INTRODUCTION TO CALCULUS LIMITS
|
|
|
- Anabel Dorsey
- 5 years ago
- Views:
Transcription
1 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 minima problems. Integral Calculus, dealing wit areas and volumes, and approimate areas under and between curves. LIMITS Consider te function f( ). Te it of te function is defined as te value of (ordinate) tat is approaced b te curve, as te value of (or abscissa) approaces a particular number, sa of value ' a '. For eample, if we approac an value of 3 from te left or te rigt, on te curve f ( ) we see tat in bot cases, te iting value is 9. i.e f ( ) 1 8 f (3) (3) As a furter eample, te diagram below is a grap of a piecemeal function. Note tat one end point is open, and te oter closed, above 3. f ( ) f( ) Te Scool For Ecellence 16 Trial Eam Revision Lectures Matematics Book 1 Page 17
2 Regardless, as we approac 3 from te left, te iting value is 9, but as we approac 3 from te rigt, te iting value is 13. Hence we sa tat te it ere does not eist. i.e. f ( ) does not eist. Limits (and ence tangents) can onl be found on curves tat 3 are continuous at te point in question. Note: Do not confuse tis wit te functional value f (3), wic in tis case is clearl defined as 13. EVALUATING LIMITS In eac of te cases below, ou sould initiall make te substitution of te given value, in an attempt to evaluate te it. Depending on wat results from tis ten determines te necessar strateg. Note tat in all cases, te use of a calculator to substitute a number ver close to te given value will give ou a ver good idea of wat to epect as a final answer ver and in an eam if ou forget te formal metod! CASE I: You substituted into te epression, and te result was well defined. In oter words, te resulting epression could be evaluated do so! QUESTION 18 Find (4 1). (4 1) 4() 1 7 QUESTION 19 Find ( 1) 5 5 ( 1) 1 6 Te last epression was a "Quotient". Quotients of epressions often create difficult. Imagine ou substituted into suc an epression, and te result was one of te following... Te Scool For Ecellence 16 Trial Eam Revision Lectures Matematics Book 1 Page 18
3 CASE II: Zero Constant ie.. c Te it is. QUESTION 5 Find (5) (5) CASE III: Constant Zero c ie.. Te it is undefined since it approaces ±. QUESTION 1 Find. + ( ) + ( ) + 4 ± Te Scool For Ecellence 16 Trial Eam Revision Lectures Matematics Book 1 Page 19
4 CASE IV: Infinit Constant ie.. c Te it is undefined since it approaces ±. QUESTION 9 + Find ( ) 7 7 CASE V: Constant Infinit ie.. c Te it is. QUESTION 3 8 Find. 8 8 ( ) Te Scool For Ecellence 16 Trial Eam Revision Lectures Matematics Book 1 Page
5 CASE VI: Zero Zero ie.. You sould simplif te algebraic fraction b first factoring, and ten cancelling, before re-substituting te given value. QUESTION Find CASE VII: Infinit Infinit ie.. Divide ever term in te epression b te igest power of present in te denominator, simplif eac term, ten use te rule for Case V. QUESTION 5 5 Find ( is te igest power of in te denominator). Te Scool For Ecellence 16 Trial Eam Revision Lectures Matematics Book 1 Page 1
6 WATCH OUT! If ever in doubt about te correct tecnique, or in doubt of our answer, simpl substitute a value for tat is close to te given value, into te given formula, on our calculator. For eample, for sa, use.99 or 3.1, and for 3, use some large value like 1,,. THE DERIVATIVE OF A FUNCTION In graping, ou ave used tables of values, to plot points (along wit oter algebraic tools suc as finding an aes intercepts etc) and to draw sketces of functions. Te original function f( ) is used to find te values or ordinates of points, on te grap, so tat we can see its sape and position. Te "derivative", or derived function is written as d or '( ) d f. As we are assuming ere tat te independent variable in te function is, ten te derivative is found "wit respect to ". Te derivative is related to, and found from, te original function f( ). Te derivative is used to find a different feature of te grap it is used to find te gradient. Te process of finding te derivative is called "differentiation". Te Scool For Ecellence 16 Trial Eam Revision Lectures Matematics Book 1 Page
7 QUESTION 6 Sketc te derived function d d for te given function,. + 1 d d Te Scool For Ecellence 16 Trial Eam Revision Lectures Matematics Book 1 Page 3
8 QUESTION 7 Sketc te derived function d for te given function,. d d d Te Scool For Ecellence 16 Trial Eam Revision Lectures Matematics Book 1 Page 4
9 QUESTION 8 Sketc te derived function d d for te given function,. 3 d d DIFFERENTIABILITY A function is said to be "differentiable" at a particular point if it satisfies two conditions te grap of te function at te point in question must be continuous (te curve is not broken at te point), and te grap at te point must also be smoot (te curve must ave te same slope on te left and approac to te point as it does on te rigt and approac). Te Scool For Ecellence 16 Trial Eam Revision Lectures Matematics Book 1 Page 5
10 ESTIMATING THE GRADIENT OF THE TANGENT If we ad an accurate grap of a function, we could draw a tangent at te point in question, judging tis purel b ee. B coosing two points on te tangent and reading off te coordinates of te two points, we 1 would use te gradient formula m to calculate te gradient of te tangent at tat 1 particular point. If we ad a rule for te function, witout an accurate grap, we could use te following metod. Sa we wanted te gradient of f ( ) at te point were 3. We could use a second point on te curve, ver close to te first and ten appl te gradient formula. Let's use te two points at 3 and 3.1. Note tat tis cannot give te eact gradient at 3 - just someting close. f ( ) f (3.1) (3.1) 9.61 f (3) (3) It is te gradient of tis secant tat we are using. Tis is wat we reall want te gradient of te tangent. Tis is because te slope of te tangent and te slope of te curve are te same. m Tis value represents te gradient of te secant joining te two points, and not of te curve or tangent. But we migt guess tat te gradient is around 6, and in fact we will sortl find out it is eactl 6. Te Scool For Ecellence 16 Trial Eam Revision Lectures Matematics Book 1 Page 6
11 DIFFERENTIATION USING FIRST PRINCIPLES f( ) f ( + ) f ( ) + Te gradient of te secant is given b: m 1 1 f ( + ) f( ) ( + ) Hence te gradient of te tangent (and ence te curve) is: d d f '( ) f ( + ) f( ) METHOD: Step 1: Write te epressions for f () and f ( + ). Step : Substitute te epressions for f () and f ( + ) into te it teorem. Step 3: Epand and collect like terms. d [ f ( + ) f ( )] d Step 4: Remove as a common factor and simplif. Step 5: Substitute. Note: To find te gradient at a specific point, we substitute te value of into te derivative. Note: Te gradient at is denoted as f '(). Te Scool For Ecellence 16 Trial Eam Revision Lectures Matematics Book 1 Page 7
12 QUESTION 9 Find te derivative, using first principles, of f '( ) f + f ( ) ( ) ( + ) ( + ) f ( ). Tis answer is called te derivative or derived function or gradient function of f ( ). Note tat at 3, f '(3) (3) 6. Tis is te same answer as tat guessed in te earlier eample. QUESTION 3 Find, using first principles, te gradient function of f( ) 5+. WATCH OUT! Wen te function as two terms or more, make sure tat te function is written inside of brackets beind te minus sign. f ( ) 5 + f ( + ) ( + ) 5( + ) f '( ) f ( + ) f( ) ( ) ( 5+ ) (+ 5) + 5 (+ 5) + () 5 5 Te Scool For Ecellence 16 Trial Eam Revision Lectures Matematics Book 1 Page 8
13 DIFFERENTIATION RULES Te derivative of an algebraic term is obtained b multipling te term b te power and ten lowering te power on tat term b one. Tis rule applies for all algebraic epressions, including rational functions, providing tat n. If For eample: If n a ten d d 6 ten an n 1 d d 1 Note also tat if a ten d a d. Te derivative of a constant (a term tat does not contain an variables) is equal to zero. d If c (were c is a constant) ten d Te derivative of a sum (or of a difference) is te sum (or differences) of te individual derivatives. d If u ( ) ± v ( ) ten u'( ) v'( ) d ± QUESTION 31 Find te derivative of 5 +. WATCH OUT! "d" for differentiate, "d" for decrease te power. (Please note tat tis is onl true for a power wic is a constant onl) d d d d + d d d d ( ) (5 ) () 5+ 5 Te Scool For Ecellence 16 Trial Eam Revision Lectures Matematics Book 1 Page 9
14 Step 1: Rewrite all terms as powers of. FINDING DERIVATIVES q 3 3 p ( q ) p q ( ) p Step : Bring terms involving in te denominator (bottom of a fraction) to te top, b canging te sign on te power. For eample: 1 Note: Step 3: Simplif epressions so tat terms are separated b addition and subtraction and ten differentiate eac term individuall. Alternativel, reduce epressions down to 1 term onl (use log and inde laws). Products: Epand simple products rater tan appling te Product Rule. Quotients: Remove common factor(s) and simplif. Factorise and einate terms b cancellation. If tere is onl one term in te denominator, write eac term in te numerator over te denominator so tat individual fractions are formed. Ten simplif eac term b cancellation. For eample: (Don t use te Quotient Rule). Step 4: Differentiate. Step 5: Re-write te answer using positive powers. Bring terms wit negative powers in te numerator (top of a fraction) to te bottom, b canging te sign on eac power. Step 6: State restrictions on te values of. Te Scool For Ecellence 16 Trial Eam Revision Lectures Matematics Book 1 Page 3
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
Differentiation. Area of study Unit 2 Calculus
Differentiation 8VCE VCEco Area of stud Unit Calculus coverage In tis ca 8A 8B 8C 8D 8E 8F capter Introduction to limits Limits of discontinuous, rational and brid functions Differentiation using first
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
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
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
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
Pre-Calculus Review Preemptive Strike
Pre-Calculus Review Preemptive Strike Attaced are some notes and one assignment wit tree parts. Tese are due on te day tat we start te pre-calculus review. I strongly suggest reading troug te notes torougly
Exponentials and Logarithms Review Part 2: Exponentials
Eponentials and Logaritms Review Part : Eponentials Notice te difference etween te functions: g( ) and f ( ) In te function g( ), te variale is te ase and te eponent is a constant. Tis is called a power
The derivative function
Roberto s Notes on Differential Calculus Capter : Definition of derivative Section Te derivative function Wat you need to know already: f is at a point on its grap and ow to compute it. Wat te derivative
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))
MATH1151 Calculus Test S1 v2a
MATH5 Calculus Test 8 S va January 8, 5 Tese solutions were written and typed up by Brendan Trin Please be etical wit tis resource It is for te use of MatSOC members, so do not repost it on oter forums
Name: Sept 21, 2017 Page 1 of 1
MATH 111 07 (Kunkle), Eam 1 100 pts, 75 minutes No notes, books, electronic devices, or outside materials of an kind. Read eac problem carefull and simplif our answers. Name: Sept 21, 2017 Page 1 of 1
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
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
NUMERICAL DIFFERENTIATION. James T. Smith San Francisco State University. In calculus classes, you compute derivatives algebraically: for example,
NUMERICAL DIFFERENTIATION James T Smit San Francisco State University In calculus classes, you compute derivatives algebraically: for example, f( x) = x + x f ( x) = x x Tis tecnique requires your knowing
Chapter. Differentiation: Basic Concepts. 1. The Derivative: Slope and Rates. 2. Techniques of Differentiation. 3. The Product and Quotient Rules
Differentiation: Basic Concepts Capter 1. Te Derivative: Slope and Rates 2. Tecniques of Differentiation 3. Te Product and Quotient Rules 4. Marginal Analsis: Approimation b Increments 5. Te Cain Rule
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;
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
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
Material for Difference Quotient
Material for Difference Quotient Prepared by Stepanie Quintal, graduate student and Marvin Stick, professor Dept. of Matematical Sciences, UMass Lowell Summer 05 Preface Te following difference quotient
Math 31A Discussion Notes Week 4 October 20 and October 22, 2015
Mat 3A Discussion Notes Week 4 October 20 and October 22, 205 To prepare for te first midterm, we ll spend tis week working eamples resembling te various problems you ve seen so far tis term. In tese notes
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)
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
3.1 Extreme Values of a Function
.1 Etreme Values of a Function Section.1 Notes Page 1 One application of te derivative is finding minimum and maimum values off a grap. In precalculus we were only able to do tis wit quadratics by find
MATH1901 Differential Calculus (Advanced)
MATH1901 Dierential Calculus (Advanced) Capter 3: Functions Deinitions : A B A and B are sets assigns to eac element in A eactl one element in B A is te domain o te unction B is te codomain o te unction
2.11 That s So Derivative
2.11 Tat s So Derivative Introduction to Differential Calculus Just as one defines instantaneous velocity in terms of average velocity, we now define te instantaneous rate of cange of a function at a point
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
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
11-19 PROGRESSION. A level Mathematics. Pure Mathematics
SSaa m m pplle e UCa ni p t ter DD iff if erfe enren tiatia tiotio nn - 9 RGRSSIN decel Slevel andmatematics level Matematics ure Matematics NW FR 07 Year/S Year decel S and level Matematics Sample material
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
SECTION 1.10: DIFFERENCE QUOTIENTS LEARNING OBJECTIVES
(Section.0: Difference Quotients).0. SECTION.0: DIFFERENCE QUOTIENTS LEARNING OBJECTIVES Define average rate of cange (and average velocity) algebraically and grapically. Be able to identify, construct,
3.4 Algebraic Limits. Ex 1) lim. Ex 2)
Calculus Maimus.4 Algebraic Limits At tis point, you sould be very comfortable finding its bot grapically and numerically wit te elp of your graping calculator. Now it s time to practice finding its witout
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 ( )(
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
MTH-112 Quiz 1 Name: # :
MTH- Quiz Name: # : Please write our name in te provided space. Simplif our answers. Sow our work.. Determine weter te given relation is a function. Give te domain and range of te relation.. Does te equation
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
SECTION 3.2: DERIVATIVE FUNCTIONS and DIFFERENTIABILITY
(Section 3.2: Derivative Functions and Differentiability) 3.2.1 SECTION 3.2: DERIVATIVE FUNCTIONS and DIFFERENTIABILITY LEARNING OBJECTIVES Know, understand, and apply te Limit Definition of te Derivative
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
REVIEW LAB ANSWER KEY
REVIEW LAB ANSWER KEY. Witout using SN, find te derivative of eac of te following (you do not need to simplify your answers): a. f x 3x 3 5x x 6 f x 3 3x 5 x 0 b. g x 4 x x x notice te trick ere! x x g
Time (hours) Morphine sulfate (mg)
Mat Xa Fall 2002 Review Notes Limits and Definition of Derivative Important Information: 1 According to te most recent information from te Registrar, te Xa final exam will be eld from 9:15 am to 12:15
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
Chapter 1D - Rational Expressions
- Capter 1D Capter 1D - Rational Expressions Definition of a Rational Expression A rational expression is te quotient of two polynomials. (Recall: A function px is a polynomial in x of degree n, if tere
Limits and an Introduction to Calculus
Limits and an Introduction to Calculus. Introduction to Limits. Tecniques for Evaluating Limits. Te Tangent Line Problem. Limits at Infinit and Limits of Sequences.5 Te Area Problem In Matematics If a
= 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''
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.
1 (10) 2 (10) 3 (10) 4 (10) 5 (10) 6 (10) Total (60)
First Name: OSU Number: Last Name: Signature: OKLAHOMA STATE UNIVERSITY Department of Matematics MATH 2144 (Calculus I) Instructor: Dr. Matias Sculze MIDTERM 1 September 17, 2008 Duration: 50 minutes No
Logarithmic functions
Roberto s Notes on Differential Calculus Capter 5: Derivatives of transcendental functions Section Derivatives of Logaritmic functions Wat ou need to know alread: Definition of derivative and all basic
1. AB Calculus Introduction
1. AB Calculus Introduction Before we get into wat calculus is, ere are several eamples of wat you could do BC (before calculus) and wat you will be able to do at te end of tis course. Eample 1: On April
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
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,
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
Section 2.4: Definition of Function
Section.4: Definition of Function Objectives Upon completion of tis lesson, you will be able to: Given a function, find and simplify a difference quotient: f ( + ) f ( ), 0 for: o Polynomial functions
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,
136 Calculus and Structures
6 Calculus and Structures CHAPTER 9 THE DERIVATIVE Calculus and Structures 7 Coprigt Capter 9 THE DERIVATIVE 9. THE ORIGINS OF CALCULUS All great discoveries ave teir origins wit an important problem tat
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
MVT and Rolle s Theorem
AP Calculus CHAPTER 4 WORKSHEET APPLICATIONS OF DIFFERENTIATION MVT and Rolle s Teorem Name Seat # Date UNLESS INDICATED, DO NOT USE YOUR CALCULATOR FOR ANY OF THESE QUESTIONS In problems 1 and, state
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
1.5 Functions and Their Rates of Change
66_cpp-75.qd /6/8 4:8 PM Page 56 56 CHAPTER Introduction to Functions and Graps.5 Functions and Teir Rates of Cange Identif were a function is increasing or decreasing Use interval notation Use and interpret
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:
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)
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.
A.P. CALCULUS (AB) Outline Chapter 3 (Derivatives)
A.P. CALCULUS (AB) Outline Capter 3 (Derivatives) NAME Date Previously in Capter 2 we determined te slope of a tangent line to a curve at a point as te limit of te slopes of secant lines using tat point
. 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
232 Calculus and Structures
3 Calculus and Structures CHAPTER 17 JUSTIFICATION OF THE AREA AND SLOPE METHODS FOR EVALUATING BEAMS Calculus and Structures 33 Copyrigt Capter 17 JUSTIFICATION OF THE AREA AND SLOPE METHODS 17.1 THE
1 1. Rationalize the denominator and fully simplify the radical expression 3 3. Solution: = 1 = 3 3 = 2
MTH - Spring 04 Exam Review (Solutions) Exam : February 5t 6:00-7:0 Tis exam review contains questions similar to tose you sould expect to see on Exam. Te questions included in tis review, owever, are
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
(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
11.6 DIRECTIONAL DERIVATIVES AND THE GRADIENT VECTOR
SECTION 11.6 DIRECTIONAL DERIVATIVES AND THE GRADIENT VECTOR 633 wit speed v o along te same line from te opposite direction toward te source, ten te frequenc of te sound eard b te observer is were c is
1 Calculus. 1.1 Gradients and the Derivative. Q f(x+h) f(x)
Calculus. Gradients and te Derivative Q f(x+) δy P T δx R f(x) 0 x x+ Let P (x, f(x)) and Q(x+, f(x+)) denote two points on te curve of te function y = f(x) and let R denote te point of intersection of
Definition of the Derivative
Te Limit Definition of te Derivative Tis Handout will: Define te limit grapically and algebraically Discuss, in detail, specific features of te definition of te derivative Provide a general strategy of
CHAPTER 3: Derivatives
CHAPTER 3: Derivatives 3.1: Derivatives, Tangent Lines, and Rates of Cange 3.2: Derivative Functions and Differentiability 3.3: Tecniques of Differentiation 3.4: Derivatives of Trigonometric Functions
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
Precalculus Test 2 Practice Questions Page 1. Note: You can expect other types of questions on the test than the ones presented here!
Precalculus Test 2 Practice Questions Page Note: You can expect oter types of questions on te test tan te ones presented ere! Questions Example. Find te vertex of te quadratic f(x) = 4x 2 x. Example 2.
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.
Section 3: The Derivative Definition of the Derivative
Capter 2 Te Derivative Business Calculus 85 Section 3: Te Derivative Definition of te Derivative Returning to te tangent slope problem from te first section, let's look at te problem of finding te slope
4. The slope of the line 2x 7y = 8 is (a) 2/7 (b) 7/2 (c) 2 (d) 2/7 (e) None of these.
Mat 11. Test Form N Fall 016 Name. Instructions. Te first eleven problems are wort points eac. Te last six problems are wort 5 points eac. For te last six problems, you must use relevant metods of algebra
Calculus I, Fall Solutions to Review Problems II
Calculus I, Fall 202 - Solutions to Review Problems II. Find te following limits. tan a. lim 0. sin 2 b. lim 0 sin 3. tan( + π/4) c. lim 0. cos 2 d. lim 0. a. From tan = sin, we ave cos tan = sin cos =
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,
MAT 1800 FINAL EXAM HOMEWORK
MAT 800 FINAL EXAM HOMEWORK Read te directions to eac problem careully ALL WORK MUST BE SHOWN DO NOT USE A CALCULATOR Problems come rom old inal eams (SS4, W4, F, SS, W) Solving Equations: Let 5 Find all
5.1 We will begin this section with the definition of a rational expression. We
Basic Properties and Reducing to Lowest Terms 5.1 We will begin tis section wit te definition of a rational epression. We will ten state te two basic properties associated wit rational epressions and go
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,
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),
ESCUELA LATINOAMERICANA DE COOPERACIÓN Y DESARROLLO Especialización en Cooperación Internacional para el Desarrollo
ESCUELA LATINOAMERICANA DE COOPERACIÓN Y DESARROLLO Especialización en Cooperación Internacional para el Desarrollo A SURVIVAL KIT IN CASE O.MATHEMATICS b Marco Missaglia * Universit of Pavia September
In Leibniz notation, we write this rule as follows. DERIVATIVE OF A CONSTANT FUNCTION. For n 4 we find the derivative of f x x 4 as follows: lim
.1 DERIVATIVES OF POLYNOIALS AND EXPONENTIAL FUNCTIONS c =c slope=0 0 FIGURE 1 Te grap of ƒ=c is te line =c, so fª()=0. In tis section we learn ow to ifferentiate constant functions, power functions, polnomials,
Derivatives. By: OpenStaxCollege
By: OpenStaxCollege Te average teen in te United States opens a refrigerator door an estimated 25 times per day. Supposedly, tis average is up from 10 years ago wen te average teenager opened a refrigerator
Bob Brown Math 251 Calculus 1 Chapter 3, Section 1 Completed 1 CCBC Dundalk
Bob Brown Mat 251 Calculus 1 Capter 3, Section 1 Completed 1 Te Tangent Line Problem Te idea of a tangent line first arises in geometry in te context of a circle. But before we jump into a discussion of
2.3 Product and Quotient Rules
.3. PRODUCT AND QUOTIENT RULES 75.3 Product and Quotient Rules.3.1 Product rule Suppose tat f and g are two di erentiable functions. Ten ( g (x)) 0 = f 0 (x) g (x) + g 0 (x) See.3.5 on page 77 for a proof.
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
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
2.1 THE DEFINITION OF DERIVATIVE
2.1 Te Derivative Contemporary Calculus 2.1 THE DEFINITION OF DERIVATIVE 1 Te grapical idea of a slope of a tangent line is very useful, but for some uses we need a more algebraic definition of te derivative
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
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 () +
Math 312 Lecture Notes Modeling
Mat 3 Lecture Notes Modeling Warren Weckesser Department of Matematics Colgate University 5 7 January 006 Classifying Matematical Models An Example We consider te following scenario. During a storm, a
MATH 1020 TEST 2 VERSION A FALL 2014 ANSWER KEY. Printed Name: Section #: Instructor:
ANSWER KEY Printed Name: Section #: Instructor: Please do not ask questions during tis eam. If you consider a question to be ambiguous, state your assumptions in te margin and do te best you can to provide
Numerical Differentiation
Numerical Differentiation Finite Difference Formulas for te first derivative (Using Taylor Expansion tecnique) (section 8.3.) Suppose tat f() = g() is a function of te variable, and tat as 0 te function
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
Calculus and Vectors. Day Lesson Title Math Learning Goals Expectations
Unit Eploring Derivatives Calculus and Vectors Lesson Outline Day Lesson Title Mat Learning Goals Epectations 1 Key caracteristics of instantaneous rates of cange (TIPS4RM Lesson) Determine intervals in
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
7.1 Using Antiderivatives to find Area
7.1 Using Antiderivatives to find Area Introduction finding te area under te grap of a nonnegative, continuous function f In tis section a formula is obtained for finding te area of te region bounded between
Combining functions: algebraic methods
Combining functions: algebraic metods Functions can be added, subtracted, multiplied, divided, and raised to a power, just like numbers or algebra expressions. If f(x) = x 2 and g(x) = x + 2, clearly f(x)