Chapter 6: Sections 6.1, 6.2.1, Chapter 8: Section 8.1, 8.2 and 8.5. In Business world the study of change important

Size: px
Start display at page:

Download "Chapter 6: Sections 6.1, 6.2.1, Chapter 8: Section 8.1, 8.2 and 8.5. In Business world the study of change important"

Transcription

1 Study Unit 5 : Calculus Chapter 6: Sections 6., 6.., 6.3. Chapter 8: Section 8., 8. and 8.5 In Business world the study of change important Example: change in the sales of a company; change in the value of the rand; change in the value of shares; change in the interest rate etc. Equally important is the rate at which these changes take place. Example: If the sales of a company increased by R ,00, it is important to know whether this change occurred over one year, two years or ten years. Rate of change: changes over time, change in costs for different production quantities in a production process, etc. Change consists of two components: size and direction. Let s look at linear function : y = mx + c. Slope (m) is the change in y which corresponds to change change in y y y of one unit in the value of x: =. change in x x x

2 Slope => indication of rate of change = constant Value of the slope => size of the change Sign of the slope => direction of the change. positive sign an increase; negative sign a decrease. Let s look at a non-linear function, for example a quadratic function The size and direction are not constant, but change continuously. Use the mathematical technique of Differentiation to determine the rate of change.

3 3. Differentiation Slope of a curve = change in y / change in x = rate curve change Use differentiation to get slope at a given point dy/ = f (x) derivative of y with respect to x Pronounce this as dee-y-dee-x. Use rules of differentiation to obtain the derivative Many rules of differentiation look at just namely Basic Rule: If f(x) = x n, then f '(x) or d ( n ) n x = nx for n 0 For example n integer and positive d ( ) 4 4 d x = x x = x n integer and negative d ( ) = 3 = 3 = 4 x x x x

4 4 n is a fraction and positive d = ( x ) x = x = = x x n is a fraction and negative 3 d = ( x ) x = x = = = 3 x ( x ) x 3 3 Note:. The derivative of any constant term say a, that is a term which consists of a number only, is zero: da 0 =, where a is a constant. Example: f(x) = 4 then f '(x) = 0.

5 5 d a f x = a f '(x).. ( ) Example: f(x) = 7x 5 then f '(x) = 7 5x 4 = 35x If f( x) = gx ( ) + hx ( ), then f '( x) = g'( x) + h'( x). Example: f(x) = 7x 5 + x 3 then f '(x) = 35x 4 + 6x Example: f(x) = 4 + 8x then f '(x) = 0 + 6x Steps:. First we need to simplify the given expression so that we can use the basic rule of differentiation.. Secondly we differentiate the new expression using the d n n basic rule x = nx where n 0 of differentiation For example:.. d d 3 ( ) ( ) 3 3 x x x x x = = = = 3 4 d d = = x ( x ) x = x = = x x Discussion class example 9, 0

6 6 Question 9 Differentiate the following expression: 3 x x x Solution We can differentiate the expression using the basic d n n rule x = nx where n 0. Therefore d 3 3 x 4x + 4x+ 5 = ()(3) x (4)() x + (4)() x + 0 = + = 0 0 3x 8x 4x because x = + 3x 8x 4

7 7 Question 0 Differentiate the following expression: x ( x 4 x) Solution d The basic rule of differentiation states that x n n = nx when n 0. To make use of this rule we first need to simplify the expression so that has the same format. We can write x as x when changing from square root form to exponential form. Thus x ( x 4 x) = x ( x 4 x ) = xx 4x xx + + = x 4x x 3 = x 4x x 5 3 = x 4x x

8 8 Now we differentiate the simplified expression, using the d n n basic rule of differentiation namely x = nx when n 0: dy 5 x 4 x x = 3 x 4() x ( ) x 3 5 = 3x 8x x 5 = 3x 8x x 3

9 9 Application : Also called rate of change because slope is rate of change Slope of a tangent line at a given point derivative at that point. Minimum or maximum, vertex, turning point => slope = 0 => dy/ = 0 Discussion class example 4 (c) using differentiation

10 0 Question 4(c) Profit = 30P P 43000? Maximum profit and price The profit function derived in is a quadratic function with a = 30, b= 7800 and c= As a < 0 the shape of the function looks like a sad face and the function thus has a maximum at the function s turning point or vertex (P ; Q). The price P at the turning point, or where the profit is a maximum, is b P = = = = 30 a and thus the maximum profit : Profit = 30(30) (30) =

11 Using differentiation: Now Profit = 30P P The slope of the tangent line at the turning point is zero. Determine the slope of the tangent line by differentiating the profit function: Thus Profit ' 30() P 7800() P 0 = + Slope = 60P Now slope = 0 at the turning point: 60P = 0 60P = 7800 P P 7800 = 60 = 30 Same as previous method.

12 Marginal analysis Marginal revenue = change in revenue change in number of units slope of the revenue function. MR = dtr/dq Marginal cost = change in cost change in number of units slope of the cost function. MC =dtc/dq Discussion class example

13 3 Question What is the marginal cost when Q =0 if the total cost is given by: TC = 0Q 4 30Q + 300Q + 00? Solution The marginal cost function is the differentiated total cost function. Thus by differentiating the total cost function we can determine the marginal cost function. Now if the total cost function is TC = 0Q 4 30Q + 300Q + 00 then the marginal cost function is dtc MC Q Q dq 3 = = Now the marginal cost function s value when Q is equal 0 is MC = 80(0) 3 60(0) = =

14 4. Integration Is the reverse of differentiation y d(y)/ d(y) d( y) o Indefinite integral : different rules Steps: Simplify function before you integrate write it so that you can apply the integration rule for example o ( ax + b) = ax + b o x = x Apply basic integration rule n+ n x x = + c where n n + 0+ ax a = + c = ax + c where a is a constant Discussion class example and 3 Hint : test your answer: differentiate answer, must be equal to function integrated.

15 5 Question Evaluate the following ( + + 3)d x x x Solution To integrate the function we make use of the basic rule of n+ n x integration namely x = + cwhen n. Therefore: n + ( + + 3)d = d + d + 3d x x x x x x x x x x 3x = c x x 3x = c 3 3 x = x 3x c

16 6 Question 3 Determine Solution Q + d Q Q First simplify the function to be integrated: Q+ ( Q+ ) = Q Q = ( Q+ ) Q = Q + Q Integrate the function using rule n+ n x x = + cwhen n n + ( Q + Q ) dq= Q + Q + + Q Q = + + c Q Q = = Q Q c 3 Q = + Q + c 3 c

17 7 Definite integral: area under a given curve between two points a and b: a b n+ n+ n x x x d x= ( x= b) ( x= a) n+ n+ Steps :. Simplify the function. Integrate the function by applying the basic rule of integration 3. Calculate the value of the integrated function at the value a substitute the values a into the integrated function answer 4. Calculate the value of the integrated function at the value b substitute the values b into the integrated function answer 5. Subtract answer from answer Discussion class example 4

18 8 Question 4 Evaluate ( z+ ) dz Solution To determine a definite integral we first integrate the function, n+ n x using the basic rule x = + cwhen n, and then n + substitute the values between which the integral has to be calculated, into the integrated function. Step : Integrate function: 0 ( z+ ) d z = ( z + z ) dz z z ( + ) + 0+ z = ( + z)

19 9 Step : Substitute the values between which the integral has to be calculated: Thus Thus x= a Fx ( ) Fa ( ) Fb ( ) x= b =. z z z ( + z) = + z + z with z= with z= () ( ) = + () + ( ) = = ( ) = + =

Chapter 6: Sections 6.1, 6.2.1, Chapter 8: Section 8.1, 8.2 and 8.5. In Business world the study of change important

Chapter 6: Sections 6.1, 6.2.1, Chapter 8: Section 8.1, 8.2 and 8.5. In Business world the study of change important Study Unit 5 : Calculus Chapter 6: Sections 6., 6.., 6.. Chapter 8: Section 8., 8. and 8.5 In Business world the study of change important Eample: change in the sales of a company; change in the value

More information

REVIEW OF MATHEMATICAL CONCEPTS

REVIEW OF MATHEMATICAL CONCEPTS REVIEW OF MATHEMATICAL CONCEPTS Variables, functions and slopes: A Variable is any entity that can take different values such as: price, output, revenue, cost, etc. In economics we try to 1. Identify the

More information

DCDM BUSINESS SCHOOL FACULTY OF MANAGEMENT ECONOMIC TECHNIQUES 102 LECTURE 4 DIFFERENTIATION

DCDM BUSINESS SCHOOL FACULTY OF MANAGEMENT ECONOMIC TECHNIQUES 102 LECTURE 4 DIFFERENTIATION DCDM BUSINESS SCHOOL FACULTY OF MANAGEMENT ECONOMIC TECHNIUES 1 LECTURE 4 DIFFERENTIATION 1 Differentiation Managers are often concerned with the way that a variable changes over time Prices, for example,

More information

Homework 1. 3x 12, 61.P (x) = 3t 21 Section 1.2

Homework 1. 3x 12, 61.P (x) = 3t 21 Section 1.2 Section 1.1 Homework 1 (34, 36) Determine whether the equation defines y as a function of x. 34. x + h 2 = 1, 36. y = 3x 1 x + 2. (40, 44) Find the following for each function: (a) f(0) (b) f(1) (c) f(

More information

1 Functions and Graphs

1 Functions and Graphs 1 Functions and Graphs 1.1 Functions Cartesian Coordinate System A Cartesian or rectangular coordinate system is formed by the intersection of a horizontal real number line, usually called the x axis,

More information

Optimization Techniques

Optimization Techniques Optimization Techniques Methods for maximizing or minimizing an objective function Examples Consumers maximize utility by purchasing an optimal combination of goods Firms maximize profit by producing and

More information

Analytic Geometry and Calculus I Exam 1 Practice Problems Solutions 2/19/7

Analytic Geometry and Calculus I Exam 1 Practice Problems Solutions 2/19/7 Analytic Geometry and Calculus I Exam 1 Practice Problems Solutions /19/7 Question 1 Write the following as an integer: log 4 (9)+log (5) We have: log 4 (9)+log (5) = ( log 4 (9)) ( log (5)) = 5 ( log

More information

Section 11.3 Rates of Change:

Section 11.3 Rates of Change: Section 11.3 Rates of Change: 1. Consider the following table, which describes a driver making a 168-mile trip from Cleveland to Columbus, Ohio in 3 hours. t Time (in hours) 0 0.5 1 1.5 2 2.5 3 f(t) Distance

More information

Integration - Past Edexcel Exam Questions

Integration - Past Edexcel Exam Questions Integration - Past Edexcel Exam Questions 1. (a) Given that y = 5x 2 + 7x + 3, find i. - ii. - (b) ( 1 + 3 ) x 1 x dx. [4] 2. Question 2b - January 2005 2. The gradient of the curve C is given by The point

More information

Functions. A function is a rule that gives exactly one output number to each input number.

Functions. A function is a rule that gives exactly one output number to each input number. Functions A function is a rule that gives exactly one output number to each input number. Why it is important to us? The set of all input numbers to which the rule applies is called the domain of the function.

More information

MS 2001: Test 1 B Solutions

MS 2001: Test 1 B Solutions MS 2001: Test 1 B Solutions Name: Student Number: Answer all questions. Marks may be lost if necessary work is not clearly shown. Remarks by me in italics and would not be required in a test - J.P. Question

More information

Math 110 Final Exam General Review. Edward Yu

Math 110 Final Exam General Review. Edward Yu Math 110 Final Exam General Review Edward Yu Da Game Plan Solving Limits Regular limits Indeterminate Form Approach Infinities One sided limits/discontinuity Derivatives Power Rule Product/Quotient Rule

More information

Mathematics for Economics ECON MA/MSSc in Economics-2017/2018. Dr. W. M. Semasinghe Senior Lecturer Department of Economics

Mathematics for Economics ECON MA/MSSc in Economics-2017/2018. Dr. W. M. Semasinghe Senior Lecturer Department of Economics Mathematics for Economics ECON 53035 MA/MSSc in Economics-2017/2018 Dr. W. M. Semasinghe Senior Lecturer Department of Economics MATHEMATICS AND STATISTICS LERNING OUTCOMES: By the end of this course unit

More information

5.1 - Polynomials. Ex: Let k(x) = x 2 +2x+1. Find (and completely simplify) the following: (a) k(1) (b) k( 2) (c) k(a)

5.1 - Polynomials. Ex: Let k(x) = x 2 +2x+1. Find (and completely simplify) the following: (a) k(1) (b) k( 2) (c) k(a) c Kathryn Bollinger, March 15, 2017 1 5.1 - Polynomials Def: A function is a rule (process) that assigns to each element in the domain (the set of independent variables, x) ONE AND ONLY ONE element in

More information

OBJECTIVE Find limits of functions, if they exist, using numerical or graphical methods.

OBJECTIVE 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 information

Quadratic function and equations Quadratic function/equations, supply, demand, market equilibrium

Quadratic function and equations Quadratic function/equations, supply, demand, market equilibrium Exercises 8 Quadratic function and equations Quadratic function/equations, supply, demand, market equilibrium Objectives - know and understand the relation between a quadratic function and a quadratic

More information

Subtract 16 from both sides. Divide both sides by 9. b. Will the swing touch the ground? Explain how you know.

Subtract 16 from both sides. Divide both sides by 9. b. Will the swing touch the ground? Explain how you know. REVIEW EXAMPLES 1) Solve 9x + 16 = 0 for x. 9x + 16 = 0 9x = 16 Original equation. Subtract 16 from both sides. 16 x 9 Divide both sides by 9. 16 x Take the square root of both sides. 9 4 x i 3 Evaluate.

More information

REVIEW OF MATHEMATICAL CONCEPTS

REVIEW OF MATHEMATICAL CONCEPTS REVIEW OF MATHEMATICAL CONCEPTS 1 Variables, functions and slopes A variable is any entity that can take different values such as: price, output, revenue, cost, etc. In economics we try to 1. Identify

More information

Review for Final Review

Review for Final Review Topics Review for Final Review 1. Functions and equations and graphing: linear, absolute value, quadratic, polynomials, rational (first 1/3 of semester) 2. Simple Interest, compounded interest, and continuously

More information

Differentiation. 1. What is a Derivative? CHAPTER 5

Differentiation. 1. What is a Derivative? CHAPTER 5 CHAPTER 5 Differentiation Differentiation is a technique that enables us to find out how a function changes when its argument changes It is an essential tool in economics If you have done A-level maths,

More information

Chapter 4 Differentiation

Chapter 4 Differentiation Chapter 4 Differentiation 08 Section 4. The derivative of a function Practice Problems (a) (b) (c) 3 8 3 ( ) 4 3 5 4 ( ) 5 3 3 0 0 49 ( ) 50 Using a calculator, the values of the cube function, correct

More information

Section 11.7 The Chain Rule

Section 11.7 The Chain Rule Section.7 The Chain Rule Composition of Functions There is another way of combining two functions to obtain a new function. For example, suppose that y = fu) = u and u = gx) = x 2 +. Since y is a function

More information

SCHOOL OF DISTANCE EDUCATION

SCHOOL OF DISTANCE EDUCATION SCHOOL OF DISTANCE EDUCATION CCSS UG PROGRAMME MATHEMATICS (OPEN COURSE) (For students not having Mathematics as Core Course) MM5D03: MATHEMATICS FOR SOCIAL SCIENCES FIFTH SEMESTER STUDY NOTES Prepared

More information

SECTION 5.1: Polynomials

SECTION 5.1: Polynomials 1 SECTION 5.1: Polynomials Functions Definitions: Function, Independent Variable, Dependent Variable, Domain, and Range A function is a rule that assigns to each input value x exactly output value y =

More information

EXAMINATION #4 ANSWER KEY. I. Multiple choice (1)a. (2)e. (3)b. (4)b. (5)d. (6)c. (7)b. (8)b. (9)c. (10)b. (11)b.

EXAMINATION #4 ANSWER KEY. I. Multiple choice (1)a. (2)e. (3)b. (4)b. (5)d. (6)c. (7)b. (8)b. (9)c. (10)b. (11)b. William M. Boal Version A EXAMINATION #4 ANSWER KEY I. Multiple choice (1)a. ()e. (3)b. (4)b. (5)d. (6)c. (7)b. (8)b. (9)c. (10)b. (11)b. II. Short answer (1) a. 4 units of food b. 1/4 units of clothing

More information

EC611--Managerial Economics

EC611--Managerial Economics EC611--Managerial Economics Optimization Techniques and New Management Tools Dr. Savvas C Savvides, European University Cyprus Models and Data Model a framework based on simplifying assumptions it helps

More information

Topic 6: Optimization I. Maximisation and Minimisation Jacques (4th Edition): Chapter 4.6 & 4.7

Topic 6: Optimization I. Maximisation and Minimisation Jacques (4th Edition): Chapter 4.6 & 4.7 Topic 6: Optimization I Maximisation and Minimisation Jacques (4th Edition): Chapter 4.6 & 4.7 1 For a straight line Y=a+bX Y= f (X) = a + bx First Derivative dy/dx = f = b constant slope b Second Derivative

More information

y = F (x) = x n + c dy/dx = F`(x) = f(x) = n x n-1 Given the derivative f(x), what is F(x)? (Integral, Anti-derivative or the Primitive function).

y = F (x) = x n + c dy/dx = F`(x) = f(x) = n x n-1 Given the derivative f(x), what is F(x)? (Integral, Anti-derivative or the Primitive function). Integration Course Manual Indefinite Integration 7.-7. Definite Integration 7.-7.4 Jacques ( rd Edition) Indefinite Integration 6. Definite Integration 6. y F (x) x n + c dy/dx F`(x) f(x) n x n- Given

More information

Second Order Derivatives. Background to Topic 6 Maximisation and Minimisation

Second Order Derivatives. Background to Topic 6 Maximisation and Minimisation Second Order Derivatives Course Manual Background to Topic 6 Maximisation and Minimisation Jacques (4 th Edition): Chapter 4.6 & 4.7 Y Y=a+bX a X Y= f (X) = a + bx First Derivative dy/dx = f = b constant

More information

Math 142 Week-in-Review #4 (Sections , 4.1, and 4.2)

Math 142 Week-in-Review #4 (Sections , 4.1, and 4.2) Math 142 WIR, copyright Angie Allen, Fall 2018 1 Math 142 Week-in-Review #4 (Sections 3.1-3.3, 4.1, and 4.2) Note: This collection of questions is intended to be a brief overview of the exam material (with

More information

1) 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.

1) 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 information

MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.

MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Calculus I - Homework Chapter 2 Name MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Determine whether the graph is the graph of a function. 1) 1)

More information

Final Exam Study Guide

Final Exam Study Guide Final Exam Study Guide Final Exam Coverage: Sections 10.1-10.2, 10.4-10.5, 10.7, 11.2-11.4, 12.1-12.6, 13.1-13.2, 13.4-13.5, and 14.1 Sections/topics NOT on the exam: Sections 10.3 (Continuity, it definition

More information

Math 116: Business Calculus Chapter 4 - Calculating Derivatives

Math 116: Business Calculus Chapter 4 - Calculating Derivatives Math 116: Business Calculus Chapter 4 - Calculating Derivatives Instructor: Colin Clark Spring 2017 Exam 2 - Thursday March 9. 4.1 Techniques for Finding Derivatives. 4.2 Derivatives of Products and Quotients.

More information

MATH150-E01 Test #2 Summer 2016 Show all work. Name 1. Find an equation in slope-intercept form for the line through (4, 2) and (1, 3).

MATH150-E01 Test #2 Summer 2016 Show all work. Name 1. Find an equation in slope-intercept form for the line through (4, 2) and (1, 3). 1. Find an equation in slope-intercept form for the line through (4, 2) and (1, 3). 2. Let the supply and demand functions for sugar be given by p = S(q) = 1.4q 0.6 and p = D(q) = 2q + 3.2 where p is the

More information

EC5555 Economics Masters Refresher Course in Mathematics September 2013

EC5555 Economics Masters Refresher Course in Mathematics September 2013 EC5555 Economics Masters Refresher Course in Mathematics September 013 Lecture 3 Differentiation Francesco Feri Rationale for Differentiation Much of economics is concerned with optimisation (maximise

More information

Math 106 Calculus 1 Topics for first exam

Math 106 Calculus 1 Topics for first exam Chapter 2: Limits and Continuity Rates of change and its: Math 06 Calculus Topics for first exam Limit of a function f at a point a = the value the function should take at the point = the value that the

More information

Final Exam Review. MATH Intuitive Calculus Fall 2013 Circle lab day: Mon / Fri. Name:. Show all your work.

Final Exam Review. MATH Intuitive Calculus Fall 2013 Circle lab day: Mon / Fri. Name:. Show all your work. MATH 11012 Intuitive Calculus Fall 2013 Circle lab day: Mon / Fri Dr. Kracht Name:. 1. Consider the function f depicted below. Final Exam Review Show all your work. y 1 1 x (a) Find each of the following

More information

STUDY MATERIALS. (The content of the study material is the same as that of Chapter I of Mathematics for Economic Analysis II of 2011 Admn.

STUDY MATERIALS. (The content of the study material is the same as that of Chapter I of Mathematics for Economic Analysis II of 2011 Admn. STUDY MATERIALS MATHEMATICAL TOOLS FOR ECONOMICS III (The content of the study material is the same as that of Chapter I of Mathematics for Economic Analysis II of 2011 Admn.) & MATHEMATICAL TOOLS FOR

More information

Chapter 8: Taylor s theorem and L Hospital s rule

Chapter 8: Taylor s theorem and L Hospital s rule Chapter 8: Taylor s theorem and L Hospital s rule Theorem: [Inverse Mapping Theorem] Suppose that a < b and f : [a, b] R. Given that f (x) > 0 for all x (a, b) then f 1 is differentiable on (f(a), f(b))

More information

2. Which of the following is the ECONOMISTS inverse of the function y = 9/x 2 (i.e. find x as a function of y, x = f(y))

2. Which of the following is the ECONOMISTS inverse of the function y = 9/x 2 (i.e. find x as a function of y, x = f(y)) Anwers for Review Quiz #1. Material Covered. Klein 1, 2; Schaums 1, 2 1. Solve the following system of equations for x, y and z: x + y = 2 2x + 2y + z = 5 7x + y + z = 9 Answers: x = 1, y = 1, z = 1. 2.

More information

Final Exam Review Packet

Final Exam Review Packet 1 Exam 1 Material Sections A.1, A.2 and A.6 were review material. There will not be specific questions focused on this material but you should know how to: Simplify functions with exponents. Factor quadratics

More information

Final Exam Review Packet

Final Exam Review Packet 1 Exam 1 Material Sections A.1, A.2 and A.6 were review material. There will not be specific questions focused on this material but you should know how to: Simplify functions with exponents. Factor quadratics

More information

Given the table of values, determine the equation

Given the table of values, determine the equation 3.1 Properties of Quadratic Functions Recall: Standard Form f(x) = ax 2 + bx + c Factored Form f(x) = a(x r)(x s) Vertex Form f(x) = a(x h) 2 + k Given the table of values, determine the equation x y 1

More information

Graphs of Polynomial Functions

Graphs of Polynomial Functions Graphs of Polynomial Functions By: OpenStaxCollege The revenue in millions of dollars for a fictional cable company from 2006 through 2013 is shown in [link]. Year 2006 2007 2008 2009 2010 2011 2012 2013

More information

Sect 2.4 Linear Functions

Sect 2.4 Linear Functions 36 Sect 2.4 Linear Functions Objective 1: Graphing Linear Functions Definition A linear function is a function in the form y = f(x) = mx + b where m and b are real numbers. If m 0, then the domain and

More information

ANSWERS, Homework Problems, Fall 2014: Lectures Now You Try It, Supplemental problems in written homework, Even Answers. 24x + 72 (x 2 6x + 4) 4

ANSWERS, Homework Problems, Fall 2014: Lectures Now You Try It, Supplemental problems in written homework, Even Answers. 24x + 72 (x 2 6x + 4) 4 ANSWERS, Homework Problems, Fall 014: Lectures 19 35 Now You Try It, Supplemental problems in written homework, Even Answers Lecture 19 1. d [ 4 ] dx x 6x + 4) 3 = 4x + 7 x 6x + 4) 4. a) P 0) = 800 b)

More information

b. /(x) = -x^-;r^ r(x)= -SX"" f'(^^^ ^ -^x"- "3.^ ^ Mth 241 Test 2 Extra Practice: ARC and IRC Name:

b. /(x) = -x^-;r^ r(x)= -SX f'(^^^ ^ -^x- 3.^ ^ Mth 241 Test 2 Extra Practice: ARC and IRC Name: Mth 241 Test 2 Extra Practice: ARC and IRC Name: This is not a practice exam. This is a supplement to the activities v\/e do in class and the homework. 1. Use the graph to answer the following questions.

More information

Study Guide - Part 2

Study Guide - Part 2 Math 116 Spring 2015 Study Guide - Part 2 1. Which of the following describes the derivative function f (x) of a quadratic function f(x)? (A) Cubic (B) Quadratic (C) Linear (D) Constant 2. Find the derivative

More information

Mathematical Economics: Lecture 2

Mathematical Economics: Lecture 2 Mathematical Economics: Lecture 2 Yu Ren WISE, Xiamen University September 25, 2012 Outline 1 Number Line The number line, origin (Figure 2.1 Page 11) Number Line Interval (a, b) = {x R 1 : a < x < b}

More information

What is on today. 1 Linear approximation. MA 123 (Calculus I) Lecture 17: November 2, 2017 Section A2. Professor Jennifer Balakrishnan,

What is on today. 1 Linear approximation. MA 123 (Calculus I) Lecture 17: November 2, 2017 Section A2. Professor Jennifer Balakrishnan, Professor Jennifer Balakrishnan, jbala@bu.edu What is on today 1 Linear approximation 1 1.1 Linear approximation and concavity....................... 2 1.2 Change in y....................................

More information

Math 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) 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 information

Math Final Solutions - Spring Jaimos F Skriletz 1

Math Final Solutions - Spring Jaimos F Skriletz 1 Math 160 - Final Solutions - Spring 2011 - Jaimos F Skriletz 1 Answer each of the following questions to the best of your ability. To receive full credit, answers must be supported by a sufficient amount

More information

MA 137 Calculus 1 with Life Science Applications Linear Approximations (Section 4.8)

MA 137 Calculus 1 with Life Science Applications Linear Approximations (Section 4.8) MA 137 Calculus 1 with Life Science Applications Linear Approximations (Section 4.8) Alberto Corso alberto.corso@uky.edu Department of Mathematics University of Kentucky October 28, 2015 1/12 Tangent Line

More information

MATH 236 ELAC FALL 2017 CA 9 NAME: SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question.

MATH 236 ELAC FALL 2017 CA 9 NAME: SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. MATH 236 ELAC FALL 207 CA 9 NAME: SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. ) 27 p 3 27 p 3 ) 2) If 9 t 3 4t 9-2t = 3, find t. 2) Solve the equation.

More information

MA Lesson 12 Notes Section 3.4 of Calculus part of textbook

MA Lesson 12 Notes Section 3.4 of Calculus part of textbook MA 15910 Lesson 1 Notes Section 3.4 of Calculus part of textbook Tangent Line to a curve: To understand the tangent line, we must first discuss a secant line. A secant line will intersect a curve at more

More information

Section 6-1 Antiderivatives and Indefinite Integrals

Section 6-1 Antiderivatives and Indefinite Integrals Name Date Class Section 6-1 Antiderivatives and Indefinite Integrals Goal: To find antiderivatives and indefinite integrals of functions using the formulas and properties Theorem 1 Antiderivatives If the

More information

The Graph of a Quadratic Function. Quadratic Functions & Models. The Graph of a Quadratic Function. The Graph of a Quadratic Function

The Graph of a Quadratic Function. Quadratic Functions & Models. The Graph of a Quadratic Function. The Graph of a Quadratic Function 8/1/015 The Graph of a Quadratic Function Quadratic Functions & Models Precalculus.1 The Graph of a Quadratic Function The Graph of a Quadratic Function All parabolas are symmetric with respect to a line

More information

Essential Mathematics for Economics and Business, 4 th Edition CHAPTER 6 : WHAT IS THE DIFFERENTIATION.

Essential Mathematics for Economics and Business, 4 th Edition CHAPTER 6 : WHAT IS THE DIFFERENTIATION. Essential Mathematics for Economics and Business, 4 th Edition CHAPTER 6 : WHAT IS THE DIFFERENTIATION. John Wiley and Sons 13 Slopes/rates of change Recall linear functions For linear functions slope

More information

Lecture 6: Sections 2.2 and 2.3 Polynomial Functions, Quadratic Models

Lecture 6: Sections 2.2 and 2.3 Polynomial Functions, Quadratic Models L6-1 Lecture 6: Sections 2.2 and 2.3 Polynomial Functions, Quadratic Models Polynomial Functions Def. A polynomial function of degree n is a function of the form f(x) = a n x n + a n 1 x n 1 +... + a 1

More information

A Library of Functions

A Library of Functions LibraryofFunctions.nb 1 A Library of Functions Any study of calculus must start with the study of functions. Functions are fundamental to mathematics. In its everyday use the word function conveys to us

More information

2.8 Linear Approximations and Differentials

2.8 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 information

MA 181 Lecture Chapter 7 College Algebra and Calculus by Larson/Hodgkins Limits and Derivatives

MA 181 Lecture Chapter 7 College Algebra and Calculus by Larson/Hodgkins Limits and Derivatives 7.5) Rates of Change: Velocity and Marginals MA 181 Lecture Chapter 7 College Algebra and Calculus by Larson/Hodgkins Limits and Derivatives Previously we learned two primary applications of derivatives.

More information

Final Exam A Name. 20 i C) Solve the equation by factoring. 4) x2 = x + 30 A) {-5, 6} B) {5, 6} C) {1, 30} D) {-5, -6} -9 ± i 3 14

Final Exam A Name. 20 i C) Solve the equation by factoring. 4) x2 = x + 30 A) {-5, 6} B) {5, 6} C) {1, 30} D) {-5, -6} -9 ± i 3 14 Final Exam A Name First, write the value(s) that make the denominator(s) zero. Then solve the equation. 1 1) x + 3 + 5 x - 3 = 30 (x + 3)(x - 3) 1) A) x -3, 3; B) x -3, 3; {4} C) No restrictions; {3} D)

More information

Part A: Short Answer Questions

Part A: Short Answer Questions Math 111 Practice Exam Your Grade: Fall 2015 Total Marks: 160 Instructor: Telyn Kusalik Time: 180 minutes Name: Part A: Short Answer Questions Answer each question in the blank provided. 1. If a city grows

More information

A) (-1, -1, -2) B) No solution C) Infinite solutions D) (1, 1, 2) A) (6, 5, -3) B) No solution C) Infinite solutions D) (1, -3, -7)

A) (-1, -1, -2) B) No solution C) Infinite solutions D) (1, 1, 2) A) (6, 5, -3) B) No solution C) Infinite solutions D) (1, -3, -7) Algebra st Semester Final Exam Review Multiple Choice. Write an equation that models the data displayed in the Interest-Free Loan graph that is provided. y = x + 80 y = -0x + 800 C) y = 0x 00 y = 0x +

More information

EconS 301. Math Review. Math Concepts

EconS 301. Math Review. Math Concepts EconS 301 Math Review Math Concepts Functions: Functions describe the relationship between input variables and outputs y f x where x is some input and y is some output. Example: x could number of Bananas

More information

Make graph of g by adding c to the y-values. on the graph of f by c. multiplying the y-values. even-degree polynomial. graph goes up on both sides

Make graph of g by adding c to the y-values. on the graph of f by c. multiplying the y-values. even-degree polynomial. graph goes up on both sides Reference 1: Transformations of Graphs an En Behavior of Polynomial Graphs Transformations of graphs aitive constant constant on the outsie g(x) = + c Make graph of g by aing c to the y-values on the graph

More information

Department 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 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 information

Integrated Math 10 Quadratic Functions Unit Test January 2013

Integrated Math 10 Quadratic Functions Unit Test January 2013 1. Answer the following question, which deal with general properties of quadratics. a. Solve the quadratic equation 0 x 9 (K) b. Fully factor the quadratic expression 3x 15x 18 (K) c. Determine the equation

More information

3.1 Derivative Formulas for Powers and Polynomials

3.1 Derivative Formulas for Powers and Polynomials 3.1 Derivative Formulas for Powers and Polynomials First, recall that a derivative is a function. We worked very hard in 2.2 to interpret the derivative of a function visually. We made the link, in Ex.

More information

Increasing or Decreasing Nature of a Function

Increasing or Decreasing Nature of a Function Öğr. Gör. Volkan ÖĞER FBA 1021 Calculus 1/ 46 Increasing or Decreasing Nature of a Function Examining the graphical behavior of functions is a basic part of mathematics and has applications to many areas

More information

MATH The Derivative as a Function - Section 3.2. The derivative of f is the function. f x h f x. f x lim

MATH 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 information

1.4 Linear Functions of Several Variables

1.4 Linear Functions of Several Variables .4 Linear Functions of Several Variables Question : What is a linear function of several independent variables? Question : What do the coefficients of the variables tell us? Question : How do you find

More information

DEFINITION OF A DERIVATIVE

DEFINITION OF A DERIVATIVE DEFINITION OF A DERIVATIVE Section 2.1 Calculus AP/Dual, Revised 2017 viet.dang@umbleisd.net 2.1: Definition of a Derivative 1 DEFINITION A. Te derivative of a function allows you to find te SLOPE OF THE

More information

c) i) f(x) 3[2(x 4)] 6

c) i) f(x) 3[2(x 4)] 6 Answers CHAPTER 1 Prerequisite Skills, pages 1. a) 7 5 11 d) 5 e) 8x 7 f) 1x 7. a) 1 10 6 d) 0 e) 4x 18x f) 18x 9x 1. a) m, b m _ 1, b _ m 5, b 7 d) m 5, b 11 e) m _ 1, b 1 4. a) y x 5 y 4x 4 y 4x 1 d)

More information

Study Unit 2 : Linear functions Chapter 2 : Sections and 2.6

Study Unit 2 : Linear functions Chapter 2 : Sections and 2.6 1 Study Unit 2 : Linear functions Chapter 2 : Sections 2.1 2.4 and 2.6 1. Function Humans = relationships Function = mathematical form of a relationship Temperature and number of ice cream sold Independent

More information

MAT1300 Final Review. Pieter Hofstra. December 4, 2009

MAT1300 Final Review. Pieter Hofstra. December 4, 2009 December 4, 2009 Sections from the book to study (8th Edition) Chapter 0: 0.1: Real line and Order 0.2: Absolute Value and Distance 0.3: Exponents and Radicals 0.4: Factoring Polynomials (you may omit

More information

Unit #4 : Interpreting Derivatives, Local Linearity, Marginal Rates

Unit #4 : Interpreting Derivatives, Local Linearity, Marginal Rates Unit #4 : Interpreting Derivatives, Local Linearity, Marginal Rates Goals: Develop natural language interpretations of the derivative Create and use linearization/tangent line formulae Describe marginal

More information

Purdue University Study Guide for MA for students who plan to obtain credit in MA by examination.

Purdue University Study Guide for MA for students who plan to obtain credit in MA by examination. Purdue University Study Guide for MA 224 for students who plan to obtain credit in MA 224 by examination. Textbook: Applied Calculus For Business, Economics, and the Social and Life Sciences, Expanded

More information

Business and Life Calculus

Business and Life Calculus Business and Life Calculus George Voutsadakis Mathematics and Computer Science Lake Superior State University LSSU Math 2 George Voutsadakis (LSSU) Calculus For Business and Life Sciences Fall 203 / 55

More information

Final Exam C Name i D) 2. Solve the equation by factoring. 4) x2 = x + 72 A) {1, 72} B) {-8, 9} C) {-8, -9} D) {8, 9} 9 ± i

Final Exam C Name i D) 2. Solve the equation by factoring. 4) x2 = x + 72 A) {1, 72} B) {-8, 9} C) {-8, -9} D) {8, 9} 9 ± i Final Exam C Name First, write the value(s) that make the denominator(s) zero. Then solve the equation. 7 ) x + + 3 x - = 6 (x + )(x - ) ) A) No restrictions; {} B) x -, ; C) x -; {} D) x -, ; {2} Add

More information

MA 151, Applied Calculus, Fall 2017, Final Exam Preview Identify each of the graphs below as one of the following functions:

MA 151, Applied Calculus, Fall 2017, Final Exam Preview Identify each of the graphs below as one of the following functions: MA 5, Applie Calculus, Fall 207, Final Exam Preview Basic Functions. Ientify each of the graphs below as one of the following functions: x 3 4x 2 + x x 2 e x x 3 ln(x) (/2) x 2 x 4 + 4x 2 + 0 5x + 0 5x

More information

Math 106 Answers to Exam 1a Fall 2015

Math 106 Answers to Exam 1a Fall 2015 Math 06 Answers to Exam a Fall 05.. Find the derivative of the following functions. Do not simplify your answers. (a) f(x) = ex cos x x + (b) g(z) = [ sin(z ) + e z] 5 Using the quotient rule on f(x) and

More information

MAT137 Calculus! Lecture 6

MAT137 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 information

Math 1120 Calculus, section 2 Test 1

Math 1120 Calculus, section 2 Test 1 February 6, 203 Name The problems count as marked. The total number of points available is 49. Throughout this test, show your work. Using a calculator to circumvent ideas discussed in class will generally

More information

x C) y = - A) $20000; 14 years B) $28,000; 14 years C) $28,000; 28 years D) $30,000; 15 years

x C) y = - A) $20000; 14 years B) $28,000; 14 years C) $28,000; 28 years D) $30,000; 15 years Dr. Lee - Math 35 - Calculus for Business - Review of 3 - Show Complete Work for Each Problem MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Find

More information

Math 1071 Final Review Sheet The following are some review questions to help you study. They do not

Math 1071 Final Review Sheet The following are some review questions to help you study. They do not Math 1071 Final Review Sheet The following are some review questions to help you study. They do not They do The exam represent the entirety of what you could be expected to know on the exam; reflect distribution

More information

3. Find the slope of the tangent line to the curve given by 3x y e x+y = 1 + ln x at (1, 1).

3. Find the slope of the tangent line to the curve given by 3x y e x+y = 1 + ln x at (1, 1). 1. Find the derivative of each of the following: (a) f(x) = 3 2x 1 (b) f(x) = log 4 (x 2 x) 2. Find the slope of the tangent line to f(x) = ln 2 ln x at x = e. 3. Find the slope of the tangent line to

More information

Math 142 (Summer 2018) Business Calculus 5.8 Notes

Math 142 (Summer 2018) Business Calculus 5.8 Notes Math 142 (Summer 2018) Business Calculus 5.8 Notes Implicit Differentiation and Related Rates Why? We have learned how to take derivatives of functions, and we have seen many applications of this. However

More information

32. Use a graphing utility to find the equation of the line of best fit. Write the equation of the line rounded to two decimal places, if necessary.

32. Use a graphing utility to find the equation of the line of best fit. Write the equation of the line rounded to two decimal places, if necessary. Pre-Calculus A Final Review Part 2 Calculator Name 31. The price p and the quantity x sold of a certain product obey the demand equation: p = x + 80 where r = xp. What is the revenue to the nearest dollar

More information

Math 0320 Final Exam Review

Math 0320 Final Exam Review Math 0320 Final Exam Review SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. Factor out the GCF using the Distributive Property. 1) 6x 3 + 9x 1) Objective:

More information

Question 1. (8 points) The following diagram shows the graphs of eight equations.

Question 1. (8 points) The following diagram shows the graphs of eight equations. MAC 2233/-6 Business Calculus, Spring 2 Final Eam Name: Date: 5/3/2 Time: :am-2:nn Section: Show ALL steps. One hundred points equal % Question. (8 points) The following diagram shows the graphs of eight

More information

Note: The zero function f(x) = 0 is a polynomial function. It has no degree and no leading coefficient. Sep 15 2:51 PM

Note: The zero function f(x) = 0 is a polynomial function. It has no degree and no leading coefficient. Sep 15 2:51 PM 2.1 Linear and Quadratic Name: Functions and Modeling Objective: Students will be able to recognize and graph linear and quadratic functions, and use these functions to model situations and solve problems.

More information

Chapter 1 Linear Equations and Graphs

Chapter 1 Linear Equations and Graphs Chapter 1 Linear Equations and Graphs Section R Linear Equations and Inequalities Important Terms, Symbols, Concepts 1.1. Linear Equations and Inequalities A first degree, or linear, equation in one variable

More information

The marks achieved in this section account for 50% of your final exam result.

The marks achieved in this section account for 50% of your final exam result. Section D The marks achieved in this section account for 50% of your final exam result. Full algebraic working must be clearly shown. Instructions: This section has two parts. Answer ALL questions in part

More information

Lecture 2. Derivative. 1 / 26

Lecture 2. Derivative. 1 / 26 Lecture 2. Derivative. 1 / 26 Basic Concepts Suppose we wish to nd the rate at which a given function f (x) is changing with respect to x when x = c. The simplest idea is to nd the average rate of change

More information

INTERNET MAT 117 Review Problems. (1) Let us consider the circle with equation. (b) Find the center and the radius of the circle given above.

INTERNET MAT 117 Review Problems. (1) Let us consider the circle with equation. (b) Find the center and the radius of the circle given above. INTERNET MAT 117 Review Problems (1) Let us consider the circle with equation x 2 + y 2 + 2x + 3y + 3 4 = 0. (a) Find the standard form of the equation of the circle given above. (b) Find the center and

More information

ANSWERS, Homework Problems, Spring 2014 Now You Try It, Supplemental problems in written homework, Even Answers R.6 8) 27, 30) 25

ANSWERS, Homework Problems, Spring 2014 Now You Try It, Supplemental problems in written homework, Even Answers R.6 8) 27, 30) 25 ANSWERS, Homework Problems, Spring 2014, Supplemental problems in written homework, Even Answers Review Assignment: Precalculus Even Answers to Sections R1 R7 R.1 24) 4a 2 16ab + 16b 2 R.2 24) Prime 5x

More information

Higher Portfolio Quadratics and Polynomials

Higher Portfolio Quadratics and Polynomials Higher Portfolio Quadratics and Polynomials Higher 5. Quadratics and Polynomials Section A - Revision Section This section will help you revise previous learning which is required in this topic R1 I have

More information