Math 130 Winter 2010 Quotient/Product/Chain Rule Practice January 28, 2010

Save this PDF as:
 WORD  PNG  TXT  JPG

Size: px
Start display at page:

Download "Math 130 Winter 2010 Quotient/Product/Chain Rule Practice January 28, 2010"

Transcription

1 Math Winter 0 Quotient/Product/Chain Rule Practice January 8, 0 1. Find the derivative of the following functions: a) gx) = x + 4x) 5 + x x + 1. To do this problem we treat each part seperately. F the first piece we use the fact that if we have fx) n then the derivatives is nfx) n 1 g x). F the second piece we us the quotient rule with u, the numerat equal to x and v = x + 1. Thus u = 1 and v = 1. Putting this together we get: g x) = 5fx)) 4 f x) + vu uv = 5x + 4) 4 x + 1)1) x1) x + 4) + x + 1) = 5x + 4) 4 x + 4) + v 1 x + 1) b) fx) = x + x 4x 5 x ) To do this problem we will apply the product rule with: u = u = ) 1 x + x ) ) 1 x + x x + 1 ) v = 4x 5 x ) v = 0x 4 + 6x 4 Here we have used rewritten the cube root as a power of 1 and rewritten x as x. To compute u we used the generalized power rule f functions that look like gx) n and f v it only requires our standard power rule that we learned a while ago. Putting this all together we have: f x) = uv + uv ) 1 = x + x = x + x ) ) x + 1 x + 1 ) 4x 5 x ) + x + x ) 1 0x 4 + 6x 4) ) 4x 5 x ) + x + x ) 1 0x 4 + 6x 4) 1

2 c) hx) = x4 + x x. F this problem we might first rewrite the function as: hx) = x 4 ) 1 + x x To compute the derivative we then apply the generalized power rule which gives: h x) = 1 ) x 4 ) 1 + x d x dx [ x 4 ] + x x We are not done though because we still have to compute the d dx [ ]. Which means compute the derivative of what is inside the square brackets. To do the derivative of what is inside the square brackets we have to use the quotient rule. Here we have: u = x 4 + x v = x u = 4x + v = 1 Applying this we have: ) 1 x h 4 ) 1 + x x )4x + ) x 4 ) + x)1) x) = x x ) One could simplify this me but I will not do that here.

3 . We have not yet learned this but if fx) = x then the derivative is f x) = x ln). Use this fact and the chain rule find the following derivatives a) hx) = 4x F this problem we first think of hx) as the composition of two function hx) = fgx)). Here fx) = x and gx) = 4x. You are given that f x) = x ln) although we now know this fact from class. g x) = 4. If we apply the chain rule we have: b) gx) = x + x ) 5 h x) = f gx))g x) = gx) ln)g x) = 4x ln)4) = 4) 4x ln) F this problem we apply the generalized power rule. When we compute the derivative of the inside function we have to use the fact that is given in the instructions to compute the derivative of x. When we do this we obtain: g x) = 5 x + x ) 4 x ln) + x) c) kx) = 6x x) This problem is extremely similar to the first problem on this page except that gx) = 6x x here and thus g x) = 1x. So we have: k x) = f gx))g x) = gx) ln)g x) = 6x x) ln)1x ) = 1x ) 6x x) ln) d) lx) = xx x + 1 F this problem we must use the quotient rule with: u = x x v = x + 1 u = 1) x + x x ln) v = 1 = x + x x ln) Here to compute u I had to use the product rule and the derivative of x which was given in the instructions. Applying the quotient rule we have: l x) = vu uv v = x + 1) x + x x ln)) x x 1) x + 1)

4 . Suppose the cost in dollars of manufacturing x items is given by : and the demand equation is given by: C = 000x x = p. Here p represents the price of one item. In terms of the demand x, a) find an expression f the revenue R; The Revenue is equal to the price, p, times the demand, x. So we have: Rx) = x p, but we want a function of x only so we must eliminate p from this expression. To do so we use the demand equation that relates x and p. We need to solve this equation f p in terms of x so we first square both sides giving: x = p Then subtracting we have: Dividing by gives: p = x. p = 000 x. If we use this to substitute into the revenue expression we obtain: ) Rx) = x 000 x = 000x x b) find an expression f the profit P ; The profit is the Revenue minus the Cost. Using the answer from a) we have: P x) = Rx) CX) c) find an expression f the marginal profit. = 000x x 000x 5000 = 8000x x 5000 The marginal profit is the derivative of the profit function. We can find this using the power rules we developed. Namely, P x) = 8000 x P x) = 8000 x 4

5 d) determine the value of the marginal profit when 1 items are in demand. Give a practical interpretation of this quantity. Should the company sell me less than this number? Why? We just need to evaluate P 1) which is: P 1) = ) = 1768 dollars item This means that if one was to go from selling 1 to 1 items then the profit would decrease by approximately $1,768. Thus the company certainly would not want to increase production. In fact if they decrease production from 1 to 11 then the profit would increase by approximately $1,768 so they would want to decrease production. e) find the value of x that makes the marginal profit equal to 0. What is the practical interpretation of this number? Should the company sell me less than this number x? Why? We set P x) = 0 P x) = 8000 x = 0 Now we solve this expression f x., 8000 x = 0., 8000 = x 4000 = x ± 4000 = x Since we are only interested in positive values of x it represents the number of items produced and sold). Then we take x = items. Since the marginal profit at this number of items is 0 then increasing decreasing the number of items will not change the profit, thus you would want to stay at this level of production. There is slightly me to this because we have to make sure that we are indeed at a maximum profit and not a minimum profit, but in this case we are indeed at a maximum when x = items) 5

The composition g f of the functions f and g is the function (g f)(x) = g(f(x)). This means, "do the function f to x, then do g to the result.

The composition g f of the functions f and g is the function (g f)(x) = g(f(x)). This means, do the function f to x, then do g to the result. 30 5.6 The chain rule The composition g f of the functions f and g is the function (g f)(x) = g(f(x)). This means, "do the function f to x, then do g to the result." Example. g(x) = x 2 and f(x) = (3x+1).

More information

Math 120 Final Exam Practice Problems, Form: A

Math 120 Final Exam Practice Problems, Form: A Math 120 Final Exam Practice Problems, Form: A Name: While every attempt was made to be complete in the types of problems given below, we make no guarantees about the completeness of the problems. Specifically,

More information

Integrals of Rational Functions

Integrals of Rational Functions Integrals of Rational Functions Scott R. Fulton Overview A rational function has the form where p and q are polynomials. For example, r(x) = p(x) q(x) f(x) = x2 3 x 4 + 3, g(t) = t6 + 4t 2 3, 7t 5 + 3t

More information

Chapter 7 - Roots, Radicals, and Complex Numbers

Chapter 7 - Roots, Radicals, and Complex Numbers Math 233 - Spring 2009 Chapter 7 - Roots, Radicals, and Complex Numbers 7.1 Roots and Radicals 7.1.1 Notation and Terminology In the expression x the is called the radical sign. The expression under the

More information

1 Lecture: Integration of rational functions by decomposition

1 Lecture: Integration of rational functions by decomposition Lecture: Integration of rational functions by decomposition into partial fractions Recognize and integrate basic rational functions, except when the denominator is a power of an irreducible quadratic.

More information

Recitation 4. 24xy for 0 < x < 1, 0 < y < 1, x + y < 1 0 elsewhere

Recitation 4. 24xy for 0 < x < 1, 0 < y < 1, x + y < 1 0 elsewhere Recitation. Exercise 3.5: If the joint probability density of X and Y is given by xy for < x

More information

Marginal Cost. Example 1: Suppose the total cost in dollars per week by ABC Corporation for 2

Marginal Cost. Example 1: Suppose the total cost in dollars per week by ABC Corporation for 2 Math 114 Marginal Functions in Economics Marginal Cost Suppose a business owner is operating a plant that manufactures a certain product at a known level. Sometimes the business owner will want to know

More information

2 Integrating Both Sides

2 Integrating Both Sides 2 Integrating Both Sides So far, the only general method we have for solving differential equations involves equations of the form y = f(x), where f(x) is any function of x. The solution to such an equation

More information

1 Calculus of Several Variables

1 Calculus of Several Variables 1 Calculus of Several Variables Reading: [Simon], Chapter 14, p. 300-31. 1.1 Partial Derivatives Let f : R n R. Then for each x i at each point x 0 = (x 0 1,..., x 0 n) the ith partial derivative is defined

More information

Homework # 3 Solutions

Homework # 3 Solutions Homework # 3 Solutions February, 200 Solution (2.3.5). Noting that and ( + 3 x) x 8 = + 3 x) by Equation (2.3.) x 8 x 8 = + 3 8 by Equations (2.3.7) and (2.3.0) =3 x 8 6x2 + x 3 ) = 2 + 6x 2 + x 3 x 8

More information

MATH SOLUTIONS TO PRACTICE FINAL EXAM. (x 2)(x + 2) (x 2)(x 3) = x + 2. x 2 x 2 5x + 6 = = 4.

MATH SOLUTIONS TO PRACTICE FINAL EXAM. (x 2)(x + 2) (x 2)(x 3) = x + 2. x 2 x 2 5x + 6 = = 4. MATH 55 SOLUTIONS TO PRACTICE FINAL EXAM x 2 4.Compute x 2 x 2 5x + 6. When x 2, So x 2 4 x 2 5x + 6 = (x 2)(x + 2) (x 2)(x 3) = x + 2 x 3. x 2 4 x 2 x 2 5x + 6 = 2 + 2 2 3 = 4. x 2 9 2. Compute x + sin

More information

Definition 2.1 The line x = a is a vertical asymptote of the function y = f(x) if y approaches ± as x approaches a from the right or left.

Definition 2.1 The line x = a is a vertical asymptote of the function y = f(x) if y approaches ± as x approaches a from the right or left. Vertical and Horizontal Asymptotes Definition 2.1 The line x = a is a vertical asymptote of the function y = f(x) if y approaches ± as x approaches a from the right or left. This graph has a vertical asymptote

More information

6.4 Logarithmic Equations and Inequalities

6.4 Logarithmic Equations and Inequalities 6.4 Logarithmic Equations and Inequalities 459 6.4 Logarithmic Equations and Inequalities In Section 6.3 we solved equations and inequalities involving exponential functions using one of two basic strategies.

More information

3.1. Solving linear equations. Introduction. Prerequisites. Learning Outcomes. Learning Style

3.1. Solving linear equations. Introduction. Prerequisites. Learning Outcomes. Learning Style Solving linear equations 3.1 Introduction Many problems in engineering reduce to the solution of an equation or a set of equations. An equation is a type of mathematical expression which contains one or

More information

Simplify the rational expression. Find all numbers that must be excluded from the domain of the simplified rational expression.

Simplify the rational expression. Find all numbers that must be excluded from the domain of the simplified rational expression. MAC 1105 Final Review Simplify the rational expression. Find all numbers that must be excluded from the domain of the simplified rational expression. 1) 8x 2-49x + 6 x - 6 A) 1, x 6 B) 8x - 1, x 6 x -

More information

Rational Exponents. Squaring both sides of the equation yields. and to be consistent, we must have

Rational Exponents. Squaring both sides of the equation yields. and to be consistent, we must have 8.6 Rational Exponents 8.6 OBJECTIVES 1. Define rational exponents 2. Simplify expressions containing rational exponents 3. Use a calculator to estimate the value of an expression containing rational exponents

More information

Solving Quadratic Equations

Solving Quadratic Equations 9.3 Solving Quadratic Equations by Using the Quadratic Formula 9.3 OBJECTIVES 1. Solve a quadratic equation by using the quadratic formula 2. Determine the nature of the solutions of a quadratic equation

More information

1.5. Factorisation. Introduction. Prerequisites. Learning Outcomes. Learning Style

1.5. Factorisation. Introduction. Prerequisites. Learning Outcomes. Learning Style Factorisation 1.5 Introduction In Block 4 we showed the way in which brackets were removed from algebraic expressions. Factorisation, which can be considered as the reverse of this process, is dealt with

More information

TOPIC 3: CONTINUITY OF FUNCTIONS

TOPIC 3: CONTINUITY OF FUNCTIONS TOPIC 3: CONTINUITY OF FUNCTIONS. Absolute value We work in the field of real numbers, R. For the study of the properties of functions we need the concept of absolute value of a number. Definition.. Let

More information

QUADRATIC EQUATIONS AND FUNCTIONS

QUADRATIC EQUATIONS AND FUNCTIONS Douglas College Learning Centre QUADRATIC EQUATIONS AND FUNCTIONS Quadratic equations and functions are very important in Business Math. Questions related to quadratic equations and functions cover a wide

More information

3.1. RATIONAL EXPRESSIONS

3.1. RATIONAL EXPRESSIONS 3.1. RATIONAL EXPRESSIONS RATIONAL NUMBERS In previous courses you have learned how to operate (do addition, subtraction, multiplication, and division) on rational numbers (fractions). Rational numbers

More information

Microeconomic Theory: Basic Math Concepts

Microeconomic Theory: Basic Math Concepts Microeconomic Theory: Basic Math Concepts Matt Van Essen University of Alabama Van Essen (U of A) Basic Math Concepts 1 / 66 Basic Math Concepts In this lecture we will review some basic mathematical concepts

More information

CHAPTER FIVE. Solutions for Section 5.1. Skill Refresher. Exercises

CHAPTER FIVE. Solutions for Section 5.1. Skill Refresher. Exercises CHAPTER FIVE 5.1 SOLUTIONS 265 Solutions for Section 5.1 Skill Refresher S1. Since 1,000,000 = 10 6, we have x = 6. S2. Since 0.01 = 10 2, we have t = 2. S3. Since e 3 = ( e 3) 1/2 = e 3/2, we have z =

More information

1 Lecture 19: Implicit differentiation

1 Lecture 19: Implicit differentiation Lecture 9: Implicit differentiation. Outline The technique of implicit differentiation Tangent lines to a circle Examples.2 Implicit differentiation Suppose we have two quantities or variables x and y

More information

20. Product rule, Quotient rule

20. Product rule, Quotient rule 20. Prouct rule, 20.1. Prouct rule Prouct rule, Prouct rule We have seen that the erivative of a sum is the sum of the erivatives: [f(x) + g(x)] = x x [f(x)] + x [(g(x)]. One might expect from this that

More information

ALGEBRA REVIEW LEARNING SKILLS CENTER. Exponents & Radicals

ALGEBRA REVIEW LEARNING SKILLS CENTER. Exponents & Radicals ALGEBRA REVIEW LEARNING SKILLS CENTER The "Review Series in Algebra" is taught at the beginning of each quarter by the staff of the Learning Skills Center at UC Davis. This workshop is intended to be an

More information

a 1 x + a 0 =0. (3) ax 2 + bx + c =0. (4)

a 1 x + a 0 =0. (3) ax 2 + bx + c =0. (4) ROOTS OF POLYNOMIAL EQUATIONS In this unit we discuss polynomial equations. A polynomial in x of degree n, where n 0 is an integer, is an expression of the form P n (x) =a n x n + a n 1 x n 1 + + a 1 x

More information

Solving Logarithmic Equations

Solving Logarithmic Equations Solving Logarithmic Equations Deciding How to Solve Logarithmic Equation When asked to solve a logarithmic equation such as log (x + 7) = or log (7x + ) = log (x + 9), the first thing we need to decide

More information

Polynomials. Key Terms. quadratic equation parabola conjugates trinomial. polynomial coefficient degree monomial binomial GCF

Polynomials. Key Terms. quadratic equation parabola conjugates trinomial. polynomial coefficient degree monomial binomial GCF Polynomials 5 5.1 Addition and Subtraction of Polynomials and Polynomial Functions 5.2 Multiplication of Polynomials 5.3 Division of Polynomials Problem Recognition Exercises Operations on Polynomials

More information

Lagrange Interpolation is a method of fitting an equation to a set of points that functions well when there are few points given.

Lagrange Interpolation is a method of fitting an equation to a set of points that functions well when there are few points given. Polynomials (Ch.1) Study Guide by BS, JL, AZ, CC, SH, HL Lagrange Interpolation is a method of fitting an equation to a set of points that functions well when there are few points given. Sasha s method

More information

Lecture Notes on Elasticity of Substitution

Lecture Notes on Elasticity of Substitution Lecture Notes on Elasticity of Substitution Ted Bergstrom, UCSB Economics 20A October 26, 205 Today s featured guest is the elasticity of substitution. Elasticity of a function of a single variable Before

More information

Lecture 3 : The Natural Exponential Function: f(x) = exp(x) = e x. y = exp(x) if and only if x = ln(y)

Lecture 3 : The Natural Exponential Function: f(x) = exp(x) = e x. y = exp(x) if and only if x = ln(y) Lecture 3 : The Natural Exponential Function: f(x) = exp(x) = Last day, we saw that the function f(x) = ln x is one-to-one, with domain (, ) and range (, ). We can conclude that f(x) has an inverse function

More information

Polynomials. Solving Equations by Using the Zero Product Rule

Polynomials. Solving Equations by Using the Zero Product Rule mil23264_ch05_303-396 9:21:05 06:16 PM Page 303 Polynomials 5.1 Addition and Subtraction of Polynomials and Polynomial Functions 5.2 Multiplication of Polynomials 5.3 Division of Polynomials 5.4 Greatest

More information

FINAL EXAM SECTIONS AND OBJECTIVES FOR COLLEGE ALGEBRA

FINAL EXAM SECTIONS AND OBJECTIVES FOR COLLEGE ALGEBRA FINAL EXAM SECTIONS AND OBJECTIVES FOR COLLEGE ALGEBRA 1.1 Solve linear equations and equations that lead to linear equations. a) Solve the equation: 1 (x + 5) 4 = 1 (2x 1) 2 3 b) Solve the equation: 3x

More information

MATH 34A REVIEW FOR MIDTERM 2, WINTER 2012. 1. Lines. (1) Find the equation of the line passing through (2,-1) and (-2,9). y = 5

MATH 34A REVIEW FOR MIDTERM 2, WINTER 2012. 1. Lines. (1) Find the equation of the line passing through (2,-1) and (-2,9). y = 5 MATH 34A REVIEW FOR MIDTERM 2, WINTER 2012 ANSWERS 1. Lines (1) Find the equation of the line passing through (2,-1) and (-2,9). y = 5 2 x + 4. (2) Find the equation of the line which meets the x-axis

More information

Calculus with Analytic Geometry I Exam 5-Take Home Part Due: Monday, October 3, 2011; 12PM

Calculus with Analytic Geometry I Exam 5-Take Home Part Due: Monday, October 3, 2011; 12PM NAME: Calculus with Analytic Geometry I Exam 5-Take Home Part Due: Monday, October 3, 2011; 12PM INSTRUCTIONS. As usual, show work where appropriate. As usual, use equal signs properly, write in full sentences,

More information

The Method of Partial Fractions Math 121 Calculus II Spring 2015

The Method of Partial Fractions Math 121 Calculus II Spring 2015 Rational functions. as The Method of Partial Fractions Math 11 Calculus II Spring 015 Recall that a rational function is a quotient of two polynomials such f(x) g(x) = 3x5 + x 3 + 16x x 60. The method

More information

This is a square root. The number under the radical is 9. (An asterisk * means multiply.)

This is a square root. The number under the radical is 9. (An asterisk * means multiply.) Page of Review of Radical Expressions and Equations Skills involving radicals can be divided into the following groups: Evaluate square roots or higher order roots. Simplify radical expressions. Rationalize

More information

A. Break even analysis

A. Break even analysis Lecture (50 minutes) Eighth week lessons Function (continued) & Quadratic Equations (Divided into 3 lectures of 50 minutes each) a) Break even analysis ) Supply, Demand and market equilirium. c) Class

More information

3.2. Solving quadratic equations. Introduction. Prerequisites. Learning Outcomes. Learning Style

3.2. Solving quadratic equations. Introduction. Prerequisites. Learning Outcomes. Learning Style Solving quadratic equations 3.2 Introduction A quadratic equation is one which can be written in the form ax 2 + bx + c = 0 where a, b and c are numbers and x is the unknown whose value(s) we wish to find.

More information

2 is the BASE 5 is the EXPONENT. Power Repeated Standard Multiplication. To evaluate a power means to find the answer in standard form.

2 is the BASE 5 is the EXPONENT. Power Repeated Standard Multiplication. To evaluate a power means to find the answer in standard form. Grade 9 Mathematics Unit : Powers and Exponent Rules Sec.1 What is a Power 5 is the BASE 5 is the EXPONENT The entire 5 is called a POWER. 5 = written as repeated multiplication. 5 = 3 written in standard

More information

Definition of Vertical Asymptote The line x = a is called a vertical asymptote of f (x) if at least one of the following is true: f (x) =

Definition of Vertical Asymptote The line x = a is called a vertical asymptote of f (x) if at least one of the following is true: f (x) = Vertical Asymptotes Definition of Vertical Asymptote The line x = a is called a vertical asymptote of f (x) if at least one of the following is true: lim f (x) = x a lim f (x) = lim x a lim f (x) = x a

More information

A. Factoring Method - Some, but not all quadratic equations can be solved by factoring.

A. Factoring Method - Some, but not all quadratic equations can be solved by factoring. DETAILED SOLUTIONS AND CONCEPTS - QUADRATIC EQUATIONS Prepared by Ingrid Stewart, Ph.D., College of Southern Nevada Please Send Questions and Comments to ingrid.stewart@csn.edu. Thank you! PLEASE NOTE

More information

Equations, Inequalities & Partial Fractions

Equations, Inequalities & Partial Fractions Contents Equations, Inequalities & Partial Fractions.1 Solving Linear Equations 2.2 Solving Quadratic Equations 1. Solving Polynomial Equations 1.4 Solving Simultaneous Linear Equations 42.5 Solving Inequalities

More information

Math 1314 Lesson 8: Business Applications: Break Even Analysis, Equilibrium Quantity/Price

Math 1314 Lesson 8: Business Applications: Break Even Analysis, Equilibrium Quantity/Price Math 1314 Lesson 8: Business Applications: Break Even Analysis, Equilibrium Quantity/Price Cost functions model the cost of producing goods or providing services. Examples: rent, utilities, insurance,

More information

Chapter 8. Exponential and Logarithmic Functions

Chapter 8. Exponential and Logarithmic Functions Chapter 8 Exponential and Logarithmic Functions This unit defines and investigates exponential and logarithmic functions. We motivate exponential functions by their similarity to monomials as well as their

More information

Lecture Notes on Elasticity of Substitution

Lecture Notes on Elasticity of Substitution Lecture Notes on Elasticity of Substitution Ted Bergstrom, UCSB Economics 210A March 3, 2011 Today s featured guest is the elasticity of substitution. Elasticity of a function of a single variable Before

More information

Functions and Equations

Functions and Equations Centre for Education in Mathematics and Computing Euclid eworkshop # Functions and Equations c 014 UNIVERSITY OF WATERLOO Euclid eworkshop # TOOLKIT Parabolas The quadratic f(x) = ax + bx + c (with a,b,c

More information

Microeconomics Sept. 16, 2010 NOTES ON CALCULUS AND UTILITY FUNCTIONS

Microeconomics Sept. 16, 2010 NOTES ON CALCULUS AND UTILITY FUNCTIONS DUSP 11.203 Frank Levy Microeconomics Sept. 16, 2010 NOTES ON CALCULUS AND UTILITY FUNCTIONS These notes have three purposes: 1) To explain why some simple calculus formulae are useful in understanding

More information

Basic Integration Formulas and the Substitution Rule

Basic Integration Formulas and the Substitution Rule Basic Integration Formulas and the Substitution Rule The second fundamental theorem of integral calculus Recall from the last lecture the second fundamental theorem of integral calculus. Theorem Let f(x)

More information

correct-choice plot f(x) and draw an approximate tangent line at x = a and use geometry to estimate its slope comment The choices were:

correct-choice plot f(x) and draw an approximate tangent line at x = a and use geometry to estimate its slope comment The choices were: Topic 1 2.1 mode MultipleSelection text How can we approximate the slope of the tangent line to f(x) at a point x = a? This is a Multiple selection question, so you need to check all of the answers that

More information

Student Activity: To investigate the Average Value of a Function

Student Activity: To investigate the Average Value of a Function Student Activity: To investigate the Average Value of a Function Use in connection with the interactive file, Average Value 3, on the Student s CD. 1. Click all the boxes in the interactive file. Move

More information

Lecture 2. Marginal Functions, Average Functions, Elasticity, the Marginal Principle, and Constrained Optimization

Lecture 2. Marginal Functions, Average Functions, Elasticity, the Marginal Principle, and Constrained Optimization Lecture 2. Marginal Functions, Average Functions, Elasticity, the Marginal Principle, and Constrained Optimization 2.1. Introduction Suppose that an economic relationship can be described by a real-valued

More information

Math 131, Lecture 19: The Chain Rule

Math 131, Lecture 19: The Chain Rule Math 131, Lecture 19: The Chain Rule Charles Staats Wednesday, 9 November 2011 Important Note: There have been a few changes to Assignment 17. Don t use the version from Lecture 18. 1 Notes on the quiz

More information

Math 265 (Butler) Practice Midterm II B (Solutions)

Math 265 (Butler) Practice Midterm II B (Solutions) Math 265 (Butler) Practice Midterm II B (Solutions) 1. Find (x 0, y 0 ) so that the plane tangent to the surface z f(x, y) x 2 + 3xy y 2 at ( x 0, y 0, f(x 0, y 0 ) ) is parallel to the plane 16x 2y 2z

More information

1.7. Partial Fractions. 1.7.1. Rational Functions and Partial Fractions. A rational function is a quotient of two polynomials: R(x) = P (x) Q(x).

1.7. Partial Fractions. 1.7.1. Rational Functions and Partial Fractions. A rational function is a quotient of two polynomials: R(x) = P (x) Q(x). .7. PRTIL FRCTIONS 3.7. Partial Fractions.7.. Rational Functions and Partial Fractions. rational function is a quotient of two polynomials: R(x) = P (x) Q(x). Here we discuss how to integrate rational

More information

Sample Problems. Practice Problems

Sample Problems. Practice Problems Lecture Notes Partial Fractions page Sample Problems Compute each of the following integrals.. x dx. x + x (x + ) (x ) (x ) dx 8. x x dx... x (x + ) (x + ) dx x + x x dx x + x x + 6x x dx + x 6. 7. x (x

More information

Introduction Proof by unique factorization in Z Proof with Gaussian integers Proof by geometry Applications. Pythagorean Triples

Introduction Proof by unique factorization in Z Proof with Gaussian integers Proof by geometry Applications. Pythagorean Triples Pythagorean Triples Keith Conrad University of Connecticut August 4, 008 Introduction We seek positive integers a, b, and c such that a + b = c. Plimpton 3 Babylonian table of Pythagorean triples (1800

More information

COMPASS Numerical Skills/Pre-Algebra Preparation Guide. Introduction Operations with Integers Absolute Value of Numbers 13

COMPASS Numerical Skills/Pre-Algebra Preparation Guide. Introduction Operations with Integers Absolute Value of Numbers 13 COMPASS Numerical Skills/Pre-Algebra Preparation Guide Please note that the guide is for reference only and that it does not represent an exact match with the assessment content. The Assessment Centre

More information

Basic numerical skills: POWERS AND LOGARITHMS

Basic numerical skills: POWERS AND LOGARITHMS 1. Introduction (easy) Basic numerical skills: POWERS AND LOGARITHMS Powers and logarithms provide a powerful way of representing large and small quantities, and performing complex calculations. Understanding

More information

More Quadratic Equations

More Quadratic Equations More Quadratic Equations Math 99 N1 Chapter 8 1 Quadratic Equations We won t discuss quadratic inequalities. Quadratic equations are equations where the unknown appears raised to second power, and, possibly

More information

c. Given your answer in part (b), what do you anticipate will happen in this market in the long-run?

c. Given your answer in part (b), what do you anticipate will happen in this market in the long-run? Perfect Competition Questions Question 1 Suppose there is a perfectly competitive industry where all the firms are identical with identical cost curves. Furthermore, suppose that a representative firm

More information

5-3 Polynomial Functions. not in one variable because there are two variables, x. and y

5-3 Polynomial Functions. not in one variable because there are two variables, x. and y y. 5-3 Polynomial Functions State the degree and leading coefficient of each polynomial in one variable. If it is not a polynomial in one variable, explain why. 1. 11x 6 5x 5 + 4x 2 coefficient of the

More information

Solving Exponential Equations

Solving Exponential Equations Solving Exponential Equations Deciding How to Solve Exponential Equations When asked to solve an exponential equation such as x + 6 = or x = 18, the first thing we need to do is to decide which way is

More information

1.6 The Order of Operations

1.6 The Order of Operations 1.6 The Order of Operations Contents: Operations Grouping Symbols The Order of Operations Exponents and Negative Numbers Negative Square Roots Square Root of a Negative Number Order of Operations and Negative

More information

6. Differentiating the exponential and logarithm functions

6. Differentiating the exponential and logarithm functions 1 6. Differentiating te exponential and logaritm functions We wis to find and use derivatives for functions of te form f(x) = a x, were a is a constant. By far te most convenient suc function for tis purpose

More information

HOMEWORK 4 SOLUTIONS. All questions are from Vector Calculus, by Marsden and Tromba

HOMEWORK 4 SOLUTIONS. All questions are from Vector Calculus, by Marsden and Tromba HOMEWORK SOLUTIONS All questions are from Vector Calculus, by Marsden and Tromba Question :..6 Let w = f(x, y) be a function of two variables, and let x = u + v, y = u v. Show that Solution. By the chain

More information

Using a table of derivatives

Using a table of derivatives Using a table of derivatives In this unit we construct a Table of Derivatives of commonly occurring functions. This is done using the knowledge gained in previous units on differentiation from first principles.

More information

Math 1314 Lesson 8: Business Applications: Break Even Analysis, Equilibrium Quantity/Price

Math 1314 Lesson 8: Business Applications: Break Even Analysis, Equilibrium Quantity/Price Math 1314 Lesson 8: Business Applications: Break Even Analysis, Equilibrium Quantity/Price Cost functions model the cost of producing goods or providing services. Examples: rent, utilities, insurance,

More information

Homework #7 Solutions

Homework #7 Solutions Homework #7 Solutions Problems Bolded problems are worth 2 points. Section 3.4: 2, 6, 14, 16, 24, 36, 38, 42 Chapter 3 Review (pp. 159 162): 24, 34, 36, 54, 66 Etra Problem 3.4.2. If f () = 2 ( 3 + 5),

More information

5.4 The Quadratic Formula

5.4 The Quadratic Formula Section 5.4 The Quadratic Formula 481 5.4 The Quadratic Formula Consider the general quadratic function f(x) = ax + bx + c. In the previous section, we learned that we can find the zeros of this function

More information

2.1 Increasing, Decreasing, and Piecewise Functions; Applications

2.1 Increasing, Decreasing, and Piecewise Functions; Applications 2.1 Increasing, Decreasing, and Piecewise Functions; Applications Graph functions, looking for intervals on which the function is increasing, decreasing, or constant, and estimate relative maxima and minima.

More information

Cost-Volume-Profit Analysis

Cost-Volume-Profit Analysis Cost-Volume-Profit Analysis Cost-volume-profit (CVP) analysis is used to determine how changes in costs and volume affect a company's operating income and net income. In performing this analysis, there

More information

Linear and quadratic Taylor polynomials for functions of several variables.

Linear and quadratic Taylor polynomials for functions of several variables. ams/econ 11b supplementary notes ucsc Linear quadratic Taylor polynomials for functions of several variables. c 010, Yonatan Katznelson Finding the extreme (minimum or maximum) values of a function, is

More information

TOPIC 4: DERIVATIVES

TOPIC 4: DERIVATIVES TOPIC 4: DERIVATIVES 1. The derivative of a function. Differentiation rules 1.1. The slope of a curve. The slope of a curve at a point P is a measure of the steepness of the curve. If Q is a point on the

More information

x 2 + y 2 = 1 y 1 = x 2 + 2x y = x 2 + 2x + 1

x 2 + y 2 = 1 y 1 = x 2 + 2x y = x 2 + 2x + 1 Implicit Functions Defining Implicit Functions Up until now in this course, we have only talked about functions, which assign to every real number x in their domain exactly one real number f(x). The graphs

More information

MBA Jump Start Program

MBA Jump Start Program MBA Jump Start Program Module 2: Mathematics Thomas Gilbert Mathematics Module Online Appendix: Basic Mathematical Concepts 2 1 The Number Spectrum Generally we depict numbers increasing from left to right

More information

Factoring Polynomials

Factoring Polynomials UNIT 11 Factoring Polynomials You can use polynomials to describe framing for art. 396 Unit 11 factoring polynomials A polynomial is an expression that has variables that represent numbers. A number can

More information

Differentiation and Integration

Differentiation and Integration This material is a supplement to Appendix G of Stewart. You should read the appendix, except the last section on complex exponentials, before this material. Differentiation and Integration Suppose we have

More information

Section 4.4. Using the Fundamental Theorem. Difference Equations to Differential Equations

Section 4.4. Using the Fundamental Theorem. Difference Equations to Differential Equations Difference Equations to Differential Equations Section 4.4 Using the Fundamental Theorem As we saw in Section 4.3, using the Fundamental Theorem of Integral Calculus reduces the problem of evaluating a

More information

8 Polynomials Worksheet

8 Polynomials Worksheet 8 Polynomials Worksheet Concepts: Quadratic Functions The Definition of a Quadratic Function Graphs of Quadratic Functions - Parabolas Vertex Absolute Maximum or Absolute Minimum Transforming the Graph

More information

Identify how changes in volume affect costs

Identify how changes in volume affect costs Chapter 18 Identify how changes in volume affect Total variable change in direct proportion to changes in the volume of activity Unit variable cost remains constant Units produced 3 5 Total direct materials

More information

x 2 x 2 cos 1 x x2, lim 1. If x > 0, multiply all three parts by x > 0, we get: x x cos 1 x x, lim lim x cos 1 lim = 5 lim sin 5x

x 2 x 2 cos 1 x x2, lim 1. If x > 0, multiply all three parts by x > 0, we get: x x cos 1 x x, lim lim x cos 1 lim = 5 lim sin 5x Homework 4 3.4,. Show that x x cos x x holds for x 0. Solution: Since cos x, multiply all three parts by x > 0, we get: x x cos x x, and since x 0 x x 0 ( x ) = 0, then by Sandwich theorem, we get: x 0

More information

Polynomials and Quadratics

Polynomials and Quadratics Polynomials and Quadratics Want to be an environmental scientist? Better be ready to get your hands dirty!.1 Controlling the Population Adding and Subtracting Polynomials............703.2 They re Multiplying

More information

Partial Fractions. p(x) q(x)

Partial Fractions. p(x) q(x) Partial Fractions Introduction to Partial Fractions Given a rational function of the form p(x) q(x) where the degree of p(x) is less than the degree of q(x), the method of partial fractions seeks to break

More information

7.1 Graphs of Quadratic Functions in Vertex Form

7.1 Graphs of Quadratic Functions in Vertex Form 7.1 Graphs of Quadratic Functions in Vertex Form Quadratic Function in Vertex Form A quadratic function in vertex form is a function that can be written in the form f (x) = a(x! h) 2 + k where a is called

More information

Using Proportions to Solve Percent Problems I

Using Proportions to Solve Percent Problems I RP7-1 Using Proportions to Solve Percent Problems I Pages 46 48 Standards: 7.RP.A. Goals: Students will write equivalent statements for proportions by keeping track of the part and the whole, and by solving

More information

Unit 7: Radical Functions & Rational Exponents

Unit 7: Radical Functions & Rational Exponents Date Period Unit 7: Radical Functions & Rational Exponents DAY 0 TOPIC Roots and Radical Expressions Multiplying and Dividing Radical Expressions Binomial Radical Expressions Rational Exponents 4 Solving

More information

Guide to SRW Section 1.7: Solving inequalities

Guide to SRW Section 1.7: Solving inequalities Guide to SRW Section 1.7: Solving inequalities When you solve the equation x 2 = 9, the answer is written as two very simple equations: x = 3 (or) x = 3 The diagram of the solution is -6-5 -4-3 -2-1 0

More information

i is a root of the quadratic equation.

i is a root of the quadratic equation. 13 14 SEMESTER EXAMS 1. This question assesses the student s understanding of a quadratic function written in vertex form. y a x h k where the vertex has the coordinates V h, k a) The leading coefficient

More information

Find all of the real numbers x that satisfy the algebraic equation:

Find all of the real numbers x that satisfy the algebraic equation: Appendix C: Factoring Algebraic Expressions Factoring algebraic equations is the reverse of expanding algebraic expressions discussed in Appendix B. Factoring algebraic equations can be a great help when

More information

Week 2 Quiz: Equations and Graphs, Functions, and Systems of Equations

Week 2 Quiz: Equations and Graphs, Functions, and Systems of Equations Week Quiz: Equations and Graphs, Functions, and Systems of Equations SGPE Summer School 014 June 4, 014 Lines: Slopes and Intercepts Question 1: Find the slope, y-intercept, and x-intercept of the following

More information

Basic Properties of Rational Expressions

Basic Properties of Rational Expressions Basic Properties of Rational Expressions A fraction is not defined when the denominator is zero! Examples: Simplify and use Mathematics Writing Style. a) x + 8 b) x 9 x 3 Solution: a) x + 8 (x + 4) x +

More information

Separable First Order Differential Equations

Separable First Order Differential Equations Separable First Order Differential Equations Form of Separable Equations which take the form = gx hy or These are differential equations = gxĥy, where gx is a continuous function of x and hy is a continuously

More information

Solving Rational Equations

Solving Rational Equations Lesson M Lesson : Student Outcomes Students solve rational equations, monitoring for the creation of extraneous solutions. Lesson Notes In the preceding lessons, students learned to add, subtract, multiply,

More information

1 Functions, Graphs and Limits

1 Functions, Graphs and Limits 1 Functions, Graphs and Limits 1.1 The Cartesian Plane In this course we will be dealing a lot with the Cartesian plane (also called the xy-plane), so this section should serve as a review of it and its

More information

3.3 Applications of Linear Functions

3.3 Applications of Linear Functions 3.3 Applications of Linear Functions A function f is a linear function if The graph of a linear function is a line with slope m and y-intercept b. The rate of change of a linear function is the slope m.

More information

1.4. Removing Brackets. Introduction. Prerequisites. Learning Outcomes. Learning Style

1.4. Removing Brackets. Introduction. Prerequisites. Learning Outcomes. Learning Style Removing Brackets 1. Introduction In order to simplify an expression which contains brackets it is often necessary to rewrite the expression in an equivalent form but without any brackets. This process

More information

Math 432 HW 2.5 Solutions

Math 432 HW 2.5 Solutions Math 432 HW 2.5 Solutions Assigned: 1-10, 12, 13, and 14. Selected for Grading: 1 (for five points), 6 (also for five), 9, 12 Solutions: 1. (2y 3 + 2y 2 ) dx + (3y 2 x + 2xy) dy = 0. M/ y = 6y 2 + 4y N/

More information