MATH CALCULUS I 2.1: Derivatives and Rates of Change
|
|
|
- Britton Norris
- 5 years ago
- Views:
Transcription
1 MATH 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
2 Introduction Our main goal in tis section is to define and determine te relationsips between several quantities: Velocity (average and instantaneous) Slope (of secant lines and tangent lines) Rate of Cange (average and instantaneous) Derivative of a Function We will see tat velocity is actually a special case of a rate of cange, so we will concentrate on te last tree concepts. D.L. Wite (Kent State University) 2 / 1
3 Velocity Recall tat as an introduction to limits, we considered average velocity on an interval and defined instantaneous velocity at a specific time as a limit of average velocities on smaller and smaller time intervals. More precisely, suppose an object is moving along a straigt line so tat its position at time t is given by s(t). Te average velocity on te interval t a to t a + is given by v avg s t s(a + ) s(a) (a + ) a s(a + ) s(a). Te instantaneous velocity at time t a is ten s(a + ) s(a) v(a) lim v avg lim. 0 0 Te instantaneous velocity v(a) is a measure of ow fast te position is canging wit respect to time at te time t a. In oter words, it is te rate of cange of position wit respect to time. D.L. Wite (Kent State University) 3 / 1
4 Rates of Cange More generally, we make te following definition: Definition Let y f (x) be a function and let a be a number in te domain of f. Te average rate of cange of y wit respect to x on te interval x a to x a + is R avg y x f (a + ) f (a). Te instantaneous rate of cange of y wit respect to x at te point x a is defined to be f (a + ) f (a) R inst lim R avg lim. 0 0 D.L. Wite (Kent State University) 4 / 1
5 Rates of Cange Notes: Average or instantaneous velocity is te average or instantaneous rate of cange of position wit respect to time. Te units of te rate of cange of y wit respect to x are y-units per x-unit. Notice tat te slope of te line passing troug te points P(a, f (a)) and Q(a +, f (a + )) on te grap of f is m y x f (a + ) f (a) (a + ) a f (a + ) f (a) R avg. Tus average rate of cange on an interval is te slope of te line passing troug te points on te grap at te endpoints of te interval. D.L. Wite (Kent State University) 5 / 1
6 Tangent Lines Let y f (x) be a function and let a be a number in te domain of f. A line passing troug two points on te grap is called a secant line. For example, te following figure illustrates te secant line troug te points P(a, f (a)) and Q(a +, f (a + )) on te grap of f : 2.1 Figure 6 Notice tat te rise ( y) is f (a + ) f (a) and te run ( x) is, f (a + ) f (a) and so te slope of te secant line is, wic is te average rate of cange of y wit respect to x on te interval [a, a + ]. D.L. Wite (Kent State University) 6 / 1
7 Tangent Lines Now instantaneous rate of cange of y wit respect to x is found by taking te limit as 0 of te average rate of cange on [a, a + ]. Tis is also te limit as 0 of te slope of te secant line troug P(a, f (a)) and Q(a +, f (a + )). Wat is te grapical meaning of tis limit? As 0, te point Q moves along te grap toward te point P, as sown in te following figure: 2.1 Figure 5 In te limit, we tink of Q moving all te way to P and te secant line becomes a tangent line at te point P(a, f (a)) (line t in te figure). D.L. Wite (Kent State University) 7 / 1
8 Tangent Lines Definition Let y f (x) be a function and let a be a number in te domain of f. Te tangent line to te grap of f at x a is te line passing troug te point (a, f (a)) wit slope f (a + ) f (a) m lim. 0 Notes: f (a+) f (a) Te tangent line exists only if lim 0 exists. By te Point-Slope formula, te equation of te tangent line is y f (a) m(x a). Tis says tat te instantaneous rate of cange of y wit respect to x at x a is equal to te slope of te tangent line to te grap of f at x a. D.L. Wite (Kent State University) 8 / 1
9 Tangent Lines Slope of Secant Line Troug (a, f (a)) and (a +, f (a +)) f (a + ) f (a) Average Rate of Cange on [a, a + ] Slope of Tangent Line at x a lim 0 f (a + ) f (a) Instantaneous Rate of Cange at x a D.L. Wite (Kent State University) 9 / 1
10 Te Derivative Te expression lim f (a+) f (a) 0 as come up in enoug different situations (instantaneous velocity, instantaneous rate of cange, slope of a tangent line) tat we want to give it its own name. Definition Let y f (x) be a function and let a be a number in te domain of f. Te derivative of f evaluated at x a is if tis limit exists. f f (a + ) f (a) (a) lim, 0 We can now add f (a) to our cart of related quantities: D.L. Wite (Kent State University) 10 / 1
11 Te Derivative Slope of Secant Line Troug (a, f (a)) and (a +, f (a +)) f (a + ) f (a) Average Rate of Cange on [a, a + ] Slope of Tangent Line at x a f (a) lim 0 f (a+) f (a) Instantaneous Rate of Cange at x a D.L. Wite (Kent State University) 11 / 1
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
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)
The Derivative The rate of change
Calculus Lia Vas Te Derivative Te rate of cange Knowing and understanding te concept of derivative will enable you to answer te following questions. Let us consider a quantity wose size is described by
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
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
. 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
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))
= 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 (
Main Points: 1. Limit of Difference Quotients. Prep 2.7: Derivatives and Rates of Change. Names of collaborators:
Name: Section: Names of collaborators: Main Points:. Definition of derivative as limit of difference quotients. Interpretation of derivative as slope of grap. Interpretation of derivative as instantaneous
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.
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,
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
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:
f a h f a h h lim lim
Te Derivative Te derivative of a function f at a (denoted f a) is f a if tis it exists. An alternative way of defining f a is f a x a fa fa fx fa x a Note tat te tangent line to te grap of f at te point
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
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
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,
Derivatives and Rates of Change
Section.1 Derivatives and Rates of Cange 2016 Kiryl Tsiscanka Derivatives and Rates of Cange Measuring te Rate of Increase of Blood Alcool Concentration Biomedical scientists ave studied te cemical and
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
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;
MATH CALCULUS I 4.1: Area and Distance
MATH 12002 - CALCULUS I 4.1: Area and Distance Professor Donald L. White Department of Mathematical Sciences Kent State University D.L. White (Kent State University) 1 / 8 The Area and Distance Problems
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
Higher Derivatives. Differentiable Functions
Calculus 1 Lia Vas Higer Derivatives. Differentiable Functions Te second derivative. Te derivative itself can be considered as a function. Te instantaneous rate of cange of tis function is te second derivative.
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,
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
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
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
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
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 + )
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
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,
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
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
Polynomial Functions. Linear Functions. Precalculus: Linear and Quadratic Functions
Concepts: definition of polynomial functions, linear functions tree representations), transformation of y = x to get y = mx + b, quadratic functions axis of symmetry, vertex, x-intercepts), transformations
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
5. (a) Find the slope of the tangent line to the parabola y = x + 2x
MATH 141 090 Homework Solutions Fall 00 Section.6: Pages 148 150 3. Consider te slope of te given curve at eac of te five points sown (see text for figure). List tese five slopes in decreasing order and
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 () +
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
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
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
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
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
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
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,
1. Which one of the following expressions is not equal to all the others? 1 C. 1 D. 25x. 2. Simplify this expression as much as possible.
004 Algebra Pretest answers and scoring Part A. Multiple coice questions. Directions: Circle te letter ( A, B, C, D, or E ) net to te correct answer. points eac, no partial credit. Wic one of te following
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
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
(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
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
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
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,
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),
Practice Problem Solutions: Exam 1
Practice Problem Solutions: Exam 1 1. (a) Algebraic Solution: Te largest term in te numerator is 3x 2, wile te largest term in te denominator is 5x 2 3x 2 + 5. Tus lim x 5x 2 2x 3x 2 x 5x 2 = 3 5 Numerical
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
Section 2: The Derivative Definition of the Derivative
Capter 2 Te Derivative Applied Calculus 80 Section 2: Te Derivative Definition of te Derivative Suppose we drop a tomato from te top of a 00 foot building and time its fall. Time (sec) Heigt (ft) 0.0 00
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
MAT 1339-S14 Class 2
MAT 1339-S14 Class 2 July 07, 2014 Contents 1 Rate of Cange 1 1.5 Introduction to Derivatives....................... 1 2 Derivatives 5 2.1 Derivative of Polynomial function.................... 5 2.2 Te
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
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
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
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
2.1 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
Gradient Descent etc.
1 Gradient Descent etc EE 13: Networked estimation and control Prof Kan) I DERIVATIVE Consider f : R R x fx) Te derivative is defined as d fx) = lim dx fx + ) fx) Te cain rule states tat if d d f gx) )
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
Finding and Using Derivative The shortcuts
Calculus 1 Lia Vas Finding and Using Derivative Te sortcuts We ave seen tat te formula f f(x+) f(x) (x) = lim 0 is manageable for relatively simple functions like a linear or quadratic. For more complex
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
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.
1. Questions (a) through (e) refer to the graph of the function f given below. (A) 0 (B) 1 (C) 2 (D) 4 (E) does not exist
Mat 1120 Calculus Test 2. October 18, 2001 Your name Te multiple coice problems count 4 points eac. In te multiple coice section, circle te correct coice (or coices). You must sow your work on te oter
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
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
Math 34A Practice Final Solutions Fall 2007
Mat 34A Practice Final Solutions Fall 007 Problem Find te derivatives of te following functions:. f(x) = 3x + e 3x. f(x) = x + x 3. f(x) = (x + a) 4. Is te function 3t 4t t 3 increasing or decreasing wen
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
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
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
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
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 ( )(
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
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
Lab 6 Derivatives and Mutant Bacteria
Lab 6 Derivatives and Mutant Bacteria Date: September 27, 20 Assignment Due Date: October 4, 20 Goal: In tis lab you will furter explore te concept of a derivative using R. You will use your knowledge
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
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
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
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
10 Derivatives ( )
Instructor: Micael Medvinsky 0 Derivatives (.6-.8) Te tangent line to te curve yf() at te point (a,f(a)) is te line l m + b troug tis point wit slope Alternatively one can epress te slope as f f a m lim
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
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
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
1 2 x Solution. The function f x is only defined when x 0, so we will assume that x 0 for the remainder of the solution. f x. f x h f x.
Problem. Let f x x. Using te definition of te derivative prove tat f x x Solution. Te function f x is only defined wen x 0, so we will assume tat x 0 for te remainder of te solution. By te definition of
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,.
MATH1131/1141 Calculus Test S1 v8a
MATH/ Calculus Test 8 S v8a October, 7 Tese solutions were written by Joann Blanco, typed by Brendan Trin and edited by Mattew Yan and Henderson Ko Please be etical wit tis resource It is for te use of
Differentiation Rules and Formulas
Differentiation Rules an Formulas Professor D. Olles December 1, 01 1 Te Definition of te Derivative Consier a function y = f(x) tat is continuous on te interval a, b]. Ten, te slope of te secant line
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
Solve exponential equations in one variable using a variety of strategies. LEARN ABOUT the Math. What is the half-life of radon?
8.5 Solving Exponential Equations GOAL Solve exponential equations in one variable using a variety of strategies. LEARN ABOUT te Mat All radioactive substances decrease in mass over time. Jamie works in
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
Preface. Here are a couple of warnings to my students who may be here to get a copy of what happened on a day that you missed.
Preface Here are my online notes for my course tat I teac ere at Lamar University. Despite te fact tat tese are my class notes, tey sould be accessible to anyone wanting to learn or needing a refreser
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
First we will go over the following derivative rule. Theorem
Tuesday, Feb 1 Tese slides will cover te following 1 d [cos(x)] = sin(x) iger-order derivatives 3 tangent line problems 4 basic differential equations First we will go over te following derivative rule
Solutions to the Multivariable Calculus and Linear Algebra problems on the Comprehensive Examination of January 31, 2014
Solutions to te Multivariable Calculus and Linear Algebra problems on te Compreensive Examination of January 3, 24 Tere are 9 problems ( points eac, totaling 9 points) on tis portion of te examination.
Slopes of Secant and!angent (ines - 2omework
Slopes o Secant and!angent (ines - omework. For te unction ( x) x +, ind te ollowing. Conirm c) on your calculator. between x and x" at x. at x. ( )! ( ) 4! + +!. For te unction ( x) x!, ind te ollowing.
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