of change. 0. Limits: Numerical and Graphical Approaches
More informationContinuity at a Point
Continuity at a Point When we eplored the limit of f() as approaches c, the emphasis was on the function values close to = c rather than what happens to the function at = c. We will now consider the following
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 informationCalculus. Weijiu Liu. Department of Mathematics University of Central Arkansas 201 Donaghey Avenue, Conway, AR 72035, USA
Calculus Weijiu Liu Department of Mathematics University of Central Arkansas 201 Donaghey Avenue, Conway, AR 72035, USA 1 Opening Welcome to your Calculus I class! My name is Weijiu Liu. I will guide you
More informationThis Week. Professor Christopher Hoffman Math 124
This Week Sections 2.1-2.3,2.5,2.6 First homework due Tuesday night at 11:30 p.m. Average and instantaneous velocity worksheet Tuesday available at http://www.math.washington.edu/ m124/ (under week 2)
More informationSolving Linear and Rational Inequalities Algebraically. Definition 22.1 Two inequalities are equivalent if they have the same solution set.
Inequalities Concepts: Equivalent Inequalities Solving Linear and Rational Inequalities Algebraically Approximating Solutions to Inequalities Graphically (Section 4.4).1 Equivalent Inequalities Definition.1
More informationConcepts of graphs of functions:
Concepts of graphs of functions: 1) Domain where the function has allowable inputs (this is looking to find math no-no s): Division by 0 (causes an asymptote) ex: f(x) = 1 x There is a vertical asymptote
More informationMath 261 Calculus I. Test 1 Study Guide. Name. Decide whether the limit exists. If it exists, find its value. 1) lim x 1. f(x) 2) lim x -1/2 f(x)
Math 261 Calculus I Test 1 Study Guide Name Decide whether the it exists. If it exists, find its value. 1) x 1 f(x) 2) x -1/2 f(x) Complete the table and use the result to find the indicated it. 3) If
More informationLimits Involving Infinity (Horizontal and Vertical Asymptotes Revisited)
Limits Involving Infinity (Horizontal and Vertical Asymptotes Revisited) Limits as Approaches Infinity At times you ll need to know the behavior of a function or an epression as the inputs get increasingly
More information3.5 Continuity of a Function One Sided Continuity Intermediate Value Theorem... 23
Chapter 3 Limit and Continuity Contents 3. Definition of Limit 3 3.2 Basic Limit Theorems 8 3.3 One sided Limit 4 3.4 Infinite Limit, Limit at infinity and Asymptotes 5 3.4. Infinite Limit and Vertical
More informationLimits at Infinity. Horizontal Asymptotes. Definition (Limits at Infinity) Horizontal Asymptotes
Limits at Infinity If a function f has a domain that is unbounded, that is, one of the endpoints of its domain is ±, we can determine the long term behavior of the function using a it at infinity. Definition
More informationLimits and Continuity
Limits and Continuity MATH 151 Calculus for Management J. Robert Buchanan Department of Mathematics Fall 2018 Objectives After this lesson we will be able to: Determine the left-hand and right-hand limits
More informationLesson Objectives. Lesson 32 - Limits. Fast Five. Fast Five - Limits and Graphs 1/19/17. Calculus - Mr Santowski
Lesson 32 - Limits Calculus - Mr Santowski 1/19/17 Mr. Santowski - Calculus & IBHL 1 Lesson Objectives! 1. Define limits! 2. Use algebraic, graphic and numeric (AGN) methods to determine if a limit exists!
More informationContinuity. To handle complicated functions, particularly those for which we have a reasonable formula or formulas, we need a more precise definition.
Continuity Intuitively, a function is continuous if its graph can be traced on paper in one motion without lifting the pencil from the paper. Thus the graph has no tears or holes. To handle complicated
More informationSolving Polynomial and Rational Inequalities Algebraically. Approximating Solutions to Inequalities Graphically
10 Inequalities Concepts: Equivalent Inequalities Solving Polynomial and Rational Inequalities Algebraically Approximating Solutions to Inequalities Graphically (Section 4.6) 10.1 Equivalent Inequalities
More informationChapter 2: Limits & Continuity
Name: Date: Period: AP Calc AB Mr. Mellina Chapter 2: Limits & Continuity Sections: v 2.1 Rates of Change of Limits v 2.2 Limits Involving Infinity v 2.3 Continuity v 2.4 Rates of Change and Tangent Lines
More informationAP Calculus AB. Limits & Continuity.
1 AP Calculus AB Limits & Continuity 2015 10 20 www.njctl.org 2 Table of Contents click on the topic to go to that section Introduction The Tangent Line Problem Definition of a Limit and Graphical Approach
More 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 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 information6.1 Polynomial Functions
6.1 Polynomial Functions Definition. A polynomial function is any function p(x) of the form p(x) = p n x n + p n 1 x n 1 + + p 2 x 2 + p 1 x + p 0 where all of the exponents are non-negative integers and
More informationDo now as a warm up: Is there some number a, such that this limit exists? If so, find the value of a and find the limit. If not, explain why not.
Do now as a warm up: Is there some number a, such that this limit exists? If so, find the value of a and find the limit. If not, explain why not. 1 Continuity and One Sided Limits To say that a function
More informationMATH 113: ELEMENTARY CALCULUS
MATH 3: ELEMENTARY CALCULUS Please check www.ualberta.ca/ zhiyongz for notes updation! 6. Rates of Change and Limits A fundamental philosophical truth is that everything changes. In physics, the change
More information2.1 The Tangent and Velocity Problems
2.1 The Tangent and Velocity Problems Ex: When you jump off a swing, where do you go? Ex: Can you approximate this line with another nearby? How would you get a better approximation? Ex: A cardiac monitor
More informationPreCalculus: Semester 1 Final Exam Review
Name: Class: Date: ID: A PreCalculus: Semester 1 Final Exam Review Short Answer 1. Determine whether the relation represents a function. If it is a function, state the domain and range. 9. Find the domain
More informationLecture 20: Further graphing
Lecture 20: Further graphing Nathan Pflueger 25 October 2013 1 Introduction This lecture does not introduce any new material. We revisit the techniques from lecture 12, which give ways to determine the
More informationMTH4100 Calculus I. Lecture notes for Week 4. Thomas Calculus, Sections 2.4 to 2.6. Rainer Klages
MTH4100 Calculus I Lecture notes for Week 4 Thomas Calculus, Sections 2.4 to 2.6 Rainer Klages School of Mathematical Sciences Queen Mary University of London Autumn 2009 One-sided its and its at infinity
More informationDuVal High School Summer Review Packet AP Calculus
DuVal High School Summer Review Packet AP Calculus Welcome to AP Calculus AB. This packet contains background skills you need to know for your AP Calculus. My suggestion is, you read the information and
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 informationOctober 27, 2018 MAT186 Week 3 Justin Ko. We use the following notation to describe the limiting behavior of functions.
October 27, 208 MAT86 Week 3 Justin Ko Limits. Intuitive Definitions of Limits We use the following notation to describe the iting behavior of functions.. (Limit of a Function A it is written as f( = L
More informationInduction, sequences, limits and continuity
Induction, sequences, limits and continuity Material covered: eclass notes on induction, Chapter 11, Section 1 and Chapter 2, Sections 2.2-2.5 Induction Principle of mathematical induction: Let P(n) be
More information5.3 Other Algebraic Functions
5.3 Other Algebraic Functions 397 5.3 Other Algebraic Functions This section serves as a watershed for functions which are combinations of polynomial, and more generally, rational functions, with the operations
More informationCalculus. Contents. Paul Sutcliffe. Office: CM212a.
Calculus Paul Sutcliffe Office: CM212a. www.maths.dur.ac.uk/~dma0pms/calc/calc.html Books One and several variables calculus, Salas, Hille & Etgen. Calculus, Spivak. Mathematical methods in the physical
More informationFinding Limits Analytically
Finding Limits Analytically Most of this material is take from APEX Calculus under terms of a Creative Commons License In this handout, we explore analytic techniques to compute its. Suppose that f(x)
More information