The derivative of a function f is a new function defined by. f f (x + h) f (x)
|
|
- Peter Edwards
- 5 years ago
- Views:
Transcription
1 Derivatives
2 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 f (x) at every point in (a, b).
3 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 f (x) at every point in (a, b). Q. How is tis te same or ifferent from wat we were oing yesteray wit tangent lines?
4 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 f (x) at every point in (a, b). Q. How is tis te same or ifferent from wat we were oing yesteray wit tangent lines? A. Yesteray, we were calculating erivatives at iniviual points, an getting numbers for answers. Toay, we ll calculate te erivative function, an get out answers wit variables in tem (o all te points at once).
5 Example: let f (x) = x 2 Derivatives at a point: If I first ask wat is f (2)?, I coul calculate f (2) 0 (2 + ) = 4.
6 Example: let f (x) = x 2 Derivatives at a point: If I first ask wat is f (2)?, I coul calculate f (2) 0 (2 + ) = 4. But ten, if I ask wat is f (3)? we ave to o it all over again.
7 Example: let f (x) = x 2 Derivatives at a point: If I first ask wat is f (2)?, I coul calculate f (2) 0 (2 + ) = 4. But ten, if I ask wat is f (3)? we ave to o it all over again. Toay s goal: Write own a function f (x) wic as all te erivatives-at-a-point collecte togeter.
8 If a is a number, (like 2 or 3) ten f (a) = f (a + ) f (a) lim }{{} 0 }{{ } gets ri a function of of te s a s an s
9 If a is a number, (like 2 or 3) ten f (a) = f (a + ) f (a) lim = number }{{} 0 }{{ } gets ri of te s a function of a s an s
10 If a is a number, (like 2 or 3) ten f (a) = But x is a variable, so f (x) = f (a + ) f (a) lim }{{} 0 }{{ } gets ri a function of of te s a s an s f (x + ) f (x) lim }{{} 0 }{{ } gets ri a function of of te s x s an s = number
11 If a is a number, (like 2 or 3) ten f (a) = But x is a variable, so f (a + ) f (a) lim }{{} 0 }{{ } gets ri a function of of te s a s an s = number f (x) = f (x + ) f (x) lim = function of x s }{{} 0 }{{ } gets ri of te s a function of x s an s
12 Starting simple Suppose we consier te piecewise linear function x x 1 f (x) = 1 1 < x < 3 x x
13 Starting simple Suppose we consier te piecewise linear function x x 1 f (x) = 1 1 < x < 3 x x
14 Starting simple Suppose we consier te piecewise linear function x x 1 f (x) = 1 1 < x < 3 x x
15 Starting simple Suppose we consier te piecewise linear function x x 1 f (x) = 1 1 < x < 3 x x
16 Starting simple Suppose we consier te piecewise linear function x x 1 f (x) = 1 1 < x < 3 x x
17 Starting simple Suppose we consier te piecewise linear function x x 1 f (x) = 1 1 < x < 3 x x Te erivative is: 1 x < 1 f (x) = 0 1 < x < < x
18 Anoter example Wat is te erivative of f (x) = x? Write own te piecewise function an sketc it on te grap
19 Anoter example Wat is te erivative of f (x) = x? Write own te piecewise function an sketc it on te grap f (x) = { 1 x < 0 1 x > 0
20 Lines In general, if m an b are constants, an f (x) = mx + b b m=slope f (x) = m te slope of te tangent line = slope of te line
21 Roug sape of te erivative if f (x) is increasing f (x) is positive! if f (x) is ecreasing f (x) is negative!
22 Matc em up! Here are graps of two functions an teir erivatives. Wic are wic?
23 Matc em up! Here are graps of two functions an teir erivatives. Wic are wic?
24 Matc em up! Here are graps of two functions an teir erivatives. Wic are wic? f (x) g(x) g (x) f (x)
25 A little more notation Back to wat lim 0 f (x + ) f (x) means:
26 A little more notation Back to wat lim 0 f (x + ) f (x) means: Rename = x an f (x + ) f (x) = y ( means cange )
27 A little more notation Back to wat lim 0 f (x + ) f (x) means: Rename = x an f (x + ) f (x) = y ( means cange ) Δy Δx So m = f (x+) f (x) = y x.
28 A little more notation Back to wat lim 0 f (x + ) f (x) means: Rename = x an f (x + ) f (x) = y ( means cange ) Δy Δx So m = f (x+) f (x) = y x.
29 A little more notation Back to wat lim 0 f (x + ) f (x) means: Rename = x an f (x + ) f (x) = y ( means cange ) Δy Δx So m = f (x+) f (x) = y x.
30 A little more notation Back to wat lim 0 f (x + ) f (x) means: Rename = x an f (x + ) f (x) = y ( means cange ) Δx Δy So m = f (x+) f (x) = y x.
31 A little more notation Back to wat lim 0 f (x + ) f (x) means: Rename = x an f (x + ) f (x) = y ( means cange ) Δx Δy So m = f (x+) f (x) = y x.
32 A little more notation Back to wat lim 0 f (x + ) f (x) means: Rename = x an f (x + ) f (x) = y ( means cange ) Δy Δx So m = f (x+) f (x) = y x.
33 A little more notation Back to wat lim 0 f (x + ) f (x) means: Rename = x an f (x + ) f (x) = y ( means cange ) As 0, x an y get infinitely small. Δy Δx So m = f (x+) f (x) = y x.
34 A little more notation Back to wat lim 0 f (x + ) f (x) means: Rename = x an f (x + ) f (x) = y ( means cange ) As 0, x an y get infinitely small. x x y y x y So m = f (x+) f (x) = y x. y y x x infinitesimals
35 Leibniz notation One way to write te erivative of f (x) versus x is f (x). Anoter way to write it is f (x) = f x = x f (x).
36 Leibniz notation One way to write te erivative of f (x) versus x is f (x). Anoter way to write it is f (x) = f x = x f (x). Derivatives at a point: f (a) means te erivative of f (x) evaluate at a. Anoter way to write it is f (a) = f x = x=a x x=a f (x)
37 Leibniz notation One way to write te erivative of f (x) versus x is f (x). Anoter way to write it is f (x) = f x = x f (x). Derivatives at a point: f (a) means te erivative of f (x) evaluate at a. Anoter way to write it is f (a) = f x = x=a x x=a f (x) Example: We can write te erivative of x 2 as x x 2 an te erivative of x 2 at x = 5 as x x 2 x=5
38 Go to work: builing our first erivative rule. Example 1: Wat is te erivative of x 2?
39 Go to work: builing our first erivative rule. Example 1: Wat is te erivative of x 2? x x 2 f (x + ) f (x) (x + ) 2 (x) 2 0 0
40 Go to work: builing our first erivative rule. Example 1: Wat is te erivative of x 2? x x 2 f (x + ) f (x) (x + ) 2 (x) x 2 + 2x + 2 x 2 0
41 Go to work: builing our first erivative rule. Example 1: Wat is te erivative of x 2? x x 2 f (x + ) f (x) (x + ) 2 (x) x 2 + 2x + 2 x 2 0 2x + 2
42 Go to work: builing our first erivative rule. Example 1: Wat is te erivative of x 2? x x 2 f (x + ) f (x) (x + ) 2 (x) x 2 + 2x + 2 x 2 0 2x + 0 2x + 2
43 Go to work: builing our first erivative rule. Example 1: Wat is te erivative of x 2? x x 2 f (x + ) f (x) (x + ) 2 (x) x 2 + 2x + 2 x 2 0 2x + = 2x 0 2x + 2
44 Go to work: builing our first erivative rule. Example 1: Wat is te erivative of x 2? x x 2 f (x + ) f (x) (x + ) 2 (x) x 2 + 2x + 2 x 2 0 2x + 2 2x + = 2x (so 0 x x 2 = 2 5) x=5
45 Go to work: builing our first erivative rule. Example 1: Wat is te erivative of x 2? x x 2 f (x + ) f (x) (x + ) 2 (x) x 2 + 2x + 2 x 2 0 2x + 2 2x + = 2x (so 0 x x 2 = 2 5) x=5 By taking limits, fill in te rest of te table: f (x) 1 x x 2 x 3 1 x 1 x 2 x 3 x f (x) 2x Hints: For 1 x 2, fin a common enominator, an ten expan. For 3 x, try multiplying an iviing by ( 3 x + ) 2 + ( 3 x + )( 3 x) + ( 3 x) 2.
46 Go to work: builing our first erivative rule. Example 1: Wat is te erivative of x 2? x x 2 f (x + ) f (x) (x + ) 2 (x) x 2 + 2x + 2 x 2 0 2x + 2 2x + = 2x (so 0 x x 2 = 2 5) x=5 By taking limits, fill in te rest of te table: f (x) 1 x x 2 x 3 1 x 1 x 2 x 3 x f (x) 0 1 2x 3x 2 1 x 2 2 x x 1 3( 3 x) 2 Hints: For 1 x 2, fin a common enominator, an ten expan. For 3 x, try multiplying an iviing by ( 3 x + ) 2 + ( 3 x + )( 3 x) + ( 3 x) 2.
47 f (x) f (x) x 0 = 1 0 x 1 = x 1 x 2 2x x 3 3x 2 1 x = x 1 x 2 1 = x 2 2x 3 x 2 x = x 1/2 (1/2)x 1/2 3 x = x 1/3 (1/3)x 2/3
48 f (x) f (x) x 0 = 1 0 x 1 = x 1 x 2 2x x 3 3x 2 1 x = x 1 x 2 1 = x 2 2x 3 x 2 x = x 1/2 (1/2)x 1/2 3 x = x 1/3 (1/3)x 2/3 Power rule: x x a = ax a 1
49 Use te power rule to take consecutive erivatives of x 5/2 : x 5/2 x x x x
50 Use te power rule to take consecutive erivatives of x 5/2 : x 5/2 x 5 2 x 3/2 x x x
51 Use te power rule to take consecutive erivatives of x 5/2 : x 5/2 x 5 2 x 3/2 x x 1/2 x x 1/2 x ( 1 2) x 3/2
52 Use te power rule to take consecutive erivatives of x 5/2 : x 5/2 x 5 2 x 3/2 1 st erivative x x 1/2 2 n erivative x x 1/2 3 r erivative x ( 1 2) x 3/2 4 t erivative
53 Use te power rule to take consecutive erivatives of x 5/2 : x 5/2 x 5 2 x 3/2 1 st erivative = f (x) = x x 2 x x 1/2 2 n erivative = f (x) = 2 x 2 x 2 x x 1/2 3 r erivative = f (3) (x) = 3 x 3 x 2 x ( 1 2) x 3/2 4 t erivative = f (4) (x) = 4 x 4 x 2
54 Use te power rule to take consecutive erivatives of x 5/2 : x 5/2 x 5 2 x 3/2 1 st erivative = f (x) = x x 2 x x 1/2 2 n erivative = f (x) = 2 x 2 x 2 x x 1/2 3 r erivative = f (3) (x) = 3 x 3 x 2 x ( 1 2) x 3/2 4 t erivative = f (4) (x) = 4 x 4 x 2 Definition: Te n t erivative of f (x) is x x... f (x) = n }{{ x} x n f (x) = f (n) (x). n
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 informationWarmup quick expansions
Pascal s triangle: 1 1 1 1 2 1 Warmup quick expansions To build Pascal s triangle: Start with 1 s on the end. Add two numbers above to get a new entry. For example, the circled 3 is the sum of the 1 and
More information. 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 informationf 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
More informationFunction 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 information0.1 Differentiation Rules
0.1 Differentiation Rules From our previous work we ve seen tat it can be quite a task to calculate te erivative of an arbitrary function. Just working wit a secon-orer polynomial tings get pretty complicate
More informationDifferential Calculus: Differentiation (First Principles, Rules) and Sketching Graphs (Grade 12) *
OpenStax-CNX moule: m39313 1 Differential Calculus: Differentiation (First Principles, Rules) an Sketcing Graps (Grae 12) * Free Hig Scool Science Texts Project Tis work is prouce by OpenStax-CNX an license
More informationPolynomial 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
More information2.4 Exponential Functions and Derivatives (Sct of text)
2.4 Exponential Functions an Derivatives (Sct. 2.4 2.6 of text) 2.4. Exponential Functions Definition 2.4.. Let a>0 be a real number ifferent tan. Anexponential function as te form f(x) =a x. Teorem 2.4.2
More informationdoes NOT exist. WHAT IF THE NUMBER X APPROACHES CANNOT BE PLUGGED INTO F(X)??????
MATH 000 Miterm Review.3 Te it of a function f ( ) L Tis means tat in a given function, f(), as APPROACHES c, a constant, it will equal te value L. Tis is c only true if f( ) f( ) L. Tat means if te verticle
More informationThe 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 informationMAT01A1: Differentiation of Polynomials & Exponential Functions + the Product & Quotient Rules
MAT01A1: Differentiation of Polynomials & Exponential Functions + te Prouct & Quotient Rules Dr Craig 22 Marc 2017 Semester Test 1 Scripts will be available for collection from Tursay morning. For marking
More informationThe 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
More informationUsing the definition of the derivative of a function is quite tedious. f (x + h) f (x)
Derivative Rules Using te efinition of te erivative of a function is quite teious. Let s prove some sortcuts tat we can use. Recall tat te efinition of erivative is: Given any number x for wic te limit
More informationf(x + h) f(x) f (x) = lim
Introuction 4.3 Some Very Basic Differentiation Formulas If a ifferentiable function f is quite simple, ten it is possible to fin f by using te efinition of erivative irectly: f () 0 f( + ) f() However,
More informationSection 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(a 1 m. a n m = < a 1/N n
Notes on a an log a Mat 9 Fall 2004 Here is an approac to te eponential an logaritmic functions wic avois any use of integral calculus We use witout proof te eistence of certain limits an assume tat certain
More information1. 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 informationSection 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 informationBob 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
More informationRules of Differentiation
LECTURE 2 Rules of Differentiation At te en of Capter 2, we finally arrive at te following efinition of te erivative of a function f f x + f x x := x 0 oing so only after an extene iscussion as wat te
More informationLesson 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 information160 Chapter 3: Differentiation
3. Differentiation Rules 159 3. Differentiation Rules Tis section introuces a few rules tat allow us to ifferentiate a great variety of functions. By proving tese rules ere, we can ifferentiate functions
More informationChapter 3 Definitions and Theorems
Chapter 3 Definitions an Theorems (from 3.1) Definition of Tangent Line with slope of m If f is efine on an open interval containing c an the limit Δy lim Δx 0 Δx = lim f (c + Δx) f (c) = m Δx 0 Δx exists,
More informationContinuity 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 informationDifferentiation 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
More informationTangent 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 informationMAT1A01: Differentiation of Polynomials & Exponential Functions + the Product & Quotient Rules
MAT1A01: Differentiation of Polynomials & Exponential Functions + te Prouct & Quotient Rules Dr Craig 22 Marc 2016 Semester Test 1 Results ave been publise on Blackboar uner My Graes. Scripts will be available
More informationMATH 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 informationIn 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,
More information158 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 informationWe name Functions f (x) or g(x) etc.
Section 2 1B: Function Notation Bot of te equations y 2x +1 and y 3x 2 are functions. It is common to ave two or more functions in terms of x in te same problem. If I ask you wat is te value for y if x
More informationSection 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
More informationDEFINITION 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 informationFunction 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 informationExam 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 informationSECTION 2.1 BASIC CALCULUS REVIEW
Tis capter covers just te very basics of wat you will nee moving forwar onto te subsequent capters. Tis is a summary capter, an will not cover te concepts in-ept. If you ve never seen calculus before,
More informationCalculus 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 informationIntroduction 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. h I B. Average velocity can be interpreted as the slope of a tangent line. I C. The difference quotient program finds the exact value of f ( a)
Capter Review Packet (questions - ) KEY. In eac case determine if te information or statement is correct (C) or incorrect (I). If it is incorrect, include te correction. f ( a ) f ( a) I A. represents
More informationLesson 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 informationDifferentiation Rules. Oct
Differentiation Rules Oct 10 2011 Differentiability versus Continuity Theorem If f (a) exists, then f is continuous at a. A function whose erivative exists at every point of an interval is continuous an
More informationSection 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 information2.2 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 information1 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 information1. 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 informationLIMITS 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 informationChapter 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 information1 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
More informationDerivatives. 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 informationPrecalculus 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.
More informationThis file is /conf/snippets/setheader.pg you can use it as a model for creating files which introduce each problem set.
Yanimov Almog WeBWorK assignment number Sections 3. 3.2 is ue : 08/3/207 at 03:2pm CDT. Te (* replace wit url for te course ome page *) for te course contains te syllabus, graing policy an oter information.
More informationREVIEW 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
More informationMathematics 123.3: Solutions to Lab Assignment #5
Matematics 3.3: Solutions to Lab Assignment #5 Find te derivative of te given function using te definition of derivative. State te domain of te function and te domain of its derivative..: f(x) 6 x Solution:
More informationMAT 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 informationA: Derivatives of Circular Functions. ( x) The central angle measures one radian. Arc Length of r
4: Derivatives of Circular Functions an Relate Rates Before we begin, remember tat we will (almost) always work in raians. Raians on't ivie te circle into parts; tey measure te size of te central angle
More information2.11 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
More informationDifferentiation Rules c 2002 Donald Kreider and Dwight Lahr
Dierentiation Rules c 00 Donal Kreier an Dwigt Lar Te Power Rule is an example o a ierentiation rule. For unctions o te orm x r, were r is a constant real number, we can simply write own te erivative rater
More information10 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
More informationMVT 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
More informationPreface. 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
More informationHigher 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.
More informationDerivative 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 informationDifferential 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= 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 informationMathematics 105 Calculus I. Exam 1. February 13, Solution Guide
Matematics 05 Calculus I Exam February, 009 Your Name: Solution Guide Tere are 6 total problems in tis exam. On eac problem, you must sow all your work, or oterwise torougly explain your conclusions. Tere
More informationDerivatives. 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
More informationAverage 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 informationMain 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
More informationLines, 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 informationMA119-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 informationCalculus 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 informationSection 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
More informationDerivatives 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
More informationlim 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 informationSection 3.1: Derivatives of Polynomials and Exponential Functions
Section 3.1: Derivatives of Polynomials and Exponential Functions In previous sections we developed te concept of te derivative and derivative function. Te only issue wit our definition owever is tat it
More informationDerivatives of trigonometric functions
Derivatives of trigonometric functions 2 October 207 Introuction Toay we will ten iscuss te erivates of te si stanar trigonometric functions. Of tese, te most important are sine an cosine; te erivatives
More informationTHE 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 informationKey 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 informationMaterial 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
More information2.1 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
More informationMath 102 TEST CHAPTERS 3 & 4 Solutions & Comments Fall 2006
Mat 102 TEST CHAPTERS 3 & 4 Solutions & Comments Fall 2006 f(x+) f(x) 10 1. For f(x) = x 2 + 2x 5, find ))))))))) and simplify completely. NOTE: **f(x+) is NOT f(x)+! f(x+) f(x) (x+) 2 + 2(x+) 5 ( x 2
More informationExponentials 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
More informationLecture 4 : General Logarithms and Exponentials. a x = e x ln a, a > 0.
For a > 0 an x any real number, we efine Lecture 4 : General Logarithms an Exponentials. a x = e x ln a, a > 0. The function a x is calle the exponential function with base a. Note that ln(a x ) = x ln
More informationExam 2 Review Solutions
Exam Review Solutions 1. True or False, an explain: (a) There exists a function f with continuous secon partial erivatives such that f x (x, y) = x + y f y = x y False. If the function has continuous secon
More information1 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
More informationMAT 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
More informationMath 212-Lecture 9. For a single-variable function z = f(x), the derivative is f (x) = lim h 0
3.4: Partial Derivatives Definition Mat 22-Lecture 9 For a single-variable function z = f(x), te derivative is f (x) = lim 0 f(x+) f(x). For a function z = f(x, y) of two variables, to define te derivatives,
More information1. 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
More informationSECTION 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
More informationSome 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 informationPractice 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
More informationExponential and logarithmic functions (pp ) () Supplement October 14, / 1. a and b positive real numbers and x and y real numbers.
MA123, Supplement Exponential and logaritmic functions pp. 315-319) Capter s Goal: Review te properties of exponential and logaritmic functions. Learn ow to differentiate exponential and logaritmic functions.
More information1 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
More informationGradient 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) )
More informationName: Answer Key No calculators. Show your work! 1. (21 points) All answers should either be,, a (finite) real number, or DNE ( does not exist ).
Mat - Final Exam August 3 rd, Name: Answer Key No calculators. Sow your work!. points) All answers sould eiter be,, a finite) real number, or DNE does not exist ). a) Use te grap of te function to evaluate
More information5. (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
More informationName: 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
More informationMath 242: Principles of Analysis Fall 2016 Homework 7 Part B Solutions
Mat 22: Principles of Analysis Fall 206 Homework 7 Part B Solutions. Sow tat f(x) = x 2 is not uniformly continuous on R. Solution. Te equation is equivalent to f(x) = 0 were f(x) = x 2 sin(x) 3. Since
More informationMath 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