Limits and Derivatives

Size: px
Start display at page:

Download "Limits and Derivatives"

Transcription

1 New Progress in Senior Matematics Module Book Etended Part Solution Guide Limits and Derivatives pp. p..... pp. p. a b c Undeined 9 d. a [ g ] g b [ g ] c g g g g g 9. a [ 9] 9 p. p. p. a b c d e g g b c. a Q p. Hong Kong Educational Publising Co.

2 Limits and Derivatives. a b Q c Q b 9. a b p. p.. a g u p.9 u u y b g u u y g g c g u e y e u e g g 9 Hong Kong Educational Publising Co. g g. a [ ][ ] b [ ][ ]. a [ ] b [ ] p. p.

3 New Progress in Senior Matematics Module Book Etended Part Solution Guide.9 a. a b e p. e e e b ln ln[ ] ln ln or ln p. [ ] ' Te slope o te tangent to te curve y at pp. Eample.T p. a [ g ] g b [ g ] g c g g g g Eample.T p. a b Eample.T p. Q Eample.T p. Eample.T p. a 9 Hong Kong Educational Publising Co.

4 Limits and Derivatives b b [ ] 9 Eample.T p. a g u lnu y ln ln g g b g u u y g [ g ] c g u u e y e e g g Eample.T p.9 [ ][ ] Eample.T p. a [ ] Eample.9T p. a e e e e b ln ln ln[ ] ln Eample.T p. a [ ] ' b Te slope o te tangent to te curve y at 9 Hong Kong Educational Publising Co.

5 New Progress in Senior Matematics Module Book Etended Part Solution Guide. pp a e e b lne c d p.9.. a I c, ten c Tereore is, c. b is undeined a [ g ]. pp. p. g b [ g ] g p c g g Hong Kong Educational Publising Co.

6 Limits and Derivatives g d g g g. g. g. g [ ] e. g ln. g u u y. g u u y [ g ]. g u lnu y ln [ g ] g g ln[ g ] [ g ].. [ ]. [ ][ ] 9. [ ]. [ ][ ]. Q 9. g u e y e u e g. g. Q 9 9 Hong Kong Educational Publising Co.

7 New Progress in Senior Matematics Module Book Etended Part Solution Guide. Q 9. Q. a g g p.... b c e e e e e g g g ln g ln ln ln e e e.. [ ] [ ].. e e e e ln ln ln[ ] ln. ln ln[ ] ln ln Hong Kong Educational Publising Co.

8 Hong Kong Educational Publising Co. Limits and Derivatives 9. 9 ] [.. ] [ p. p..

9 Hong Kong Educational Publising Co. New Progress in Senior Matematics Module Book Etended Part Solution Guide.... a b c Te slope o te tangent at. a ] [ b c Te slope o te tangent at. a ] [ b c Te slope o te tangent at 9. a b

10 Hong Kong Educational Publising Co. Limits and Derivatives c Te slope o te tangent at p. 9. a b rom a c. a b rom a. a b c Yes Since or any and, or any and.. a ] [ ] [

11 New Progress in Senior Matematics Module Book Etended Part Solution Guide b Yes, is a constant. Since y is a straigt line, te slopes o any points on te grap are te same. For m c, slope o te straigt line m. Q.. [ ][ ]... Q pp.9 9 p.9. Q [ ] e e e e e e e e e. ln ln ln ln Hong Kong Educational Publising Co.

12 Limits and Derivatives. ln[ ]. a. a ln ln[ ] ln [ ] b c Te slope o te tangent at b.... p.9 c Te slope o te tangent at 9. [ ] 9 Hong Kong Educational Publising Co.

13 Hong Kong Educational Publising Co. New Progress in Senior Matematics Module Book Etended Part Solution Guide. ] [. 9 9] [. ] ln [ ln ] ln[ ] [ ln. a b c Te slope o te tangent at. a b d d ] [. a ] [ b d d

14 Hong Kong Educational Publising Co. Limits and Derivatives. a b d d roma c Slope o tangent at Etended Question p.9. a b i ii Open-ended Question p.9. Since, must be a actor o. b a Let b a b a, a b and c

MATH CALCULUS I 2.1: Derivatives and Rates of Change

MATH CALCULUS I 2.1: Derivatives and Rates of Change MATH 12002 - CALCULUS I 2.1: Derivatives and Rates of Cange Professor Donald L. Wite Department of Matematical Sciences Kent State University D.L. Wite (Kent State University) 1 / 1 Introduction Our main

More information

Calculus I Homework: The Derivative as a Function Page 1

Calculus I Homework: The Derivative as a Function Page 1 Calculus I Homework: Te Derivative as a Function Page 1 Example (2.9.16) Make a careful sketc of te grap of f(x) = sin x and below it sketc te grap of f (x). Try to guess te formula of f (x) from its grap.

More information

Tangent Lines-1. Tangent Lines

Tangent Lines-1. Tangent Lines Tangent Lines- Tangent Lines In geometry, te tangent line to a circle wit centre O at a point A on te circle is defined to be te perpendicular line at A to te line OA. Te tangent lines ave te special property

More information

Math Module Preliminary Test Solutions

Math Module Preliminary Test Solutions SSEA Summer 207 Mat Module Preliminar Test Solutions. [3 points] Find all values of tat satisf =. Solution: = ( ) = ( ) = ( ) =. Tis means ( ) is positive. Tat is, 0, wic implies. 2. [6 points] Find all

More information

Example: When describing where a function is increasing, decreasing or constant we use the x- axis values.

Example: When describing where a function is increasing, decreasing or constant we use the x- axis values. Business Calculus Lecture Notes (also Calculus With Applications or Business Math II) Chapter 3 Applications o Derivatives 31 Increasing and Decreasing Functions Inormal Deinition: A unction is increasing

More information

Chapter 2. Limits and Continuity 16( ) 16( 9) = = 001. Section 2.1 Rates of Change and Limits (pp ) Quick Review 2.1

Chapter 2. Limits and Continuity 16( ) 16( 9) = = 001. Section 2.1 Rates of Change and Limits (pp ) Quick Review 2.1 Capter Limits and Continuity Section. Rates of Cange and Limits (pp. 969) Quick Review..... f ( ) ( ) ( ) 0 ( ) f ( ) f ( ) sin π sin π 0 f ( ). < < < 6. < c c < < c 7. < < < < < 8. 9. 0. c < d d < c

More information

LIMITS AND DERIVATIVES CONDITIONS FOR THE EXISTENCE OF A LIMIT

LIMITS AND DERIVATIVES CONDITIONS FOR THE EXISTENCE OF A LIMIT LIMITS AND DERIVATIVES Te limit of a function is defined as te value of y tat te curve approaces, as x approaces a particular value. Te limit of f (x) as x approaces a is written as f (x) approaces, as

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

Chapter 2 Limits and Continuity. Section 2.1 Rates of Change and Limits (pp ) Section Quick Review 2.1

Chapter 2 Limits and Continuity. Section 2.1 Rates of Change and Limits (pp ) Section Quick Review 2.1 Section. 6. (a) N(t) t (b) days: 6 guppies week: 7 guppies (c) Nt () t t t ln ln t ln ln ln t 8. 968 Tere will be guppies ater ln 8.968 days, or ater nearly 9 days. (d) Because it suggests te number o

More information

Differentiation. introduction to limits

Differentiation. introduction to limits 9 9A Introduction to limits 9B Limits o discontinuous, rational and brid unctions 9C Dierentiation using i rst principles 9D Finding derivatives b rule 9E Antidierentiation 9F Deriving te original unction

More information

Section 2.1 The Definition of the Derivative. We are interested in finding the slope of the tangent line at a specific point.

Section 2.1 The Definition of the Derivative. We are interested in finding the slope of the tangent line at a specific point. Popper 6: Review of skills: Find tis difference quotient. f ( x ) f ( x) if f ( x) x Answer coices given in audio on te video. Section.1 Te Definition of te Derivative We are interested in finding te slope

More information

Continuity. Example 1

Continuity. Example 1 Continuity MATH 1003 Calculus and Linear Algebra (Lecture 13.5) Maoseng Xiong Department of Matematics, HKUST A function f : (a, b) R is continuous at a point c (a, b) if 1. x c f (x) exists, 2. f (c)

More information

Some Review Problems for First Midterm Mathematics 1300, Calculus 1

Some Review Problems for First Midterm Mathematics 1300, Calculus 1 Some Review Problems for First Midterm Matematics 00, Calculus. Consider te trigonometric function f(t) wose grap is sown below. Write down a possible formula for f(t). Tis function appears to be an odd,

More information

Outline. MS121: IT Mathematics. Limits & Continuity Rates of Change & Tangents. Is there a limit to how fast a man can run?

Outline. MS121: IT Mathematics. Limits & Continuity Rates of Change & Tangents. Is there a limit to how fast a man can run? Outline MS11: IT Matematics Limits & Continuity & 1 Limits: Atletics Perspective Jon Carroll Scool of Matematical Sciences Dublin City University 3 Atletics Atletics Outline Is tere a limit to ow fast

More information

(a) At what number x = a does f have a removable discontinuity? What value f(a) should be assigned to f at x = a in order to make f continuous at a?

(a) At what number x = a does f have a removable discontinuity? What value f(a) should be assigned to f at x = a in order to make f continuous at a? Solutions to Test 1 Fall 016 1pt 1. Te grap of a function f(x) is sown at rigt below. Part I. State te value of eac limit. If a limit is infinite, state weter it is or. If a limit does not exist (but is

More information

Click here to see an animation of the derivative

Click here to see an animation of the derivative Differentiation Massoud Malek Derivative Te concept of derivative is at te core of Calculus; It is a very powerful tool for understanding te beavior of matematical functions. It allows us to optimize functions,

More information

= h. Geometrically this quantity represents the slope of the secant line connecting the points

= h. Geometrically this quantity represents the slope of the secant line connecting the points Section 3.7: Rates of Cange in te Natural and Social Sciences Recall: Average rate of cange: y y y ) ) ), ere Geometrically tis quantity represents te slope of te secant line connecting te points, f (

More information

SFU UBC UNBC Uvic Calculus Challenge Examination June 5, 2008, 12:00 15:00

SFU UBC UNBC Uvic Calculus Challenge Examination June 5, 2008, 12:00 15:00 SFU UBC UNBC Uvic Calculus Callenge Eamination June 5, 008, :00 5:00 Host: SIMON FRASER UNIVERSITY First Name: Last Name: Scool: Student signature INSTRUCTIONS Sow all your work Full marks are given only

More information

Section 2.7 Derivatives and Rates of Change Part II Section 2.8 The Derivative as a Function. at the point a, to be. = at time t = a is

Section 2.7 Derivatives and Rates of Change Part II Section 2.8 The Derivative as a Function. at the point a, to be. = at time t = a is Mat 180 www.timetodare.com Section.7 Derivatives and Rates of Cange Part II Section.8 Te Derivative as a Function Derivatives ( ) In te previous section we defined te slope of te tangent to a curve wit

More information

Calculus I Practice Exam 1A

Calculus I Practice Exam 1A Calculus I Practice Exam A Calculus I Practice Exam A Tis practice exam empasizes conceptual connections and understanding to a greater degree tan te exams tat are usually administered in introductory

More information

Exam 1 Solutions. x(x 2) (x + 1)(x 2) = x

Exam 1 Solutions. x(x 2) (x + 1)(x 2) = x Eam Solutions Question (0%) Consider f() = 2 2 2 2. (a) By calculating relevant its, determine te equations of all vertical asymptotes of te grap of f(). If tere are none, say so. f() = ( 2) ( + )( 2)

More information

MA119-A Applied Calculus for Business Fall Homework 4 Solutions Due 9/29/ :30AM

MA119-A Applied Calculus for Business Fall Homework 4 Solutions Due 9/29/ :30AM MA9-A Applied Calculus for Business 006 Fall Homework Solutions Due 9/9/006 0:0AM. #0 Find te it 5 0 + +.. #8 Find te it. #6 Find te it 5 0 + + = (0) 5 0 (0) + (0) + =.!! r + +. r s r + + = () + 0 () +

More information

Function Composition and Chain Rules

Function Composition and Chain Rules Function Composition and s James K. Peterson Department of Biological Sciences and Department of Matematical Sciences Clemson University Marc 8, 2017 Outline 1 Function Composition and Continuity 2 Function

More information

Honors Calculus Midterm Review Packet

Honors Calculus Midterm Review Packet Name Date Period Honors Calculus Midterm Review Packet TOPICS THAT WILL APPEAR ON THE EXAM Capter Capter Capter (Sections. to.6) STRUCTURE OF THE EXAM Part No Calculators Miture o multiple-coice, matcing,

More information

Continuity and Differentiability Worksheet

Continuity and Differentiability Worksheet Continuity and Differentiability Workseet (Be sure tat you can also do te grapical eercises from te tet- Tese were not included below! Typical problems are like problems -3, p. 6; -3, p. 7; 33-34, p. 7;

More information

1 Lecture 13: The derivative as a function.

1 Lecture 13: The derivative as a function. 1 Lecture 13: Te erivative as a function. 1.1 Outline Definition of te erivative as a function. efinitions of ifferentiability. Power rule, erivative te exponential function Derivative of a sum an a multiple

More information

JANE PROFESSOR WW Prob Lib1 Summer 2000

JANE PROFESSOR WW Prob Lib1 Summer 2000 JANE PROFESSOR WW Prob Lib Summer 2000 Sample WeBWorK problems. WeBWorK assignment Derivatives due 2//06 at 2:00 AM..( pt) If f x 9, find f 0. 2.( pt) If f x 7x 27, find f 5. 3.( pt) If f x 7 4x 5x 2,

More information

Average Rate of Change

Average Rate of Change Te Derivative Tis can be tougt of as an attempt to draw a parallel (pysically and metaporically) between a line and a curve, applying te concept of slope to someting tat isn't actually straigt. Te slope

More information

Used to estimate energy loss due to friction in pipe. D = internal diameter of pipe (feet) L = length of pipe (feet) Penn State-Harrisburg

Used to estimate energy loss due to friction in pipe. D = internal diameter of pipe (feet) L = length of pipe (feet) Penn State-Harrisburg Module b: Flow in Pipes Darcy-Weisbac Robert Pitt University o Alabama and Sirley Clark Penn State-Harrisburg Darcy-Weisbac can be written or low (substitute V Q/A, were A (π/4)d in te above equation):

More information

Section 15.6 Directional Derivatives and the Gradient Vector

Section 15.6 Directional Derivatives and the Gradient Vector Section 15.6 Directional Derivatives and te Gradient Vector Finding rates of cange in different directions Recall tat wen we first started considering derivatives of functions of more tan one variable,

More information

The Derivative as a Function

The Derivative as a Function Section 2.2 Te Derivative as a Function 200 Kiryl Tsiscanka Te Derivative as a Function DEFINITION: Te derivative of a function f at a number a, denoted by f (a), is if tis limit exists. f (a) f(a + )

More information

Key Concepts. Important Techniques. 1. Average rate of change slope of a secant line. You will need two points ( a, the formula: to find value

Key Concepts. Important Techniques. 1. Average rate of change slope of a secant line. You will need two points ( a, the formula: to find value AB Calculus Unit Review Key Concepts Average and Instantaneous Speed Definition of Limit Properties of Limits One-sided and Two-sided Limits Sandwic Teorem Limits as x ± End Beaviour Models Continuity

More information

University Mathematics 2

University Mathematics 2 University Matematics 2 1 Differentiability In tis section, we discuss te differentiability of functions. Definition 1.1 Differentiable function). Let f) be a function. We say tat f is differentiable at

More information

THE IDEA OF DIFFERENTIABILITY FOR FUNCTIONS OF SEVERAL VARIABLES Math 225

THE IDEA OF DIFFERENTIABILITY FOR FUNCTIONS OF SEVERAL VARIABLES Math 225 THE IDEA OF DIFFERENTIABILITY FOR FUNCTIONS OF SEVERAL VARIABLES Mat 225 As we ave seen, te definition of derivative for a Mat 111 function g : R R and for acurveγ : R E n are te same, except for interpretation:

More information

Taylor Series and the Mean Value Theorem of Derivatives

Taylor Series and the Mean Value Theorem of Derivatives 1 - Taylor Series and te Mean Value Teorem o Derivatives Te numerical solution o engineering and scientiic problems described by matematical models oten requires solving dierential equations. Dierential

More information

MATHEMATICS. The assessment objectives of the Compulsory Part are to test the candidates :

MATHEMATICS. The assessment objectives of the Compulsory Part are to test the candidates : MATHEMATICS INTRODUCTION The public assessment of this subject is based on the Curriculum and Assessment Guide (Secondary 4 6) Mathematics jointly prepared by the Curriculum Development Council and the

More information

1 Limits and Continuity

1 Limits and Continuity 1 Limits and Continuity 1.0 Tangent Lines, Velocities, Growt In tion 0.2, we estimated te slope of a line tangent to te grap of a function at a point. At te end of tion 0.3, we constructed a new function

More information

The algebra of functions Section 2.2

The algebra of functions Section 2.2 Te algebra of functions Section 2.2 f(a+) f(a) Q. Suppose f (x) = 2x 2 x + 1. Find and simplify. Soln: f (a + ) f (a) = [ 2(a + ) 2 ( a + ) + 1] [ 2a 2 a + 1 ] = [ 2(a 2 + 2 + 2a) a + 1] 2a 2 + a 1 = 2a

More information

Math 261 Calculus I. Test 1 Study Guide. Name. Decide whether the limit exists. If it exists, find its value. 1) lim x 1. f(x) 2) lim x -1/2 f(x)

Math 261 Calculus I. Test 1 Study Guide. Name. Decide whether the limit exists. If it exists, find its value. 1) lim x 1. f(x) 2) lim x -1/2 f(x) Math 261 Calculus I Test 1 Study Guide Name Decide whether the it exists. If it exists, find its value. 1) x 1 f(x) 2) x -1/2 f(x) Complete the table and use the result to find the indicated it. 3) If

More information

Differential Calculus (The basics) Prepared by Mr. C. Hull

Differential Calculus (The basics) Prepared by Mr. C. Hull Differential Calculus Te basics) A : Limits In tis work on limits, we will deal only wit functions i.e. tose relationsips in wic an input variable ) defines a unique output variable y). Wen we work wit

More information

INTRODUCTION TO CALCULUS LIMITS

INTRODUCTION TO CALCULUS LIMITS Calculus can be divided into two ke areas: INTRODUCTION TO CALCULUS Differential Calculus dealing wit its, rates of cange, tangents and normals to curves, curve sketcing, and applications to maima and

More information

Math 1210 Midterm 1 January 31st, 2014

Math 1210 Midterm 1 January 31st, 2014 Mat 110 Midterm 1 January 1st, 01 Tis exam consists of sections, A and B. Section A is conceptual, wereas section B is more computational. Te value of every question is indicated at te beginning of it.

More information

Mathematics 5 Worksheet 11 Geometry, Tangency, and the Derivative

Mathematics 5 Worksheet 11 Geometry, Tangency, and the Derivative Matematics 5 Workseet 11 Geometry, Tangency, and te Derivative Problem 1. Find te equation of a line wit slope m tat intersects te point (3, 9). Solution. Te equation for a line passing troug a point (x

More information

Public Assessment of the HKDSE Mathematics Examination

Public Assessment of the HKDSE Mathematics Examination Public Assessment of the HKDSE Mathematics Examination. Exam Format (a) The examination consists of one paper. (b) All questions are conventional questions. (c) The duration is hours and 30 minutes. Section

More information

Notes: DERIVATIVES. Velocity and Other Rates of Change

Notes: DERIVATIVES. Velocity and Other Rates of Change Notes: DERIVATIVES Velocity and Oter Rates of Cange I. Average Rate of Cange A.) Def.- Te average rate of cange of f(x) on te interval [a, b] is f( b) f( a) b a secant ( ) ( ) m troug a, f ( a ) and b,

More information

Recall from our discussion of continuity in lecture a function is continuous at a point x = a if and only if

Recall from our discussion of continuity in lecture a function is continuous at a point x = a if and only if Computational Aspects of its. Keeping te simple simple. Recall by elementary functions we mean :Polynomials (including linear and quadratic equations) Eponentials Logaritms Trig Functions Rational Functions

More information

Mock Exam 3. 1 Hong Kong Educational Publishing Company. Section A. 1. Reference: HKDSE Math M Q1 (a) (1 + 2x) 2 (1 - x) n

Mock Exam 3. 1 Hong Kong Educational Publishing Company. Section A. 1. Reference: HKDSE Math M Q1 (a) (1 + 2x) 2 (1 - x) n Mock Eam Mock Eam Section A. Reference: HKDSE Math M 0 Q (a) ( + ) ( - ) n nn ( ) ( + + ) n + + Coefficient of - n - n -7 n (b) Coefficient of nn ( - ) - n + (- ) - () + (). Reference: HKDSE Math M PP

More information

90 Chapter 5 Logarithmic, Exponential, and Other Transcendental Functions. Name Class. (a) (b) ln x (c) (a) (b) (c) 1 x. y e (a) 0 (b) y.

90 Chapter 5 Logarithmic, Exponential, and Other Transcendental Functions. Name Class. (a) (b) ln x (c) (a) (b) (c) 1 x. y e (a) 0 (b) y. 90 Chapter 5 Logarithmic, Eponential, and Other Transcendental Functions Test Form A Chapter 5 Name Class Date Section. Find the derivative: f ln. 6. Differentiate: y. ln y y y y. Find dy d if ey y. y

More information

Function Composition and Chain Rules

Function Composition and Chain Rules Function Composition an Cain Rules James K. Peterson Department of Biological Sciences an Department of Matematical Sciences Clemson University November 2, 2018 Outline Function Composition an Continuity

More information

PROJECTIVE REPRESENTATIONS OF QUIVERS

PROJECTIVE REPRESENTATIONS OF QUIVERS IJMMS 31:2 22 97 11 II. S1611712218192 ttp://ijmms.indawi.com Hindawi ublisin Corp. ROJECTIVE RERESENTATIONS OF QUIVERS SANGWON ARK Received 3 Auust 21 We prove tat 2 is a projective representation o a

More information

Suggested text book Ex. Pre-lesson work. Ex. 1.2 (10) 2 Let s Practise p. 3, Ex. 1.1 (6c) Problem solving: Deriving 7/10/2014

Suggested text book Ex. Pre-lesson work. Ex. 1.2 (10) 2 Let s Practise p. 3, Ex. 1.1 (6c) Problem solving: Deriving 7/10/2014 01 015 S Mathematics Teaching Schedule for CDE 1st term: 7p nd Term: 105p Total: 179p Text book: New Progress in Senior Mathematics, CK.Kwun, M.W. Cheung, M.C. Choi, S.W. Chong, Hong Kong Educational Publishing

More information

How to Find the Derivative of a Function: Calculus 1

How to Find the Derivative of a Function: Calculus 1 Introduction How to Find te Derivative of a Function: Calculus 1 Calculus is not an easy matematics course Te fact tat you ave enrolled in suc a difficult subject indicates tat you are interested in te

More information

Midterm #1B. x 8 < < x 8 < 11 3 < x < x > x < 5 or 3 2x > 5 2x < 8 2x > 2

Midterm #1B. x 8 < < x 8 < 11 3 < x < x > x < 5 or 3 2x > 5 2x < 8 2x > 2 Mat 30 College Algebra Februar 2, 2016 Midterm #1B Name: Answer Ke David Arnold Instructions. ( points) For eac o te ollowing questions, select te best answer and darken te corresponding circle on our

More information

Math 124. Section 2.6: Limits at infinity & Horizontal Asymptotes. 1 x. lim

Math 124. Section 2.6: Limits at infinity & Horizontal Asymptotes. 1 x. lim Mat 4 Section.6: Limits at infinity & Horizontal Asymptotes Tolstoy, Count Lev Nikolgevic (88-90) A man is like a fraction wose numerator is wat e is and wose denominator is wat e tinks of imself. Te larger

More information

1,3. f x x f x x. Lim. Lim. Lim. Lim Lim. y 13x b b 10 b So the equation of the tangent line is y 13x

1,3. f x x f x x. Lim. Lim. Lim. Lim Lim. y 13x b b 10 b So the equation of the tangent line is y 13x 1.5 Topics: The Derivative lutions 1. Use the limit definition of derivative (the one with x in it) to find f x given f x 4x 5x 6 4 x x 5 x x 6 4x 5x 6 f x x f x f x x0 x x0 x xx x x x x x 4 5 6 4 5 6

More information

158 Calculus and Structures

158 Calculus and Structures 58 Calculus and Structures CHAPTER PROPERTIES OF DERIVATIVES AND DIFFERENTIATION BY THE EASY WAY. Calculus and Structures 59 Copyrigt Capter PROPERTIES OF DERIVATIVES. INTRODUCTION In te last capter you

More information

Integral Calculus, dealing with areas and volumes, and approximate areas under and between curves.

Integral Calculus, dealing with areas and volumes, and approximate areas under and between curves. Calculus can be divided into two ke areas: Differential Calculus dealing wit its, rates of cange, tangents and normals to curves, curve sketcing, and applications to maima and minima problems Integral

More information

y = 3 2 x 3. The slope of this line is 3 and its y-intercept is (0, 3). For every two units to the right, the line rises three units vertically.

y = 3 2 x 3. The slope of this line is 3 and its y-intercept is (0, 3). For every two units to the right, the line rises three units vertically. Mat 2 - Calculus for Management and Social Science. Understanding te basics of lines in te -plane is crucial to te stud of calculus. Notes Recall tat te and -intercepts of a line are were te line meets

More information

g (x) = (3xe x ) = 3(xe x ) (const. mult.) = 3 ((x) e x + x(e x ) ) (product rule) = 3(e x + xe x ) = 3e x (1 + x)

g (x) = (3xe x ) = 3(xe x ) (const. mult.) = 3 ((x) e x + x(e x ) ) (product rule) = 3(e x + xe x ) = 3e x (1 + x) 1. f() = 2 + 2 + 1 Solution: Just use the power rule: f () = 2 + 2 2. g() = 3e Solution: g () = (3e ) = 3(e ) (const. mult.) = 3 (() e + (e ) ) (product rule) = 3(e + e ) = 3e (1 + ) 3. h() = ln( 2 + )

More information

MTH 119 Pre Calculus I Essex County College Division of Mathematics Sample Review Questions 1 Created April 17, 2007

MTH 119 Pre Calculus I Essex County College Division of Mathematics Sample Review Questions 1 Created April 17, 2007 MTH 9 Pre Calculus I Essex County College Division of Matematics Sample Review Questions Created April 7, 007 At Essex County College you sould be prepared to sow all work clearly and in order, ending

More information

Lines, Conics, Tangents, Limits and the Derivative

Lines, Conics, Tangents, Limits and the Derivative Lines, Conics, Tangents, Limits and te Derivative Te Straigt Line An two points on te (,) plane wen joined form a line segment. If te line segment is etended beond te two points ten it is called a straigt

More information

Chapter 9 - Solved Problems

Chapter 9 - Solved Problems Capter 9 - Solved Problems Solved Problem 9.. Consider an internally stable feedback loop wit S o (s) = s(s + ) s + 4s + Determine weter Lemma 9. or Lemma 9. of te book applies to tis system. Solutions

More information

x f(x) =x 2 +2x

x f(x) =x 2 +2x . Section P.. 3.8 5.0 7.t t 9. t 3 30t +56t 80. 5 3. 5. 6= 7.

More information

Exam 1 Review Solutions

Exam 1 Review Solutions Exam Review Solutions Please also review te old quizzes, and be sure tat you understand te omework problems. General notes: () Always give an algebraic reason for your answer (graps are not sufficient),

More information

Derivatives. if such a limit exists. In this case when such a limit exists, we say that the function f is differentiable.

Derivatives. if such a limit exists. In this case when such a limit exists, we say that the function f is differentiable. Derivatives 3. Derivatives Definition 3. Let f be a function an a < b be numbers. Te average rate of cange of f from a to b is f(b) f(a). b a Remark 3. Te average rate of cange of a function f from a to

More information

ACCESS TO SCIENCE, ENGINEERING AND AGRICULTURE: MATHEMATICS 1 MATH00030 SEMESTER /2019

ACCESS TO SCIENCE, ENGINEERING AND AGRICULTURE: MATHEMATICS 1 MATH00030 SEMESTER /2019 ACCESS TO SCIENCE, ENGINEERING AND AGRICULTURE: MATHEMATICS MATH00030 SEMESTER 208/209 DR. ANTHONY BROWN 6. Differential Calculus 6.. Differentiation from First Principles. In tis capter, we will introduce

More information

Lesson 6: The Derivative

Lesson 6: The Derivative Lesson 6: Te Derivative Def. A difference quotient for a function as te form f(x + ) f(x) (x + ) x f(x + x) f(x) (x + x) x f(a + ) f(a) (a + ) a Notice tat a difference quotient always as te form of cange

More information

Continuity and Differentiability

Continuity and Differentiability Continuity and Dierentiability Tis capter requires a good understanding o its. Te concepts o continuity and dierentiability are more or less obvious etensions o te concept o its. Section - INTRODUCTION

More information

Unit #3 Rules of Differentiation Homework Packet

Unit #3 Rules of Differentiation Homework Packet Unit #3 Rules of Differentiation Homework Packet In the table below, a function is given. Show the algebraic analysis that leads to the derivative of the function. Find the derivative by the specified

More information

1watt=1W=1kg m 2 /s 3

1watt=1W=1kg m 2 /s 3 Appendix A Matematics Appendix A.1 Units To measure a pysical quantity, you need a standard. Eac pysical quantity as certain units. A unit is just a standard we use to compare, e.g. a ruler. In tis laboratory

More information

Lesson 4 - Limits & Instantaneous Rates of Change

Lesson 4 - Limits & Instantaneous Rates of Change Lesson Objectives Lesson 4 - Limits & Instantaneous Rates of Cange SL Topic 6 Calculus - Santowski 1. Calculate an instantaneous rate of cange using difference quotients and limits. Calculate instantaneous

More information

Teaching Differentiation: A Rare Case for the Problem of the Slope of the Tangent Line

Teaching Differentiation: A Rare Case for the Problem of the Slope of the Tangent Line Teacing Differentiation: A Rare Case for te Problem of te Slope of te Tangent Line arxiv:1805.00343v1 [mat.ho] 29 Apr 2018 Roman Kvasov Department of Matematics University of Puerto Rico at Aguadilla Aguadilla,

More information

1. (a) 3. (a) 4 3 (b) (a) t = 5: 9. (a) = 11. (a) The equation of the line through P = (2, 3) and Q = (8, 11) is y 3 = 8 6

1. (a) 3. (a) 4 3 (b) (a) t = 5: 9. (a) = 11. (a) The equation of the line through P = (2, 3) and Q = (8, 11) is y 3 = 8 6 A Answers Important Note about Precision of Answers: In many of te problems in tis book you are required to read information from a grap and to calculate wit tat information. You sould take reasonable

More information

Derivative as Instantaneous Rate of Change

Derivative as Instantaneous Rate of Change 43 Derivative as Instantaneous Rate of Cange Consider a function tat describes te position of a racecar moving in a straigt line away from some starting point Let y s t suc tat t represents te time in

More information

AP Calculus Notes: Unit 1 Limits & Continuity. Syllabus Objective: 1.1 The student will calculate limits using the basic limit theorems.

AP Calculus Notes: Unit 1 Limits & Continuity. Syllabus Objective: 1.1 The student will calculate limits using the basic limit theorems. Syllabus Objective:. The student will calculate its using the basic it theorems. LIMITS how the outputs o a unction behave as the inputs approach some value Finding a Limit Notation: The it as approaches

More information

AP Calculus BC Multiple-Choice Answer Key!

AP Calculus BC Multiple-Choice Answer Key! Multiple-Choice Answer Key!!!!! "#$$%&'! "#$$%&'!!,#-! ()*+%$,#-! ()*+%$!!!!!! "!!!!! "!! 5!! 6! 7!! 8! 7! 9!!! 5:!!!!! 5! (!!!! 5! "! 5!!! 5!! 8! (!! 56! "! :!!! 59!!!!! 5! 7!!!! 5!!!!! 55! "! 6! "!!

More information

lim 1 lim 4 Precalculus Notes: Unit 10 Concepts of Calculus

lim 1 lim 4 Precalculus Notes: Unit 10 Concepts of Calculus Syllabus Objectives: 1.1 Te student will understand and apply te concept of te limit of a function at given values of te domain. 1. Te student will find te limit of a function at given values of te domain.

More information

Chapter 4: Numerical Methods for Common Mathematical Problems

Chapter 4: Numerical Methods for Common Mathematical Problems 1 Capter 4: Numerical Metods for Common Matematical Problems Interpolation Problem: Suppose we ave data defined at a discrete set of points (x i, y i ), i = 0, 1,..., N. Often it is useful to ave a smoot

More information

Derivatives of Exponentials

Derivatives of Exponentials mat 0 more on derivatives: day 0 Derivatives of Eponentials Recall tat DEFINITION... An eponential function as te form f () =a, were te base is a real number a > 0. Te domain of an eponential function

More information

ADDITIONAL MATHEMATICS

ADDITIONAL MATHEMATICS 005-CE A MATH HONG KONG CERTIFICATE OF EDUCATION EXAMINATION 005 ADDITIONAL MATHEMATICS :00 pm 5:0 pm (½ hours) This paper must be answered in English 1. Answer ALL questions in Section A and any FOUR

More information

The Gradient and Directional Derivative - (12.6)

The Gradient and Directional Derivative - (12.6) Te Gradient and Directional Deriatie - (.6). Directional Deriaties Definition: Te directional deriatie of f x,y at te point a,b andintedirectionofteunit ector u u,u, denoted as D u f a,b, is defined by

More information

0.1 The Slope of Nonlinear Function

0.1 The Slope of Nonlinear Function WEEK Reading [SB],.3-.7, pp. -38 0. Te Slope of Nonlinear Function If we want approimate a nonlinear function y = f() by a linear one around some point 0, te best approimation is te line tangent to te

More information

AP Calculus AB Free-Response Scoring Guidelines

AP Calculus AB Free-Response Scoring Guidelines Question pt The rate at which raw sewage enters a treatment tank is given by Et 85 75cos 9 gallons per hour for t 4 hours. Treated sewage is removed from the tank at the constant rate of 645 gallons per

More information

2.2 Derivative. 1. Definition of Derivative at a Point: The derivative of the function f x at x a is defined as

2.2 Derivative. 1. Definition of Derivative at a Point: The derivative of the function f x at x a is defined as . Derivative. Definition of Derivative at a Point: Te derivative of te function f at a is defined as f fa fa a lim provided te limit eists. If te limit eists, we sa tat f is differentiable at a, oterwise,

More information

Chapter 2 Limits and Continuity

Chapter 2 Limits and Continuity 4 Section. Capter Limits and Continuity Section. Rates of Cange and Limits (pp. 6) Quick Review.. f () ( ) () 4 0. f () 4( ) 4. f () sin sin 0 4. f (). 4 4 4 6. c c c 7. 8. c d d c d d c d c 9. 8 ( )(

More information

. Compute the following limits.

. Compute the following limits. Today: Tangent Lines and te Derivative at a Point Warmup:. Let f(x) =x. Compute te following limits. f( + ) f() (a) lim f( +) f( ) (b) lim. Let g(x) = x. Compute te following limits. g(3 + ) g(3) (a) lim

More information

MAT 145. Type of Calculator Used TI-89 Titanium 100 points Score 100 possible points

MAT 145. Type of Calculator Used TI-89 Titanium 100 points Score 100 possible points MAT 15 Test #2 Name Solution Guide Type of Calculator Used TI-89 Titanium 100 points Score 100 possible points Use te grap of a function sown ere as you respond to questions 1 to 8. 1. lim f (x) 0 2. lim

More information

1 The concept of limits (p.217 p.229, p.242 p.249, p.255 p.256) 1.1 Limits Consider the function determined by the formula 3. x since at this point

1 The concept of limits (p.217 p.229, p.242 p.249, p.255 p.256) 1.1 Limits Consider the function determined by the formula 3. x since at this point MA00 Capter 6 Calculus and Basic Linear Algebra I Limits, Continuity and Differentiability Te concept of its (p.7 p.9, p.4 p.49, p.55 p.56). Limits Consider te function determined by te formula f Note

More information

1. Consider the trigonometric function f(t) whose graph is shown below. Write down a possible formula for f(t).

1. Consider the trigonometric function f(t) whose graph is shown below. Write down a possible formula for f(t). . Consider te trigonometric function f(t) wose grap is sown below. Write down a possible formula for f(t). Tis function appears to be an odd, periodic function tat as been sifted upwards, so we will use

More information

MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. B) 94 C) ) A) 1 2

MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. B) 94 C) ) A) 1 2 Chapter Calculus MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Find the average rate of change of the function over the given interval. ) = 73-5

More information

The derivative of a function f is a new function defined by. f f (x + h) f (x)

The derivative of a function f is a new function defined by. f f (x + h) f (x) Derivatives Definition Te erivative of a function f is a new function efine by f f (x + ) f (x) (x). 0 We will say tat a function f is ifferentiable over an interval (a, b) if if te erivative function

More information

Math 226 Calculus Spring 2016 Practice Exam 1. (1) (10 Points) Let the differentiable function y = f(x) have inverse function x = f 1 (y).

Math 226 Calculus Spring 2016 Practice Exam 1. (1) (10 Points) Let the differentiable function y = f(x) have inverse function x = f 1 (y). Math 6 Calculus Spring 016 Practice Exam 1 1) 10 Points) Let the differentiable function y = fx) have inverse function x = f 1 y). a) Write down the formula relating the derivatives f x) and f 1 ) y).

More information

1. State whether the function is an exponential growth or exponential decay, and describe its end behaviour using limits.

1. State whether the function is an exponential growth or exponential decay, and describe its end behaviour using limits. Questions 1. State weter te function is an exponential growt or exponential decay, and describe its end beaviour using its. (a) f(x) = 3 2x (b) f(x) = 0.5 x (c) f(x) = e (d) f(x) = ( ) x 1 4 2. Matc te

More information

1 Solutions to the in class part

1 Solutions to the in class part NAME: Solutions to te in class part. Te grap of a function f is given. Calculus wit Analytic Geometry I Exam, Friday, August 30, 0 SOLUTIONS (a) State te value of f(). (b) Estimate te value of f( ). (c)

More information

2.8 The Derivative as a Function

2.8 The Derivative as a Function .8 Te Derivative as a Function Typically, we can find te derivative of a function f at many points of its domain: Definition. Suppose tat f is a function wic is differentiable at every point of an open

More information

Introduction to Derivatives

Introduction to Derivatives Introduction to Derivatives 5-Minute Review: Instantaneous Rates and Tangent Slope Recall te analogy tat we developed earlier First we saw tat te secant slope of te line troug te two points (a, f (a))

More information

Math-2A Lesson 13-3 (Analyzing Functions, Systems of Equations and Inequalities) Which functions are symmetric about the y-axis?

Math-2A Lesson 13-3 (Analyzing Functions, Systems of Equations and Inequalities) Which functions are symmetric about the y-axis? Math-A Lesson 13-3 (Analyzing Functions, Systems of Equations and Inequalities) Which functions are symmetric about the y-axis? f ( x) x x x x x x 3 3 ( x) x We call functions that are symmetric about

More information

Chapter Primer on Differentiation

Chapter Primer on Differentiation Capter 0.01 Primer on Differentiation After reaing tis capter, you soul be able to: 1. unerstan te basics of ifferentiation,. relate te slopes of te secant line an tangent line to te erivative of a function,.

More information

= 0 and states ''hence there is a stationary point'' All aspects of the proof dx must be correct (c)

= 0 and states ''hence there is a stationary point'' All aspects of the proof dx must be correct (c) Paper 1: Pure Matematics 1 Mark Sceme 1(a) (i) (ii) d d y 3 1x 4x x M1 A1 d y dx 1.1b 1.1b 36x 48x A1ft 1.1b Substitutes x = into teir dx (3) 3 1 4 Sows d y 0 and states ''ence tere is a stationary point''

More information

Exponential Decay CHEM 1231 Fall 2009 Claudia Neuhauser

Exponential Decay CHEM 1231 Fall 2009 Claudia Neuhauser Exponential Decay Propylene glycol monometyl eter (PGME) is a solvent used in surface coatings or cleaners. A study by Domoradzki et al. 003 examined ow quickly PGEM was eliminated from blood in vitro.

More information