f (x) f (a) f (a) = lim x a f (a) x a
|
|
- Preston Pope
- 6 years ago
- Views:
Transcription
1 Differentiability Revisited Recall that the function f is differentiable at a if exists and is finite. f (a) = lim x a f (x) f (a) x a Another way to say this is that the function f (x) f (a) F a (x) = x a f (a) if x = a if x = a is. That is, lim F a(x) x a Clint Lee Math 112 Lecture 11: Differentiation Product & Quotient Rules 2/26 The Wrong Product Rule Let P(x) = f (x)g(x). The Product Rule gives the rule for finding P (x). Is it true that P (x) = f (x)g (x)? In Example 37 we found the derivative of the function P(x) = x ( x 2 3 ) to be by multiplying through by the x before taking the derivative. But we can write P(x) = f (x)g(x) where f (x) and g(x). Then, f (x) and g (x). So that f (x)g (x) Clint Lee Math 112 Lecture 11: Differentiation Product & Quotient Rules 3/26
2 Derivation of the Product Rule For the function P(x) = f (x)g(x) suppose that f and g are differentiable at a. Let F a and G a be the functions given in the alternate description of differentiability. Then F a and G a are continuous at a with and if lim F a(x) x a lim G a(x) x a F a (x) G a (x) Clint Lee Math 112 Lecture 11: Differentiation Product & Quotient Rules 4/26 Derivation of the Product Rule Continued We now use Formula 1 to find P (a) as P (a) Now, using the expressions for f (x) and g(x) in terms of F a (x) and G a (X), gives P(x) = f (x)g(x) = [f (a) + F a (x) (x a)] [g(a) + G a (x) (x a)] since P(a). So we have P(x) P(a) Clint Lee Math 112 Lecture 11: Differentiation Product & Quotient Rules 5/26
3 The Product Rule The Final Form So that P(x) P(a) x a Thus, finally we have P (a) Replacing the a by x we have the Product Rule P(x) = f (x)g(x) P (x) or using operator notation d [f (x)g(x)] Clint Lee Math 112 Lecture 11: Differentiation Product & Quotient Rules 6/26 Extending the Product Rule The reason for writing the Product Rule in the given form is that the extension to products with more factors is easy. For a product with three factors we have d [f (x)g(x)h(x)] You can think of the Product Rule as simply applying the derivative to each factor in the product individually leaving the other factors alone, and repeating for every factor. Clint Lee Math 112 Lecture 11: Differentiation Product & Quotient Rules 7/26
4 Example 39 Verifying the Product Rule Verify that the Product Rule gives the correct derivative for the function P(x) = x ( x 2 3 ) from Example 37. Solution: Write the function as P(x) = f (x)g(x) where f (x) g(x). Then using the Product Rule gives and P (x) = 3x 2 3 This is the same result obtained in Example 37. This example illustrates that fact that it is not always desirable to use rules such at the Product Rule. Here it is easier to simply multiply out and then take the derivative. Clint Lee Math 112 Lecture 11: Differentiation Product & Quotient Rules 8/26 Visualizing the Product Rule Using Formula 3 for the derivative of the function P(x) = f (x)g(x) as a function of x gives P (x) = Now let f (x + h) and g(x + h). Then f and g so that using Formula 3 again gives f f (x + h) f (x) (x) = lim h 0 h g g(x + h) g(x) (x) = lim h 0 h Further note that since f and g are differentiable, they are continuous. So that and. Clint Lee Math 112 Lecture 11: Differentiation Product & Quotient Rules 9/26
5 Visualizing the Product Rule Continued In the limit for P (x) we need to simplify the expression f (x + h)g(x + h) f (x)g(x) = g(x) g f To do this consider the rectangle with sides f (x) + f and g(x) + g, divided into four regions as shown. f (x) Clint Lee Math 112 Lecture 11: Differentiation Product & Quotient Rules 10/26 Visualizing the Product Rule Continued The product g(x) g (f (x) + f ) (g(x) + g) represents the total area of the rectangle, which is the sum of the areas the four regions. Thus, from the diagram f f (x) (f (x) + f ) (g(x) + g) = And f (x + h)g(x + h) f (x)g(x) The remaining three terms represent the change in the area of the rectangle when x changes by h. Clint Lee Math 112 Lecture 11: Differentiation Product & Quotient Rules 11/26
6 Visualizing the Product Rule Continued Then the derivative is P (x) Note that the third term above comes from the small rectangle in the upper corner. Its contribution becomes negligible as h 0. This is similar to what we saw when we found the rate of change of the area of the square with respect to its side length in Example 26. Clint Lee Math 112 Lecture 11: Differentiation Product & Quotient Rules 12/26 Example 40 Using the Product Rule Differentiate each function (a) f (x) = ( x 2 + x + 1 ) ( x 2 x + 1 ) (b) y = x 2 e x (c) P(x) = ( x 3 + 3x 2 + 4x + 5 ) 2 Solution (a): We could multiply the function out, or use the Product Rule. Using the Product Rule gives f (x) Clint Lee Math 112 Lecture 11: Differentiation Product & Quotient Rules 13/26
7 Solution: Example 40(b) Solution (b): Applying the Product Rule gives dy Solution (c): Applying the Product Rule with f (x) = g(x) gives P (x) In most cases it is not necessary, or desirable, to multiply out an expression like this. Clint Lee Math 112 Lecture 11: Differentiation Product & Quotient Rules 14/26 Example 41 Rate of Change Using the Product Rule In Example 33 we considered the beat capacity of the heart, C = f (r), giving the number of millilitres of blood pumped by the heart per beat as a function of the heart rate r in beats per minute. In that example we used a table of values to estimate the rate of change of the beat capacity with respect to the heart rate when the heart rate is 70 beats per minute. We found that C(70) = 4.2 ml/beat and C (70) (ml/beat) / (beat/min). Now consider the cardiac flow rate of the heart which is the number of millilitres of blood pumped by the heart per minute. It is given by F(r) = rc(r). (a) (b) Find the value of F(70) and verify that it has the units you expect. Estimate the value of F (70). Give the units of your answer and explain what your answer means in practical terms. Clint Lee Math 112 Lecture 11: Differentiation Product & Quotient Rules 15/26
8 Solution: Example 41(a) Using the values given F(70) The units are as you would expect, since the flow rate gives. Clint Lee Math 112 Lecture 11: Differentiation Product & Quotient Rules 16/26 Solution: Example 41(b) First use the Product Rule to find an expression for F (r). Hence F (r) F (70) = 2.45 From the expression above it is apparent that the units of F (70) are the same as the units of C(70), which are ml/beat This value, and the units, mean that. Clint Lee Math 112 Lecture 11: Differentiation Product & Quotient Rules 17/26
9 The Reciprocal Rule Let g(x) be differentiable and define the reciprocal function as f (x) = 1 g(x) Find f (x) in terms of g(x) and g (x). First note that f (x)g(x) = 1, so that d [f (x)g(x)] But using the Product Rule gives d [f (x)g(x)] = Then solving for f (x) gives f (x)g(x) Clint Lee Math 112 Lecture 11: Differentiation Product & Quotient Rules 18/26 The Reciprocal Rule Continued The Reciprocal Rule is d ( 1 ) g(x) We will see that the Reciprocal Rule is a special case of a more general rule soon. Clint Lee Math 112 Lecture 11: Differentiation Product & Quotient Rules 19/26
10 The Power Rule for Negative Integer Powers Recall that we only proved the Power Rule for non-negative integer powers. We can now prove it for negative integer powers as well. For a positive integer n let f (x) = x n This is a reciprocal function with g(x) ( ) d 1 x n. So the derivative is So that d ( x n ) which is the Power Rule for the case of a negative integer power. Clint Lee Math 112 Lecture 11: Differentiation Product & Quotient Rules 20/26 The Quotient Rule Let Q(x) = f (x) g(x) Find Q (x). We could go back to the limit definition of the derivative as we did for the Product Rule. Instead, we will apply the Product and Reciprocal Rules to the function This gives Q(x) Q (x) Clint Lee Math 112 Lecture 11: Differentiation Product & Quotient Rules 21/26
11 The Quotient Rule Continued Thus, the Quotient Rule is ( ) d f (x) g(x) Most computer algebra systems such as Maple do not use the Quotient Rule in this form. They take the derivative a quotient using the Product Rule as in the second line in the development above. Clint Lee Math 112 Lecture 11: Differentiation Product & Quotient Rules 22/26 Example 42 Verifying the Quotient Rule In Example 40(b) we used the Product Rule to find the derivative of y = x 2 e x. Use the Quotient Rule to find the derivative of this function and verify that you get the same answer as you got using the Product Rule. Solution: Write the function as Then applying the Quotient Rule gives y Q (x) This is the same result that we got using the Product Rule. Clint Lee Math 112 Lecture 11: Differentiation Product & Quotient Rules 23/26
12 Example 43 Applying the Quotient Rule Let (a) Find f (x). f (x) = x 2 x (b) Determine the intervals in which f (x) > 0 and f (x) < 0. (c) Sketch the graphs of f and f and explain how the intervals found in part (b) relate to the graph of f. Solution (a): Using the quotient rule gives f (x) Clint Lee Math 112 Lecture 11: Differentiation Product & Quotient Rules 24/26 Solution: Example 43(b) Since we must solve the inequalities f (x) = 2x (x 2 + 1) 2 2x (x 2 + 1) 2 and 2x (x 2 + 1) 2 But the denominator is always, so these inequalities become and Hence f (x) > 0 on the interval f (x) < 0 on the interval Clint Lee Math 112 Lecture 11: Differentiation Product & Quotient Rules 25/26
13 Solution: Example 43(c) The graph of this function is the same as the graph in Example 31(b). The graph of f looks like this, and the graph of f looks like this. Using the same notation as in Example 37(c) we indicate intervals found in part (b) like this. So the graph of f is sloping downward on the interval (, 0) and is sloping upward on the interval (0, ). Clint Lee Math 112 Lecture 11: Differentiation Product & Quotient Rules 26/26
Limits 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 informationSection 3.3 Product and Quotient rules 1 Lecture. Dr. Abdulla Eid. College of Science. MATHS 101: Calculus I
Section 3.3 Product and Quotient rules 1 Lecture College of Science MATHS 101: Calculus I (University of Bahrain) Differentiation Rules 1 / 12 Motivation Goal: We want to derive rules to find the derivative
More informationChapter 5B - Rational Functions
Fry Texas A&M University Math 150 Chapter 5B Fall 2015 143 Chapter 5B - Rational Functions Definition: A rational function is The domain of a rational function is all real numbers, except those values
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 informationCHAPTER 2 POLYNOMIALS KEY POINTS
CHAPTER POLYNOMIALS KEY POINTS 1. Polynomials of degrees 1, and 3 are called linear, quadratic and cubic polynomials respectively.. A quadratic polynomial in x with real coefficient is of the form a x
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 informationThe Derivative Function. Differentiation
The Derivative Function If we replace a in the in the definition of the derivative the function f at the point x = a with a variable x, we get the derivative function f (x). Using Formula 2 gives f (x)
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 informationDIFFERENTIATION RULES
3 DIFFERENTIATION RULES DIFFERENTIATION RULES 3. The Product and Quotient Rules In this section, we will learn about: Formulas that enable us to differentiate new functions formed from old functions by
More informationExample 9 Algebraic Evaluation for Example 1
A Basic Principle Consider the it f(x) x a If you have a formula for the function f and direct substitution gives the indeterminate form 0, you may be able to evaluate the it algebraically. 0 Principle
More informationMHF4U. Advanced Functions Grade 12 University Mitchell District High School. Unit 3 Rational Functions & Equations 6 Video Lessons
MHF4U Advanced Functions Grade 12 University Mitchell District High School Unit 3 Rational Functions & Equations 6 Video Lessons Allow no more than 15 class days for this unit! This includes time for review
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 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 informationOBJECTIVE Find limits of functions, if they exist, using numerical or graphical methods.
1.1 Limits: A Numerical and Graphical Approach OBJECTIVE Find limits of functions, if they exist, using numerical or graphical methods. 1.1 Limits: A Numerical and Graphical Approach DEFINITION: As x approaches
More informationSection 4.3 Concavity and Curve Sketching 1.5 Lectures. Dr. Abdulla Eid. College of Science. MATHS 101: Calculus I
Section 4.3 Concavity and Curve Sketching 1.5 Lectures College of Science MATHS 101: Calculus I (University of Bahrain) Concavity 1 / 29 Concavity Increasing Function has three cases (University of Bahrain)
More informationSection 0.2 & 0.3 Worksheet. Types of Functions
MATH 1142 NAME Section 0.2 & 0.3 Worksheet Types of Functions Now that we have discussed what functions are and some of their characteristics, we will explore different types of functions. Section 0.2
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 informationChapter 12: Differentiation. SSMth2: Basic Calculus Science and Technology, Engineering and Mathematics (STEM) Strands Mr. Migo M.
Chapter 12: Differentiation SSMth2: Basic Calculus Science and Technology, Engineering and Mathematics (STEM) Strands Mr. Migo M. Mendoza Chapter 12: Differentiation Lecture 12.1: The Derivative Lecture
More informationChapter 3 Differentiation Rules
Chapter 3 Differentiation Rules Derivative constant function if c is any real number, then Example: The Power Rule: If n is a positive integer, then Example: Extended Power Rule: If r is any real number,
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 informationMath 1 Variable Manipulation Part 1 Algebraic Equations
Math 1 Variable Manipulation Part 1 Algebraic Equations 1 PRE ALGEBRA REVIEW OF INTEGERS (NEGATIVE NUMBERS) Numbers can be positive (+) or negative (-). If a number has no sign it usually means that it
More informationCh 7 Summary - POLYNOMIAL FUNCTIONS
Ch 7 Summary - POLYNOMIAL FUNCTIONS 1. An open-top box is to be made by cutting congruent squares of side length x from the corners of a 8.5- by 11-inch sheet of cardboard and bending up the sides. a)
More informationMath 142 Week-in-Review #11 (Final Exam Review: All previous sections as well as sections 6.6 and 6.7)
Math 142 Week-in-Review #11 (Final Exam Review: All previous sections as well as sections 6.6 and 6.7) Note: This review is intended to highlight the topics covered on the Final Exam (with emphasis on
More informationSolutions to Exercises, Section 2.5
Instructor s Solutions Manual, Section 2.5 Exercise 1 Solutions to Exercises, Section 2.5 For Exercises 1 4, write the domain of the given function r as a union of intervals. 1. r(x) 5x3 12x 2 + 13 x 2
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 information1.1 : (The Slope of a straight Line)
1.1 : (The Slope of a straight Line) Equations of Nonvertical Lines: A nonvertical line L has an equation of the form y mx b. The number m is called the slope of L and the point (0, b) is called the y-intercept.
More information2.6. Graphs of Rational Functions. Copyright 2011 Pearson, Inc.
2.6 Graphs of Rational Functions Copyright 2011 Pearson, Inc. Rational Functions What you ll learn about Transformations of the Reciprocal Function Limits and Asymptotes Analyzing Graphs of Rational Functions
More informationDepartment of Mathematics, University of Wisconsin-Madison Math 114 Worksheet Sections 3.1, 3.3, and 3.5
Department of Mathematics, University of Wisconsin-Madison Math 11 Worksheet Sections 3.1, 3.3, and 3.5 1. For f(x) = 5x + (a) Determine the slope and the y-intercept. f(x) = 5x + is of the form y = mx
More information1. Which one of the following points is a singular point of. f(x) = (x 1) 2/3? f(x) = 3x 3 4x 2 5x + 6? (C)
Math 1120 Calculus Test 3 November 4, 1 Name In the first 10 problems, each part counts 5 points (total 50 points) and the final three problems count 20 points each Multiple choice section Circle the correct
More informationStudent: Date: Instructor: kumnit nong Course: MATH 105 by Nong https://xlitemprodpearsoncmgcom/api/v1/print/math Assignment: CH test review 1 Find the transformation form of the quadratic function graphed
More informationCommon Core Algebra 2. Chapter 5: Rational Exponents & Radical Functions
Common Core Algebra 2 Chapter 5: Rational Exponents & Radical Functions 1 Chapter Summary This first part of this chapter introduces radicals and nth roots and how these may be written as rational exponents.
More informationUMUC MATH-107 Final Exam Information
UMUC MATH-07 Final Exam Information What should you know for the final exam? Here are some highlights of textbook material you should study in preparation for the final exam. Review this material from
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 informationMath /Foundations of Algebra/Fall 2017 Numbers at the Foundations: Real Numbers In calculus, the derivative of a function f(x) is defined
Math 400-001/Foundations of Algebra/Fall 2017 Numbers at the Foundations: Real Numbers In calculus, the derivative of a function f(x) is defined using limits. As a particular case, the derivative of f(x)
More informationFunctions and Their Graphs
Functions and Their Graphs DEFINITION Function A function from a set D to a set Y is a rule that assigns a unique (single) element ƒ(x) Y to each element x D. A symbolic way to say y is a function of x
More information3.5. Dividing Polynomials. LEARN ABOUT the Math. Selecting a strategy to divide a polynomial by a binomial
3.5 Dividing Polynomials GOAL Use a variety of strategies to determine the quotient when one polynomial is divided by another polynomial. LEARN ABOU the Math Recall that long division can be used to determine
More informationSection 10.1 Extra Practice
Section 0. Extra Practice BLM 0. For each pair of functions, determine h(x) f (x) g(x). a) f (x) x 4 g(x) f (x) x 7 g(x) 5x c) f (x) x 3x g(x) x x 5 d) f (x) (x 4) g(x) 7x. Consider the functions f (x)
More informationChapter 2 Analysis of Graphs of Functions
Chapter Analysis of Graphs of Functions Chapter Analysis of Graphs of Functions Covered in this Chapter:.1 Graphs of Basic Functions and their Domain and Range. Odd, Even Functions, and their Symmetry..
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 informationMATH 215/255 Solutions to Additional Practice Problems April dy dt
. For the nonlinear system MATH 5/55 Solutions to Additional Practice Problems April 08 dx dt = x( x y, dy dt = y(.5 y x, x 0, y 0, (a Show that if x(0 > 0 and y(0 = 0, then the solution (x(t, y(t of the
More information1) The line has a slope of ) The line passes through (2, 11) and. 6) r(x) = x + 4. From memory match each equation with its graph.
Review Test 2 Math 1314 Name Write an equation of the line satisfying the given conditions. Write the answer in standard form. 1) The line has a slope of - 2 7 and contains the point (3, 1). Use the point-slope
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 informationMaking Connections with Rational Functions and Equations
Section 3.5 Making Connections with Rational Functions and Equations When solving a problem, it's important to read carefully to determine whether a function is being analyzed (Finding key features) or
More informationSection 13.3 Concavity and Curve Sketching. Dr. Abdulla Eid. College of Science. MATHS 104: Mathematics for Business II
Section 13.3 Concavity and Curve Sketching College of Science MATHS 104: Mathematics for Business II (University of Bahrain) Concavity 1 / 18 Concavity Increasing Function has three cases (University of
More informationMATH CALCULUS I 2.2: Differentiability, Graphs, and Higher Derivatives
MATH 12002 - CALCULUS I 2.2: Differentiability, Graphs, and Higher Derivatives Professor Donald L. White Department of Mathematical Sciences Kent State University D.L. White (Kent State University) 1 /
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 informationMAT116 Final Review Session Chapter 3: Polynomial and Rational Functions
MAT116 Final Review Session Chapter 3: Polynomial and Rational Functions Quadratic Function A quadratic function is defined by a quadratic or second-degree polynomial. Standard Form f x = ax 2 + bx + c,
More information6.1. Rational Expressions and Functions; Multiplying and Dividing. Copyright 2016, 2012, 2008 Pearson Education, Inc. 1
6.1 Rational Expressions and Functions; Multiplying and Dividing 1. Define rational expressions.. Define rational functions and give their domains. 3. Write rational expressions in lowest terms. 4. Multiply
More informationUnit 4 - Equations and Inequalities - Vocabulary
12/5/17 Unit 4 Unit 4 - Equations and Inequalities - Vocabulary Begin on a new page Write the date and unit in the top corners of the page Write the title across the top line Review Vocabulary: Absolute
More informationSolutions to Math 41 First Exam October 15, 2013
Solutions to Math 41 First Exam October 15, 2013 1. (16 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
More informationChapter 2 Linear Equations and Inequalities in One Variable
Chapter 2 Linear Equations and Inequalities in One Variable Section 2.1: Linear Equations in One Variable Section 2.3: Solving Formulas Section 2.5: Linear Inequalities in One Variable Section 2.6: Compound
More informationChapter 2. Limits and Continuity 2.6 Limits Involving Infinity; Asymptotes of Graphs
2.6 Limits Involving Infinity; Asymptotes of Graphs Chapter 2. Limits and Continuity 2.6 Limits Involving Infinity; Asymptotes of Graphs Definition. Formal Definition of Limits at Infinity.. We say that
More informationKing Fahd University of Petroleum and Minerals Prep-Year Math Program Math Term 161 Recitation (R1, R2)
Math 001 - Term 161 Recitation (R1, R) Question 1: How many rational and irrational numbers are possible between 0 and 1? (a) 1 (b) Finite (c) 0 (d) Infinite (e) Question : A will contain how many elements
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 information(2) Dividing both sides of the equation in (1) by the divisor, 3, gives: =
Dividing Polynomials Prepared by: Sa diyya Hendrickson Name: Date: Let s begin by recalling the process of long division for numbers. Consider the following fraction: Recall that fractions are just division
More informationSimplifying Rational Expressions and Functions
Department of Mathematics Grossmont College October 15, 2012 Recall: The Number Types Definition The set of whole numbers, ={0, 1, 2, 3, 4,...} is the set of natural numbers unioned with zero, written
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 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 informationSOLUTIONS FOR PROBLEMS 1-30
. Answer: 5 Evaluate x x + 9 for x SOLUTIONS FOR PROBLEMS - 0 When substituting x in x be sure to do the exponent before the multiplication by to get (). + 9 5 + When multiplying ( ) so that ( 7) ( ).
More informationSummer Packet for Students Taking Introduction to Calculus in the Fall
Summer Packet for Students Taking Introduction to Calculus in the Fall Algebra 2 Topics Needed for Introduction to Calculus Need to know: à Solve Equations Linear Quadratic Absolute Value Polynomial Rational
More informationDIFFERENTIATION RULES
3 DIFFERENTIATION RULES DIFFERENTIATION RULES 3.2 The Product and Quotient Rules In this section, we will learn about: Formulas that enable us to differentiate new functions formed from old functions by
More informationMATH 121: EXTRA PRACTICE FOR TEST 2. Disclaimer: Any material covered in class and/or assigned for homework is a fair game for the exam.
MATH 121: EXTRA PRACTICE FOR TEST 2 Disclaimer: Any material covered in class and/or assigned for homework is a fair game for the exam. 1 Linear Functions 1. Consider the functions f(x) = 3x + 5 and g(x)
More informationQuadratics. SPTA Mathematics Higher Notes
H Quadratics SPTA Mathematics Higher Notes Quadratics are expressions with degree 2 and are of the form ax 2 + bx + c, where a 0. The Graph of a Quadratic is called a Parabola, and there are 2 types as
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 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 informationMath 75 Mini-Mod Due Dates Spring 2016
Mini-Mod 1 Whole Numbers Due: 4/3 1.1 Whole Numbers 1.2 Rounding 1.3 Adding Whole Numbers; Estimation 1.4 Subtracting Whole Numbers 1.5 Basic Problem Solving 1.6 Multiplying Whole Numbers 1.7 Dividing
More informationMAC 1105-College Algebra LSCC, S. Nunamaker
MAC 1105-College Algebra LSCC, S. Nunamaker Chapter 1-Graphs, Functions, and Models 1.1 Introduction to Graphing I. Reasons for using graphs A. Visual presentations enhance understanding. B. Visual presentations
More information30 Wyner Math Academy I Fall 2015
30 Wyner Math Academy I Fall 2015 CHAPTER FOUR: QUADRATICS AND FACTORING Review November 9 Test November 16 The most common functions in math at this level are quadratic functions, whose graphs are parabolas.
More informationChapter 7 Quadratic Equations
Chapter 7 Quadratic Equations We have worked with trinomials of the form ax 2 + bx + c. Now we are going to work with equations of this form ax 2 + bx + c = 0 quadratic equations. When we write a quadratic
More informationWhat makes f '(x) undefined? (set the denominator = 0)
Chapter 3A Review 1. Find all critical numbers for the function ** Critical numbers find the first derivative and then find what makes f '(x) = 0 or undefined Q: What is the domain of this function (especially
More information8. Limit Laws. lim(f g)(x) = lim f(x) lim g(x), (x) = lim x a f(x) g lim x a g(x)
8. Limit Laws 8.1. Basic Limit Laws. If f and g are two functions and we know the it of each of them at a given point a, then we can easily compute the it at a of their sum, difference, product, constant
More information5.5 Special Rights. A Solidify Understanding Task
SECONDARY MATH III // MODULE 5 MODELING WITH GEOMETRY 5.5 In previous courses you have studied the Pythagorean theorem and right triangle trigonometry. Both of these mathematical tools are useful when
More informationUnit 4: Polynomial and Rational Functions
50 Unit 4: Polynomial and Rational Functions Polynomial Functions A polynomial function y px ( ) is a function of the form p( x) ax + a x + a x +... + ax + ax+ a n n 1 n n n 1 n 1 0 where an, an 1,...,
More informationSection Properties of Rational Expressions
88 Section. - Properties of Rational Expressions Recall that a rational number is any number that can be written as the ratio of two integers where the integer in the denominator cannot be. Rational Numbers:
More informationAcademic Algebra 2. Algebra 1 Review
Academic Algebra On the following pages you will find a review of the Algebra concepts needed to successfully complete Academic Algebra. Concepts such as fractions, solving equations, inequalities, absolute
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 informationP.1 Prerequisite skills Basic Algebra Skills
P.1 Prerequisite skills Basic Algebra Skills Topics: Evaluate an algebraic expression for given values of variables Combine like terms/simplify algebraic expressions Solve equations for a specified variable
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 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 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 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 informationMath 2 Variable Manipulation Part 7 Absolute Value & Inequalities
Math 2 Variable Manipulation Part 7 Absolute Value & Inequalities 1 MATH 1 REVIEW SOLVING AN ABSOLUTE VALUE EQUATION Absolute value is a measure of distance; how far a number is from zero. In practice,
More informationA. 16 B. 16 C. 4 D What is the solution set of 4x + 8 > 16?
Algebra II Honors Summer Math Packet 2017 Name: Date: 1. Solve for x: x + 6 = 5x + 12 2. What is the value of p in the equation 8p + 2 = p 10? F. 1 G. 1 H. J.. Solve for x: 15x (x + ) = 6 11. Solve for
More informationLesson 28: Another Computational Method of Solving a Linear System
Lesson 28: Another Computational Method of Solving a Linear System Student Outcomes Students learn the elimination method for solving a system of linear equations. Students use properties of rational numbers
More informationCALCULUS II - Self Test
175 2- CALCULUS II - Self Test Instructor: Andrés E. Caicedo November 9, 2009 Name These questions are divided into four groups. Ideally, you would answer YES to all questions in group A, to most questions
More informationMath 131 Week-in-Review #11 (Final Exam Review: All previous sections as well as sections 5.5, 6.1, 6.5, and 6.7)
Math 131 Week-in-Review #11 (Final Exam Review: All previous sections as well as sections 5.5, 6.1, 6.5, and 6.7) Note: This collection of questions is intended to be a brief overview of the exam material
More informationSuppose that f is continuous on [a, b] and differentiable on (a, b). Then
Lectures 1/18 Derivatives and Graphs When we have a picture of the graph of a function f(x), we can make a picture of the derivative f (x) using the slopes of the tangents to the graph of f. In this section
More informationy 1 x 1 ) 2 + (y 2 ) 2 A circle is a set of points P in a plane that are equidistant from a fixed point, called the center.
Ch 12. Conic Sections Circles, Parabolas, Ellipses & Hyperbolas The formulas for the conic sections are derived by using the distance formula, which was derived from the Pythagorean Theorem. If you know
More informationChapter Five Notes N P U2C5
Chapter Five Notes N P UC5 Name Period Section 5.: Linear and Quadratic Functions with Modeling In every math class you have had since algebra you have worked with equations. Most of those equations have
More informationAlgebra Summer Review Packet
Name: Algebra Summer Review Packet About Algebra 1: Algebra 1 teaches students to think, reason, and communicate mathematically. Students use variables to determine solutions to real world problems. Skills
More informationMath 111: Calculus. David Perkinson
Math : Calculus David Perkinson Fall 207 Contents Week, Monday: Introduction: derivatives, integrals, and the fundamental theorem. 5 Week, Wednesday: Average speed, instantaneous speed. Definition of the
More informationReading Mathematical Expressions & Arithmetic Operations Expression Reads Note
Math 001 - Term 171 Reading Mathematical Expressions & Arithmetic Operations Expression Reads Note x A x belongs to A,x is in A Between an element and a set. A B A is a subset of B Between two sets. φ
More informationMath 141: Lecture 11
Math 141: Lecture 11 The Fundamental Theorem of Calculus and integration methods Bob Hough October 12, 2016 Bob Hough Math 141: Lecture 11 October 12, 2016 1 / 36 First Fundamental Theorem of Calculus
More informationIn general, if we start with a function f and want to reverse the differentiation process, then we are finding an antiderivative of f.
Math 1410 Worksheet #27: Section 4.9 Name: Our final application of derivatives is a prelude to what will come in later chapters. In many situations, it will be necessary to find a way to reverse the differentiation
More informationMath Review ECON 300: Spring 2014 Benjamin A. Jones MATH/CALCULUS REVIEW
MATH/CALCULUS REVIEW SLOPE, INTERCEPT, and GRAPHS REVIEW (adapted from Paul s Online Math Notes) Let s start with some basic review material to make sure everybody is on the same page. The slope of a line
More informationUnit IV Derivatives 20 Hours Finish by Christmas
Unit IV Derivatives 20 Hours Finish by Christmas Calculus There two main streams of Calculus: Differentiation Integration Differentiation is used to find the rate of change of variables relative to one
More informationUnit IV Derivatives 20 Hours Finish by Christmas
Unit IV Derivatives 20 Hours Finish by Christmas Calculus There two main streams of Calculus: Differentiation Integration Differentiation is used to find the rate of change of variables relative to one
More informationMATH 121: EXTRA PRACTICE FOR TEST 2. Disclaimer: Any material covered in class and/or assigned for homework is a fair game for the exam.
MATH 121: EXTRA PRACTICE FOR TEST 2 Disclaimer: Any material covered in class and/or assigned for homework is a fair game for the exam. 1 Linear Functions 1. Consider the functions f(x) = 3x + 5 and g(x)
More information4.2: What Derivatives Tell Us
4.2: What Derivatives Tell Us Problem Fill in the following blanks with the correct choice of the words from this list: Increasing, decreasing, positive, negative, concave up, concave down (a) If you know
More informationChapter 2.7 and 7.3. Lecture 5
Chapter 2.7 and 7.3 Chapter 2 Polynomial and Rational Functions 2.1 Complex Numbers 2.2 Quadratic Functions 2.3 Polynomial Functions and Their Graphs 2.4 Dividing Polynomials; Remainder and Factor Theorems
More information