Announcements. Related Rates (last week), Linear approximations (today) l Hôpital s Rule (today) Newton s Method Curve sketching Optimization problems
|
|
- Lizbeth Edwards
- 5 years ago
- Views:
Transcription
1 Announcements Assignment 4 is now posted. Midterm results should be available by the end of the week (assuming the scantron results are back in time). Today: Continuation of applications of derivatives: Related Rates (last week), Linear approximations (today) l Hôpital s Rule (today) Newton s Method Curve sketching Optimization problems AMAT 217 (University of Calgary) Fall / 18
2 An observation Draw the tangent line to f (x) at x = a: For values of x close to a: f (x) and the tangent line are very close to each other. For values of x far from a: f (x) and the tangent line are far from each other. AMAT 217 (University of Calgary) Fall / 18
3 We also use the terminology tangent line approximation and linear approximation. AMAT 217 (University of Calgary) Fall / 18 Tangent lines and linear approximations Geometric Interpretation of Derivative f (x) equals is the slope of the tangent line to the graph at the point (x, f (x)). Recall the point-slope formula for a straight line: y y 1 = m(x x 1 ). Using m = f (a) and (x, y) = (a, f (a)) we can write the equation of tangent line: Equation of Tangent Line at x=a y = f (a) + f (a)(x a). The previous observation is the the motivation behind the concept of linearizations. In particular, we call the tangent line formula a linearization of the function f (x) at x = a: Linearization of f(x) at x=a L(x) = f (a) + f (a)(x a). By the observation, for values of x close to a, the linearization provides a good approximation for the function. Thus, we have: For x near a: f (x) L(x).
4 Example Determine the linear approximation for f (x) = 3 x at x = 8. Use it to approximate Solution: We simply find the equation of a tangent line (and give it the name L(x)). The slope of the tangent line to f (x) at x = 8 is f (8). f (x) = 1 3 x 2/3 = 1 3x 2/3 Therefore, f (8) = 1 3(8 2/3 ) = 1 3( 3 8) = L(x) = f (8) + f (8)(x 8) = (x 8). 12 To approximate we use the fact that when x is close to 8, the tangent line gives a good approximation to the function: = f (8.1) L(8.1) = 2 + (8.1 8) = = In other words, we plug 8.1 into the equation of the tangent line to obtain the approximation for f (8.1) = AMAT 217 (University of Calgary) Fall / 18
5 Question Is the approximation an overestimate or underestimate? 120 Answer with and without using a calculator. Solution: First, by using a calculator we see that the approximation gives while the actual value is equal to , = Thus, the approximation is an overestimate of the actual value. AMAT 217 (University of Calgary) Fall / 18
6 Without a calculator: We plot the function f (x) = 3 x and the tangent line at x = 8: Based on the shape of the graph, we can conclude that the approximation for is larger than the exact value, since the tangent line lies above the function for values of x close to 8. Hence, it is an overestimate. AMAT 217 (University of Calgary) Fall / 18
7 Example Determine the linear approximation for f (x) = sin x at x = 0. Solution: We must compute the equation of the tangent line to sin x at x = 0. First we compute f (0) to get: f (0) = sin 0 = 0 Now taking the derivative of f (x) = sin x gives: f (x) = cos x Hence, m = f (a) gives: f (0) = cos 0 = 1 Therefore, the linear approximation is: L(x) = f (0) + f (0)(x 0) = 0 + (1)(x 0) = x Hence, for values of x close to 0, we have sin x x. Note: The approximation sin x x (for x close to 0) is a very important one. It is frequently used in optics to simplify formulas, and helps describe the motion of pendulums and vibrations in springs. AMAT 217 (University of Calgary) Fall / 18
8 Better approximations (Not covered in this course) We can obtain better approximations to f (x) by using higher degree polynomials. We call these approximations Taylor polynomials, and denote by T n(x) (degree n). f (x) = e x T 0 (x) = 1 T 1 (x) = 1 + x T 2 (x) = 1 + x x 2 T 3 (x) = 1 + x x x 3 T 4 (x) = 1 + x x x x 4 AMAT 217 (University of Calgary) Fall / 18
9 L Hôpital s Rule Uses derivative to compute its. Discovered by Bernoulli (but purchased by l Hôpital). Also spelled l Hospital but pronounced low-pea-tahl not la-hospital. L Hôpital s Rule Suppose we have: where a is any real number (or ± ). Then, f (x) x a g(x) = 0 ± or 0 ±, f (x) x a g(x) f (x) x a g (x). That is, if it is type 0/0 or ± / ± then all we need to do is differentiate the numerator and differentiate the denominator, then take the it. We call a it of type 0/0 or ± / ± an indeterminate form. Caution 1: Don t use the quotient rule on the function f (x)/g(x). Simply differentiate the top, then differentiate the bottom. Caution 2: Can only use for types 0 0 and ± ±. AMAT 217 (University of Calgary) Fall / 18
10 Example sin(nx) Using l Hôpital s Rule show that = 1, where n is a positive constant. x 0 nx Solution: Plugging in x = 0 gives a it of the type 0/0, since sin(0) = 0. Applying l Hôpital s Rule: sin(nx) x 0 nx H x 0 n cos(nx), Since (sin(nx)) = n cos(nx) and (nx) = n, n = n cos(0), Plug in x = 0, n = 1, Since cos(0) = 1. Notation We use the symbol H = to denote we are using l Hôpital s Rule in that step. AMAT 217 (University of Calgary) Fall / 18
11 Example Using l Hôpital s Rule, evaluate x 1 x 2 1 x 1. Solution: Plugging in x = 1 gives x 2 1 x 1 x 1 = = 0 = indeterminate form! 0 Applying l Hôpital s Rule x 2 1 H 2x x 1 x 1 x 1 1 = 2 1 = 2 Alternatively, using just algebraic simplification x 2 1 x 1 x 1 (x + 1)(x 1) x + 1 = 2 x 1 x 1 x 1 AMAT 217 (University of Calgary) Fall / 18
12 Example Evaluate Solution: e x x x 2. Taking x to gives a it of the type /. Therefore, l Hôpital s Rule applies: e x H e x x x 2 x 2x. The new it is also of type /, thus, we can apply l Hôpital s Rule again: e x x 2x H e x x 2. The last it gives a type of /2 =. Therefore, the value of the original it is + : e x x x 2 =. AMAT 217 (University of Calgary) Fall / 18
13 Other it types Sometimes when plugging in x = a we get a it of the type: 0 (± ) On these it problems, we can not apply l Hôpital s Rule directly. However, we can always turn a product of 0 and ± into one of the required forms by taking one of the functions to the bottom as follows: f (x) g(x) = g(x) 1/f (x) or f (x) g(x) = f (x) 1/g(x) Tip: Try both (one will be easier than the other). AMAT 217 (University of Calgary) Fall / 18
14 Example Evaluate x ln x. x 0 + Solution: Taking x to the bottom (as a reciprocal) gives: x ln x ln x x 0 + x 0 + 1/x. Now our it is of the form ( )/( ) and l Hôpital s Rule can be applied. Note that if we brought ln x to the bottom instead, our it would be of the form 0/0 and l Hôpital s Rule would still apply. ln x ln x) x 0 +(x x 0 + 1/x H x 0 + 1/x 1/x 2 x 2 x 0 + x x 0 +( x) Rewrite the it Derivative of top and bottom Simplifying Cancelling = 0 Taking the it AMAT 217 (University of Calgary) Fall / 18
15 In the next problem we see how to deal with indeterminate forms of the types: 0 0, 0 and 1 Side note: A common misconception is that 1 is 1, but this is incorrect. Infinity is not a real number and can be thought of as shorthand for a it. The misconception comes from the fact that 1 x = 1 for any finite power x. When dealing with its, we must allow both the base and exponent to vary as follows: 1 a 1,x ax. Now, it becomes clearer that we can get any value we want by making x and a vary simultaneously at different rates. For example, if a was approaching 1 from the right (i.e., a > 1) at a very slow rate, and x was approaching at a very fast rate, you might expect the answer to be. On the other hand, if a was approaching 1 from the left (i.e., a < 1) at a very slow rate, and x was approaching at a fast rate, you might expect the answer to be 0. The next example illustrates how a it of the type 1 can result in something unexpected. AMAT 217 (University of Calgary) Fall / 18
16 Example Show that x 1/(x 1) = e. x 1 + Solution: Plugging in x = 1 (from the right) gives a it of the type 1. To deal with this type of it we will use logarithms. Let L denote our it: L x 1/(x 1). x 1 + Now, take the natural log of both sides: ( ln L ln x 1/(x 1)). x 1 + Using log properties we have: ln x ln L x 1 + x 1. The right side it is now of the type 0/0, therefore, we can apply l Hôpital s Rule: ln x ln L x 1 + x 1 H 1/x x = 1 Thus, ln L = 1 and hence, our original it (denoted by L) is: L = e 1 = e. AMAT 217 (University of Calgary) Fall / 18
17 Example Compute Solution: x 4 + e x x x 2 + 2e x. x 4 + e x H 4x 3 + e x x x 2 + 2e x x 2x + 2e x Since / type (can take derivatives) H 12x 2 + e x x 2 + 2e x Since / type (can take derivatives) H 24x + e x x 2e x H 24 + e x x 2e x H e x x 2e x = 1 2 Since / type (can take derivatives) Since / type (can take derivatives) Since / type (can take derivatives) Cancelling and taking the it AMAT 217 (University of Calgary) Fall / 18
18 Example Compute ln(cos x) x 0 ln(cos 3x). Solution: This is a 0/0 it type, thus: ln(cos x) x 0 ln(cos 3x) H x 0 sin x cos x 3 sin(3x) cos(3x) tan x x 0 3 tan(3x) H sec 2 x x 0 9 sec 2 (3x) = 1 9 Taking derivatives of top and bottom Simplifying giving 0/0 type Taking derivatives Since sec(0) = 1 AMAT 217 (University of Calgary) Fall / 18
Tangent Lines Sec. 2.1, 2.7, & 2.8 (continued)
Tangent Lines Sec. 2.1, 2.7, & 2.8 (continued) Prove this Result How Can a Derivative Not Exist? Remember that the derivative at a point (or slope of a tangent line) is a LIMIT, so it doesn t exist whenever
More informationChapter 4: More Applications of Differentiation
Chapter 4: More Applications of Differentiation Autumn 2017 Department of Mathematics Hong Kong Baptist University 1 / 68 In the fall of 1972, President Nixon announced that, the rate of increase of inflation
More informationChapter 4: More Applications of Differentiation
Chapter 4: More Applications of Differentiation Winter 2016 Department of Mathematics Hong Kong Baptist University 1 / 61 In the fall of 1972, President Nixon announced that, the rate of increase of inflation
More informationCalculus I Announcements
Slie 1 Calculus I Announcements Office Hours: Amos Eaton 309, Monays 12:50-2:50 Exam 2 is Thursay, October 22n. The stuy guie is now on the course web page. Start stuying now, an make a plan to succee.
More informationReview for the Final Exam
Math 171 Review for the Final Exam 1 Find the limits (4 points each) (a) lim 4x 2 3; x x (b) lim ( x 2 x x 1 )x ; (c) lim( 1 1 ); x 1 ln x x 1 sin (x 2) (d) lim x 2 x 2 4 Solutions (a) The limit lim 4x
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 informationHello Future Calculus Level One Student,
Hello Future Calculus Level One Student, This assignment must be completed and handed in on the first day of class. This assignment will serve as the main review for a test on this material. The test will
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 informationLimit. Chapter Introduction
Chapter 9 Limit Limit is the foundation of calculus that it is so useful to understand more complicating chapters of calculus. Besides, Mathematics has black hole scenarios (dividing by zero, going to
More informationReview for Final. The final will be about 20% from chapter 2, 30% from chapter 3, and 50% from chapter 4. Below are the topics to study:
Review for Final The final will be about 20% from chapter 2, 30% from chapter 3, and 50% from chapter 4. Below are the topics to study: Chapter 2 Find the exact answer to a limit question by using the
More information2.2 The derivative as a Function
2.2 The derivative as a Function Recall: The derivative of a function f at a fixed number a: f a f a+h f(a) = lim h 0 h Definition (Derivative of f) For any number x, the derivative of f is f x f x+h f(x)
More information(x 3)(x + 5) = (x 3)(x 1) = x + 5. sin 2 x e ax bx 1 = 1 2. lim
SMT Calculus Test Solutions February, x + x 5 Compute x x x + Answer: Solution: Note that x + x 5 x x + x )x + 5) = x )x ) = x + 5 x x + 5 Then x x = + 5 = Compute all real values of b such that, for fx)
More informationa x a y = a x+y a x a = y ax y (a x ) r = a rx and log a (xy) = log a (x) + log a (y) log a ( x y ) = log a(x) log a (y) log a (x r ) = r log a (x).
You should prepare the following topics for our final exam. () Pre-calculus. (2) Inverses. (3) Algebra of Limits. (4) Derivative Formulas and Rules. (5) Graphing Techniques. (6) Optimization (Maxima and
More informationAP Calculus AB: Semester Review Notes Information in the box are MASTERY CONCEPTS. Be prepared to apply these concepts on your midterm.
AP Calculus AB: Semester Review Notes Information in the box are MASTERY CONCEPTS. Be prepared to apply these concepts on your midterm. Name: Date: Period: I. Limits and Continuity Definition of Average
More informationMath 180, Exam 2, Practice Fall 2009 Problem 1 Solution. f(x) = arcsin(2x + 1) = sin 1 (3x + 1), lnx
Math 80, Exam, Practice Fall 009 Problem Solution. Differentiate the functions: (do not simplify) f(x) = x ln(x + ), f(x) = xe x f(x) = arcsin(x + ) = sin (3x + ), f(x) = e3x lnx Solution: For the first
More informationFUNCTIONS AND MODELS
1 FUNCTIONS AND MODELS FUNCTIONS AND MODELS 1.6 Inverse Functions and Logarithms In this section, we will learn about: Inverse functions and logarithms. INVERSE FUNCTIONS The table gives data from an experiment
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 informationCore Mathematics 3 Differentiation
http://kumarmaths.weebly.com/ Core Mathematics Differentiation C differentiation Page Differentiation C Specifications. By the end of this unit you should be able to : Use chain rule to find the derivative
More informationMATH 408N PRACTICE FINAL
2/03/20 Bormashenko MATH 408N PRACTICE FINAL Show your work for all the problems. Good luck! () Let f(x) = ex e x. (a) [5 pts] State the domain and range of f(x). Name: TA session: Since e x is defined
More informationSANDERSON HIGH SCHOOL AP CALCULUS AB/BC SUMMER REVIEW PACKET
SANDERSON HIGH SCHOOL AP CALCULUS AB/BC SUMMER REVIEW PACKET 017-018 Name: 1. This packet is to be handed in on Monday August 8, 017.. All work must be shown on separate paper attached to the packet. 3.
More informationAim: How do we prepare for AP Problems on limits, continuity and differentiability? f (x)
Name AP Calculus Date Supplemental Review 1 Aim: How do we prepare for AP Problems on limits, continuity and differentiability? Do Now: Use the graph of f(x) to evaluate each of the following: 1. lim x
More information80 Wyner PreCalculus Spring 2017
80 Wyner PreCalculus Spring 2017 CHAPTER NINE: DERIVATIVES Review May 16 Test May 23 Calculus begins with the study of rates of change, called derivatives. For example, the derivative of velocity is acceleration
More informationDRAFT - Math 101 Lecture Note - Dr. Said Algarni
3 Differentiation Rules 3.1 The Derivative of Polynomial and Exponential Functions In this section we learn how to differentiate constant functions, power functions, polynomials, and exponential functions.
More information5.8 Indeterminate forms and L Hôpital s rule
5.8 Indeterminate forms and L Hôpital s rule Mark Woodard Furman U Fall 2009 Mark Woodard (Furman U) 5.8 Indeterminate forms and L Hôpital s rule Fall 2009 1 / 11 Outline 1 The forms 0/0 and / 2 Examples
More informationMATH 151 Engineering Mathematics I
MATH 151 Engineering Mathematics I Spring 2019, WEEK 10 JoungDong Kim Week 10 Section 4.2, 4.3, 4.4 Mean Value Theorem, How Derivatives Affect the Shape of a Graph, Indeterminate Forms and L Hospital s
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 informationDr. Z s Math151 Handout #4.7 [L Hôspital s Rule]
By Doron Zeilberger Dr Z s Math151 Handout #47 [L Hôspital s Rule] Problem Type 471 : Given certain its of certain functions f(x) g(x) at a designated point x = a determine whether the its (at that very
More informationExample. Evaluate. 3x 2 4 x dx.
3x 2 4 x 3 + 4 dx. Solution: We need a new technique to integrate this function. Notice that if we let u x 3 + 4, and we compute the differential du of u, we get: du 3x 2 dx Going back to our integral,
More informationMATH 408N PRACTICE FINAL
05/05/2012 Bormashenko MATH 408N PRACTICE FINAL Name: TA session: Show your work for all the problems. Good luck! (1) Calculate the following limits, using whatever tools are appropriate. State which results
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 informationFall 2009 Math 113 Final Exam Solutions. f(x) = 1 + ex 1 e x?
. What are the domain and range of the function Fall 9 Math 3 Final Exam Solutions f(x) = + ex e x? Answer: The function is well-defined everywhere except when the denominator is zero, which happens when
More informationPractice problems from old exams for math 132 William H. Meeks III
Practice problems from old exams for math 32 William H. Meeks III Disclaimer: Your instructor covers far more materials that we can possibly fit into a four/five questions exams. These practice tests are
More informationMath 1102: Calculus I (Math/Sci majors) MWF 3pm, Fulton Hall 230 Homework 4 Solutions
Math 0: Calculus I (Math/Sci majors) MWF 3pm, Fulton Hall 30 Homework 4 Solutions Please write neatly, and show all work. Caution: An answer with no work is wrong! Problem A. Use Weierstrass (ɛ,δ)-definition
More informationMTH Calculus with Analytic Geom I TEST 1
MTH 229-105 Calculus with Analytic Geom I TEST 1 Name Please write your solutions in a clear and precise manner. SHOW your work entirely. (1) Find the equation of a straight line perpendicular to the line
More informationWelcome to AP Calculus!!!
Welcome to AP Calculus!!! In preparation for next year, you need to complete this summer packet. This packet reviews & expands upon the concepts you studied in Algebra II and Pre-calculus. Make sure 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 informationUsing the definition of the derivative of a function is quite tedious. f (x + h) f (x)
Derivative Rules Using te efinition of te erivative of a function is quite teious. Let s prove some sortcuts tat we can use. Recall tat te efinition of erivative is: Given any number x for wic te limit
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 informationCalculus I Review Solutions
Calculus I Review Solutions. Compare and contrast the three Value Theorems of the course. When you would typically use each. The three value theorems are the Intermediate, Mean and Extreme value theorems.
More informationdx dx [x2 + y 2 ] = y d [tan x] + tan x = 2x + 2y = y sec 2 x + tan x dy dy = tan x dy dy = [tan x 2y] dy dx = 2x y sec2 x [1 + sin y] = sin(xy)
Math 7 Activit: Implicit & Logarithmic Differentiation (Solutions) Implicit Differentiation. For each of the following equations, etermine x. a. tan x = x 2 + 2 tan x] = x x x2 + 2 ] = tan x] + tan x =
More informationThe above statement is the false product rule! The correct product rule gives g (x) = 3x 4 cos x+ 12x 3 sin x. for all angles θ.
Math 7A Practice Midterm III Solutions Ch. 6-8 (Ebersole,.7-.4 (Stewart DISCLAIMER. This collection of practice problems is not guaranteed to be identical, in length or content, to the actual exam. You
More information2.8 Linear Approximation and Differentials
2.8 Linear Approximation Contemporary Calculus 1 2.8 Linear Approximation and Differentials Newton's method used tangent lines to "point toward" a root of the function. In this section we examine and use
More informationChapter 2: Differentiation
Chapter 2: Differentiation Spring 2018 Department of Mathematics Hong Kong Baptist University 1 / 82 2.1 Tangent Lines and Their Slopes This section deals with the problem of finding a straight line L
More informationMath 230 Mock Final Exam Detailed Solution
Name: Math 30 Mock Final Exam Detailed 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
More informationSection 3.6 The chain rule 1 Lecture. Dr. Abdulla Eid. College of Science. MATHS 101: Calculus I
Section 3.6 The chain rule 1 Lecture College of Science MATHS 101: Calculus I (University of Bahrain) Logarithmic Differentiation 1 / 23 Motivation Goal: We want to derive rules to find the derivative
More informationLearning Objectives for Math 165
Learning Objectives for Math 165 Chapter 2 Limits Section 2.1: Average Rate of Change. State the definition of average rate of change Describe what the rate of change does and does not tell us in a given
More informationReview Problems for Test 1
Review Problems for Test Math 6-03/06 9 9/0 007 These problems are provided to help you study The presence of a problem on this handout does not imply that there will be a similar problem on the test And
More informationCalculus II Lecture Notes
Calculus II Lecture Notes David M. McClendon Department of Mathematics Ferris State University 206 edition Contents Contents 2 Review of Calculus I 5. Limits..................................... 7.2 Derivatives...................................3
More informationChapter 2: Differentiation
Chapter 2: Differentiation Winter 2016 Department of Mathematics Hong Kong Baptist University 1 / 75 2.1 Tangent Lines and Their Slopes This section deals with the problem of finding a straight line L
More informationChapter 2 Derivatives
Contents Chapter 2 Derivatives Motivation to Chapter 2 2 1 Derivatives and Rates of Change 3 1.1 VIDEO - Definitions................................................... 3 1.2 VIDEO - Examples and Applications
More informationAPPLICATIONS OF DIFFERENTIATION
4 APPLICATIONS OF DIFFERENTIATION APPLICATIONS OF DIFFERENTIATION 4.4 Indeterminate Forms and L Hospital s Rule In this section, we will learn: How to evaluate functions whose values cannot be found at
More informationSection 3.6 The chain rule 1 Lecture. Dr. Abdulla Eid. College of Science. MATHS 101: Calculus I
Section 3.6 The chain rule 1 Lecture College of Science MATHS 101: Calculus I (University of Bahrain) Logarithmic Differentiation 1 / 1 Motivation Goal: We want to derive rules to find the derivative of
More information1.4 Techniques of Integration
.4 Techniques of Integration Recall the following strategy for evaluating definite integrals, which arose from the Fundamental Theorem of Calculus (see Section.3). To calculate b a f(x) dx. Find a function
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 informationSemester University of Sheffield. School of Mathematics and Statistics
University of Sheffield School of Mathematics and Statistics MAS140: Mathematics (Chemical) MAS15: Civil Engineering Mathematics MAS15: Essential Mathematical Skills & Techniques MAS156: Mathematics (Electrical
More informationb n x n + b n 1 x n b 1 x + b 0
Math Partial Fractions Stewart 7.4 Integrating basic rational functions. For a function f(x), we have examined several algebraic methods for finding its indefinite integral (antiderivative) F (x) = f(x)
More informationMath 180, Exam 2, Spring 2013 Problem 1 Solution
Math 80, Eam, Spring 0 Problem Solution. Find the derivative of each function below. You do not need to simplify your answers. (a) tan ( + cos ) (b) / (logarithmic differentiation may be useful) (c) +
More informationMAT01A1: Functions and Mathematical Models
MAT01A1: Functions and Mathematical Models Dr Craig 21 February 2017 Introduction Who: Dr Craig What: Lecturer & course coordinator for MAT01A1 Where: C-Ring 508 acraig@uj.ac.za Web: http://andrewcraigmaths.wordpress.com
More informationTrue or False. Circle T if the statement is always true; otherwise circle F. for all angles θ. T F. 1 sin θ
Math 90 Practice Midterm III Solutions Ch. 8-0 (Ebersole), 3.3-3.8 (Stewart) DISCLAIMER. This collection of practice problems is not guaranteed to be identical, in length or content, to the actual exam.
More information2.1 The derivative. Rates of change. m sec = y f (a + h) f (a)
2.1 The derivative Rates of change 1 The slope of a secant line is m sec = y f (b) f (a) = x b a and represents the average rate of change over [a, b]. Letting b = a + h, we can express the slope of the
More informationM155 Exam 2 Concept Review
M155 Exam 2 Concept Review Mark Blumstein DERIVATIVES Product Rule Used to take the derivative of a product of two functions u and v. u v + uv Quotient Rule Used to take a derivative of the quotient of
More information2. Theory of the Derivative
2. Theory of the Derivative 2.1 Tangent Lines 2.2 Definition of Derivative 2.3 Rates of Change 2.4 Derivative Rules 2.5 Higher Order Derivatives 2.6 Implicit Differentiation 2.7 L Hôpital s Rule 2.8 Some
More information3. Go over old quizzes (there are blank copies on my website try timing yourself!)
final exam review General Information The time and location of the final exam are as follows: Date: Tuesday, June 12th Time: 10:15am-12:15pm Location: Straub 254 The exam will be cumulative; that is, it
More informationSESSION 6 Trig. Equations and Identities. Math 30-1 R 3. (Revisit, Review and Revive)
SESSION 6 Trig. Equations and Identities Math 30-1 R 3 (Revisit, Review and Revive) 1 P a g e 2 P a g e Mathematics 30-1 Learning Outcomes Specific Outcome 5: Solve, algebraically and graphically, first
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 informationf(x) = lim x 0 + x = lim f(x) =
Infinite Limits Having discussed in detail its as x ±, we would like to discuss in more detail its where f(x) ±. Once again we would like to emphasize that ± are not numbers, so if we write f(x) = we are
More informationInterpreting Derivatives, Local Linearity, Newton s
Unit #4 : Method Interpreting Derivatives, Local Linearity, Newton s Goals: Review inverse trigonometric functions and their derivatives. Create and use linearization/tangent line formulas. Investigate
More information4.6. Indeterminate Forms and L Hôpital s Rule. 292 Chapter 4: Applications of Derivatives. Indeterminate Form 0/0
292 Chapter 4: Applications of Derivatives 4.6 Indeterminate Forms and L Hôpital s Rule HISTORICAL BIOGRAPHY Guillaume François Antoine de l Hôpital (66 74) John Bernoulli discovered a rule for calculating
More informationSection 2.1, Section 3.1 Rate of change, Tangents and Derivatives at a point
Section 2.1, Section 3.1 Rate of change, Tangents and Derivatives at a point Line through P and Q approaches to the tangent line at P as Q approaches P. That is as a + h a = h gets smaller. Slope of the
More informationSlide 1. Slide 2. Slide 3 Remark is a new function derived from called derivative. 2.2 The derivative as a Function
Slide 1 2.2 The derivative as a Function Slide 2 Recall: The derivative of a function number : at a fixed Definition (Derivative of ) For any number, the derivative of is Slide 3 Remark is a new function
More informationFinal Exam Review Exercise Set A, Math 1551, Fall 2017
Final Exam Review Exercise Set A, Math 1551, Fall 2017 This review set gives a list of topics that we explored throughout this course, as well as a few practice problems at the end of the document. A complete
More informationMAT137 Calculus! Lecture 19
MAT137 Calculus! Lecture 19 Today: L Hôpital s Rule 11.5 The Indeterminate Form (0/0) 11.6 The Indeterminate Form ( / ) + other Indeterminate Forms Test 2: Friday, Nov. 25. If you have a conflict, let
More informationCHALLENGE! (0) = 5. Construct a polynomial with the following behavior at x = 0:
TAYLOR SERIES Construct a polynomial with the following behavior at x = 0: CHALLENGE! P( x) = a + ax+ ax + ax + ax 2 3 4 0 1 2 3 4 P(0) = 1 P (0) = 2 P (0) = 3 P (0) = 4 P (4) (0) = 5 Sounds hard right?
More informationMAT 1320 Study Sheet for the final exam. Format. Topics
MAT 1320 Study Sheet for the final exam August 2015 Format The exam consists of 10 Multiple Choice questions worth 1 point each, and 5 Long Answer questions worth 30 points in total. Please make sure that
More informationRadicals: To simplify means that 1) no radicand has a perfect square factor and 2) there is no radical in the denominator (rationalize).
Summer Review Packet for Students Entering Prealculus Radicals: To simplify means that 1) no radicand has a perfect square factor and ) there is no radical in the denominator (rationalize). Recall the
More informationMAT137 Calculus! Lecture 6
MAT137 Calculus! Lecture 6 Today: 3.2 Differentiation Rules; 3.3 Derivatives of higher order. 3.4 Related rates 3.5 Chain Rule 3.6 Derivative of Trig. Functions Next: 3.7 Implicit Differentiation 4.10
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 information8.7 Taylor s Inequality Math 2300 Section 005 Calculus II. f(x) = ln(1 + x) f(0) = 0
8.7 Taylor s Inequality Math 00 Section 005 Calculus II Name: ANSWER KEY Taylor s Inequality: If f (n+) is continuous and f (n+) < M between the center a and some point x, then f(x) T n (x) M x a n+ (n
More informationMath 10b Ch. 8 Reading 1: Introduction to Taylor Polynomials
Math 10b Ch. 8 Reading 1: Introduction to Taylor Polynomials Introduction: In applications, it often turns out that one cannot solve the differential equations or antiderivatives that show up in the real
More informationO.K. But what if the chicken didn t have access to a teleporter.
The intermediate value theorem, and performing algebra on its. This is a dual topic lecture. : The Intermediate value theorem First we should remember what it means to be a continuous function: A function
More informationAPPLICATIONS OF DIFFERENTIATION
4 APPLICATIONS OF DIFFERENTIATION APPLICATIONS OF DIFFERENTIATION 4.4 Indeterminate Forms and L Hospital s Rule In this section, we will learn: How to evaluate functions whose values cannot be found at
More informationSchool of the Art Institute of Chicago. Calculus. Frank Timmes. flash.uchicago.edu/~fxt/class_pages/class_calc.
School of the Art Institute of Chicago Calculus Frank Timmes ftimmes@artic.edu flash.uchicago.edu/~fxt/class_pages/class_calc.shtml Syllabus 1 Aug 29 Pre-calculus 2 Sept 05 Rates and areas 3 Sept 12 Trapezoids
More informationDRAFT - Math 102 Lecture Note - Dr. Said Algarni
Math02 - Term72 - Guides and Exercises - DRAFT 7 Techniques of Integration A summery for the most important integrals that we have learned so far: 7. Integration by Parts The Product Rule states that if
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 informationTaylor and Maclaurin Series. Approximating functions using Polynomials.
Taylor and Maclaurin Series Approximating functions using Polynomials. Approximating f x = e x near x = 0 In order to approximate the function f x = e x near x = 0, we can use the tangent line (The Linear
More informationCalculus I Homework: Linear Approximation and Differentials Page 1
Calculus I Homework: Linear Approximation and Differentials Page Example (3..8) Find the linearization L(x) of the function f(x) = (x) /3 at a = 8. The linearization is given by which approximates the
More informationYou are being asked to create your own AP CALCULUS Survival kit. For the survival kit you will need:
You are being asked to create your own AP CALCULUS Survival kit. For the survival kit you will need: Index cards Ring (so that you can put all of your flash cards together) Hole punch (to punch holes in
More informationChapter 8 Indeterminate Forms and Improper Integrals Math Class Notes
Chapter 8 Indeterminate Forms and Improper Integrals Math 1220-004 Class Notes Section 8.1: Indeterminate Forms of Type 0 0 Fact: The it of quotient is equal to the quotient of the its. (book page 68)
More information2.3 Differentiation Formulas. Copyright Cengage Learning. All rights reserved.
2.3 Differentiation Formulas Copyright Cengage Learning. All rights reserved. Differentiation Formulas Let s start with the simplest of all functions, the constant function f (x) = c. The graph of this
More information2.8 Linear Approximations and Differentials
Arkansas Tech University MATH 294: Calculus I Dr. Marcel B. Finan 2.8 Linear Approximations and Differentials In this section we approximate graphs by tangent lines which we refer to as tangent line approximations.
More informationBloomsburg University Bloomsburg, Pennsylvania 17815
Department of Mathematics, Computer Science, and Statistics Bloomsburg University Bloomsburg, Pennsylvania 17815 L Hôpital s Rule Summary Many its may be determined by direct substitution, using a geometric
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 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 informationa k 0, then k + 1 = 2 lim 1 + 1
Math 7 - Midterm - Form A - Page From the desk of C. Davis Buenger. https://people.math.osu.edu/buenger.8/ Problem a) [3 pts] If lim a k = then a k converges. False: The divergence test states that if
More informationMath 106 Fall 2014 Exam 2.1 October 31, ln(x) x 3 dx = 1. 2 x 2 ln(x) + = 1 2 x 2 ln(x) + 1. = 1 2 x 2 ln(x) 1 4 x 2 + C
Math 6 Fall 4 Exam. October 3, 4. The following questions have to do with the integral (a) Evaluate dx. Use integration by parts (x 3 dx = ) ( dx = ) x3 x dx = x x () dx = x + x x dx = x + x 3 dx dx =
More information2. If the values for f(x) can be made as close as we like to L by choosing arbitrarily large. lim
Limits at Infinity and Horizontal Asymptotes As we prepare to practice graphing functions, we should consider one last piece of information about a function that will be helpful in drawing its graph the
More informationChapter 7: Techniques of Integration
Chapter 7: Techniques of Integration MATH 206-01: Calculus II Department of Mathematics University of Louisville last corrected September 14, 2013 1 / 43 Chapter 7: Techniques of Integration 7.1. Integration
More informationSummer 2017 Review For Students Entering AP Calculus AB/BC
Summer 2017 Review For Students Entering AP Calculus AB/BC Holy Name High School AP Calculus Summer Homework 1 A.M.D.G. AP Calculus AB Summer Review Packet Holy Name High School Welcome to AP Calculus
More informationAP Calculus (Mr. Surowski)
AP Calculus Mr. Surowski) Lesson 14 Indeterminate forms Homework from Chapter 4 and some of Chapter 8) 8.2) l Hôpital s rule and more on its; see notes below.) 5 30, 35 47, 50, 60. 1 8.3): 1 12, 13 28,
More informationStep 1: Greatest Common Factor Step 2: Count the number of terms If there are: 2 Terms: Difference of 2 Perfect Squares ( + )( - )
Review for Algebra 2 CC Radicals: r x p 1 r x p p r = x p r = x Imaginary Numbers: i = 1 Polynomials (to Solve) Try Factoring: i 2 = 1 Step 1: Greatest Common Factor Step 2: Count the number of terms If
More information