Chapter 3: Three faces of the derivative. Overview
|
|
|
- Conrad Newton
- 5 years ago
- Views:
Transcription
1 Overview We alread saw an algebraic wa of thinking about a derivative. Geometric: zooming into a line Analtic: continuit and rational functions Computational: approimations with computers
2 3. The geometric view: zooming into the graph of a function Zooming In For a smooth function, if we zoom in at a point, we see a line: In this eample, the slope of our zoomed-in line looks to be about: 2
3 3. The geometric view: zooming into the graph of a function Recall The secant line to the curve = f () through points R and Q is a line that passes through R and Q. We call the slope of the secant line the average rate of change of f () from R to Q. R Q P secant line tangent line Definition The straight line that we see when we zoom into the graph of a smooth function at some point P is called the tangent line at P. The slope of the tangent line is the instantaneous rate of change (derivative)
4 3. The geometric view: zooming into the graph of a function On the graph below, draw the secant line to the curve through points P and Q. On the graph below, draw the tangent line to the curve at point P. P Q P
5 3. The geometric view: zooming into the graph of a function Derivatives of Familiar Functions The derivative of the function f () = A is: (a) 0 (b) (c) A (d) (e) A 2 The derivative of the function f () = A is: (a) 0 (b) (c) A (d) (e) A 3 The derivative of the function f () = A + is: (a) 0 (b) (c) A (d) (e) A
6 3. The geometric view: zooming into the graph of a function km from home 6 3 = s(t) 0 8:00 8:07 8:30 It takes me half an hour to bike 6 km. So, 2 kph represents the: A. secant line to = s(t) from t = 8 : 00 to t = 8 : 30 t time B. slope of the secant line to = s(t) from t = 8 : 00 to t = 8 : 30 C. tangent line to = s(t) at t = 8 : 30t = 8 : 25 D. slope of the tangent line to = s(t) at t = 8 : 30
7 3. The geometric view: zooming into the graph of a function km from home 6 3 = s(t) 0 8:00 8:07 8:30 t time When I pedal up Crescent Rd at 8:25, the speedometer on m bike tells me I m going 5kph. This corresponds to the: A. secant line to = s(t) from t = 8 : 00 to t = 8 : 25 B. slope of the secant line to = s(t) from t = 8 : 00 to t = 8 : 25 C. tangent line to = s(t) at t = 8 : 25 D. slope of the tangent line to = s(t) att = 8 : 25
8 Photo credit: Piaba 3. The geometric view: zooming into the graph of a function Zooming in on functions that aren t smooth For a function with a cusp or a discontinuit, even though we zoom in ver closel, we don t see simpl a single straight line. Cusp: Discontinuit:
9 3. The geometric view: zooming into the graph of a function Sketch a derivative Sketch the derivative of the function shown. = f ()
10 3. The geometric view: zooming into the graph of a function Sketch a derivative Sketch the derivative of the function shown. = f () = f ()
11 3. The geometric view: zooming into the graph of a function Sketch a derivative Sketch the derivative of the function shown. = f ()
12 3. The geometric view: zooming into the graph of a function Sketch a derivative Sketch the derivative of the function shown. = f () = f ()
13 3. The geometric view: zooming into the graph of a function Sketch a derivative Sketch the derivative of the function shown. = f ()
14 3. The geometric view: zooming into the graph of a function Sketch a derivative Sketch the derivative of the function shown. = f () = f ()
15 3. The geometric view: zooming into the graph of a function Sketch a derivative Sketch the approimate behaviour of the function, then use our sketch of the function to sketch its derivative. f () =
16 3. The geometric view: zooming into the graph of a function Kinesin Kinesin: link Kinesin transports vesicles along microtubules, which have a plus and minus end Kinesin onl travels towards the plus end Kinesin can hop on and off microtubules + + More eplanation of Kinesin s walking: link
17 3. The geometric view: zooming into the graph of a function Kinesin Below is a sketch of the position of a kinesin molecule over time, as it travels around a network of microtubules of fied position. a b c t Sketch the velocit of the kinesin molecule over the same time period. 2 Describe what happened to the kinesin molecule.
18 3. The geometric view: zooming into the graph of a function Kinesin Below is a sketch of the position of a kinesin molecule over time, as it travels around a network of microtubules of fied position. velocit a b c t (0, a) The molecule travels along a microtubule at a velocit of in the positive direction. (a, b) Then, it falls off or gets stuck. (b, c) Then it travels on along a mictorubule with the opposite orientation, so it moves in the negative direction. t > c Finall, it hops onto another microtubule (possibl the same as the first) and travels in the positive direction.
19 3.2 The analtic view: calculating the derivative Overview Eplain the definition of a continuous function. Identif functions with various tpes of discontinuities. Evaluate simple limits of rational functions. Calculate the derivative of a simple function using the definition of the derivative
20 3.2 The analtic view: calculating the derivative Intuitive Continuit Intuitivel, we think of a point on a function as continuous if it s connected to the points around it. Jump Discontinuit
21 3.2 The analtic view: calculating the derivative Intuitive Continuit Intuitivel, we think of a point on a function as continuous if it s connected to the points around it. Removable Discontinuit
22 3.2 The analtic view: calculating the derivative Intuitive Continuit Intuitivel, we think of a point on a function as continuous if it s connected to the points around it. Blow-Up (Infinite) Discontinuit If a graph has a discontinuit at a point, it has no tangent line at that point (and so no derivative at that point).
23 3.2 The analtic view: calculating the derivative Intuition Failing { sin ( ) f () = if 0 0 if = 0
24 3.2 The analtic view: calculating the derivative Intuition Failing { sin ( ) f () = if 0 0 if = 0
25 3.2 The analtic view: calculating the derivative Intuition Failing f () = { 3 if is rational 3 if is irrational
26 3.2 The analtic view: calculating the derivative A More Rigorous Definition Definition A function f () is continuous at a point a in its domain if lim f () = f (a) a f (a) must eist As approaches a, there are no surprises. Jump: Limit doesn t eist (left and right don t match)
27 3.2 The analtic view: calculating the derivative Continuit Proofs Decide whether f () = Justif our result. is continuous or discontinuous at = 2. It is discontinuous, because 2 is not in the domain of f (). That is, f (2) does not eist. Eample : 2 4 if 2 For which value(s) of a is f () = 2 a if = 2 Prove our result. continuous at = 2?
28 3.2 The analtic view: calculating the derivative Rational Functions with Holes Eample 2: Let f () = ( + )( )( 2) ( )( 2) Note the domain of f () does not include = or = 2. Evaluate the following: (a) lim 0 f () (b) lim f () (c) lim 2 f () (d) lim 3 f () 2
29 3.2 The analtic view: calculating the derivative Using limits to evaluate derivatives Evaluate the derivative of f () = using the definition of the derivative, f () = lim h 0 f ( + h) f () h f f ( + h) f () + h () = lim = lim h 0 h h 0 h ( ) + h + h + = lim h 0 h + h + = lim h 0 ( + h) h( + h + ) = lim h 0 = lim h 0 ( + h + ) = = h h( + h + )
30 3.2 The analtic view: calculating the derivative Using limits to evaluate derivatives Eample 3: Find the derivatives of the following functions using the definition of a derivative: (a) f () = (b) g() = f () = lim h 0 f ( + h) f () h
31 3.2 The analtic view: calculating the derivative Concept Check True or False: If f () is not defined at =, then f () does not eist. lim True or False: If lim f () eists, then it s equal to f (). True or False: If f () is continuous at =, then f () eists. True or False: If f () eists, then f () is continuous at =.
32 3.2 The analtic view: calculating the derivative Using Graphs Eample 4: Approimate sin lim 0 using the graph below.
33 3.3 The computational view: software to the rescue! Overview. Use software to numericall compute an approimation to the derivative. 2 Eplain that the approimation replaces a (true) tangent line with an (approimating) secant line. 3 Eplain using words how the derivative shape is connected with the shape of the original function.
34 3.3 The computational view: software to the rescue! Idea: Approimate Tangent Line with Secant Line Derivative (slope of tangent line) f ( + h) f () lim h 0 h Approimate derivative Slope of secant line f ( + h) f () h
35 3.3 The computational view: software to the rescue! Approimate Derivative f () = f f ( + h) f () () = lim = 320 h 0 h h f( + h) f() h spreadsheet link
36 3.3 The computational view: software to the rescue! Approimate Derivative f () = f f ( + h0.) f () () h0. Using ver small values of h can get us an approimation of the derivative at a particular point. We can also use a spreadsheet to guess the derivative at a number of different points, using tin secant lines
37 3.3 The computational view: software to the rescue! Approimate derivative at man points Interval: [, 4] Small h: sa h = 0. f () = spreadsheet link
38 3.3 The computational view: software to the rescue! Predator-Pre Interactions: Holling Predator Response predation rate P() = pre densit Tpe III: I can hardl find the pre when the pre densit is low, but I also get satiated at high pre densit. (a) Label the asmptote on the -ais. (b) Label the half-ma activation level on the -ais. (c) At what pre densit is the predation rate changing the fastest? That is, at which densit is each additional pre animal responsible for the
Limits 4: Continuity
Limits 4: Continuit 55 Limits 4: Continuit Model : Continuit I. II. III. IV. z V. VI. z a VII. VIII. IX. Construct Your Understanding Questions (to do in class). Which is the correct value of f (a) in
Copyright PreCalculusCoach.com
Continuit, End Behavior, and Limits Assignment Determine whether each function is continuous at the given -values. Justif using the continuit test. If discontinuous, identif the tpe of discontinuit as
Continuity, End Behavior, and Limits. Unit 1 Lesson 3
Unit Lesson 3 Students will be able to: Interpret ke features of graphs and tables in terms of the quantities, and sketch graphs showing ke features given a verbal description of the relationship. Ke Vocabular:
1.2 Functions and Their Properties PreCalculus
1. Functions and Their Properties PreCalculus 1. FUNCTIONS AND THEIR PROPERTIES Learning Targets for 1. 1. Determine whether a set of numbers or a graph is a function. Find the domain of a function given
2.5 CONTINUITY. a x. Notice that Definition l implicitly requires three things if f is continuous at a:
SECTION.5 CONTINUITY 9.5 CONTINUITY We noticed in Section.3 that the it of a function as approaches a can often be found simpl b calculating the value of the function at a. Functions with this propert
Slopes and Rates of Change
Slopes and Rates of Change If a particle is moving in a straight line at a constant velocity, then the graph of the function of distance versus time is as follows s s = f(t) t s s t t = average velocity
2.1 Rates of Change and Limits AP Calculus
. Rates of Change and Limits AP Calculus. RATES OF CHANGE AND LIMITS Limits Limits are what separate Calculus from pre calculus. Using a it is also the foundational principle behind the two most important
6.4 graphs OF logarithmic FUnCTIOnS
SECTION 6. graphs of logarithmic functions 9 9 learning ObjeCTIveS In this section, ou will: Identif the domain of a logarithmic function. Graph logarithmic functions. 6. graphs OF logarithmic FUnCTIOnS
4.3 Mean-Value Theorem and Monotonicity
.3 Mean-Value Theorem and Monotonicit 1. Mean Value Theorem Theorem: Suppose that f is continuous on the interval a, b and differentiable on the interval a, b. Then there eists a number c in a, b such
1.2 Functions and Their Properties PreCalculus
1. Functions and Their Properties PreCalculus 1. FUNCTIONS AND THEIR PROPERTIES Learning Targets for 1. 1. Determine whether a set of numbers or a graph is a function. Find the domain of a function given
Derivatives 2: The Derivative at a Point
Derivatives 2: The Derivative at a Point 69 Derivatives 2: The Derivative at a Point Model 1: Review of Velocit In the previous activit we eplored position functions (distance versus time) and learned
Math Exam 1a. c) lim tan( 3x. 2) Calculate the derivatives of the following. DON'T SIMPLIFY! d) s = t t 3t
Math 111 - Eam 1a 1) Evaluate the following limits: 7 3 1 4 36 a) lim b) lim 5 1 3 6 + 4 c) lim tan( 3 ) + d) lim ( ) 100 1+ h 1 h 0 h ) Calculate the derivatives of the following. DON'T SIMPLIFY! a) y
Methods for Advanced Mathematics (C3) Coursework Numerical Methods
Woodhouse College 0 Page Introduction... 3 Terminolog... 3 Activit... 4 Wh use numerical methods?... Change of sign... Activit... 6 Interval Bisection... 7 Decimal Search... 8 Coursework Requirements on
Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Page 119 Mark Sparks 2012
Unit # Understanding the Derivative Homework Packet f ( h) f ( Find lim for each of the functions below. Then, find the equation of the tangent line to h 0 h the graph of f( at the given value of. 1. f
2.1 How Do We Measure Speed? Student Notes HH6ed. Time (sec) Position (m)
2.1 How Do We Measure Speed? Student Notes HH6ed Part I: Using a table of values for a position function The table below represents the position of an object as a function of time. Use the table to answer
1. d = 1. or Use Only in Pilot Program F Review Exercises 131
or Use Onl in Pilot Program F 0 0 Review Eercises. Limit proof Suppose f is defined for all values of near a, ecept possibl at a. Assume for an integer N 7 0, there is another integer M 7 0 such that f
Set 3: Limits of functions:
Set 3: Limits of functions: A. The intuitive approach (.): 1. Watch the video at: https://www.khanacademy.org/math/differential-calculus/it-basics-dc/formal-definition-of-its-dc/v/itintuition-review. 3.
Quick Review 4.1 (For help, go to Sections 1.2, 2.1, 3.5, and 3.6.)
Section 4. Etreme Values of Functions 93 EXPLORATION Finding Etreme Values Let f,.. Determine graphicall the etreme values of f and where the occur. Find f at these values of.. Graph f and f or NDER f,,
Derivatives of Multivariable Functions
Chapter 0 Derivatives of Multivariable Functions 0. Limits Motivating Questions In this section, we strive to understand the ideas generated b the following important questions: What do we mean b the limit
Limits and the derivative function. Limits and the derivative function
The Velocity Problem A particle is moving in a straight line. t is the time that has passed from the start of motion (which corresponds to t = 0) s(t) is the distance from the particle to the initial position
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. B) 94 C) ) A) 1 2
Chapter Calculus MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Find the average rate of change of the function over the given interval. ) = 73-5
A function from a set D to a set R is a rule that assigns a unique element in R to each element in D.
1.2 Functions and Their Properties PreCalculus 1.2 FUNCTIONS AND THEIR PROPERTIES Learning Targets for 1.2 1. Determine whether a set of numbers or a graph is a function 2. Find the domain of a function
Infinite Limits. Let f be the function given by. f x 3 x 2.
0_005.qd //0 :07 PM Page 8 SECTION.5 Infinite Limits 8, as Section.5, as + f() = f increases and decreases without bound as approaches. Figure.9 Infinite Limits Determine infinite its from the left and
In everyday speech, a continuous. Limits and Continuity. Critical Thinking Exercises
062 Chapter Introduction to Calculus Critical Thinking Eercises Make Sense? In Eercises 74 77, determine whether each statement makes sense or does not make sense, and eplain our reasoning. 74. I evaluated
Differentiation and applications
FS O PA G E PR O U N C O R R EC TE D Differentiation and applications. Kick off with CAS. Limits, continuit and differentiabilit. Derivatives of power functions.4 C oordinate geometr applications of differentiation.5
Exponential and Logarithmic Functions
Eponential and Logarithmic Functions 6 Figure Electron micrograph of E. Coli bacteria (credit: Mattosaurus, Wikimedia Commons) CHAPTER OUTLINE 6. Eponential Functions 6. Logarithmic Properties 6. Graphs
Understanding the Derivative
Chapter 1 Understanding the Derivative 1.1 How do we measure velocity? Motivating Questions In this section, we strive to understand the ideas generated by the following important questions: How is the
Finding Limits Graphically and Numerically. An Introduction to Limits
8 CHAPTER Limits and Their Properties Section Finding Limits Graphicall and Numericall Estimate a it using a numerical or graphical approach Learn different was that a it can fail to eist Stud and use
Limits and Their Properties
Chapter 1 Limits and Their Properties Course Number Section 1.1 A Preview of Calculus Objective: In this lesson you learned how calculus compares with precalculus. I. What is Calculus? (Pages 42 44) Calculus
Tangent Lines. Limits 1
Limits Tangent Lines The concept of the tangent line to a circle dates back at least to the earl das of Greek geometr, that is, at least 5 ears. The tangent line to a circle with centre O at a point A
Lecture 3 (Limits and Derivatives)
Lecture 3 (Limits and Derivatives) Continuity In the previous lecture we saw that very often the limit of a function as is just. When this is the case we say that is continuous at a. Definition: A function
Calculus I Practice Test Problems for Chapter 2 Page 1 of 7
Calculus I Practice Test Problems for Chapter Page of 7 This is a set of practice test problems for Chapter This is in no way an inclusive set of problems there can be other types of problems on the actual
3 Polynomial and Rational Functions
3 Polnomial and Rational Functions 3.1 Quadratic Functions and Models 3.2 Polnomial Functions and Their Graphs 3.3 Dividing Polnomials 3.4 Real Zeros of Polnomials 3.5 Comple Zeros and the Fundamental
Unit 10 - Graphing Quadratic Functions
Unit - Graphing Quadratic Functions PREREQUISITE SKILLS: students should be able to add, subtract and multipl polnomials students should be able to factor polnomials students should be able to identif
Pre-Calculus Module 4
Pre-Calculus Module 4 4 th Nine Weeks Table of Contents Precalculus Module 4 Unit 9 Rational Functions Rational Functions with Removable Discontinuities (1 5) End Behavior of Rational Functions (6) Rational
All work must be shown in this course for full credit. Unsupported answers may receive NO credit.
AP Calculus. Worksheet All work must be shown in this course for full credit. Unsupported answers ma receive NO credit.. What is the definition of a derivative?. What is the alternative definition of a
One of the most common applications of Calculus involves determining maximum or minimum values.
8 LESSON 5- MAX/MIN APPLICATIONS (OPTIMIZATION) One of the most common applications of Calculus involves determining maimum or minimum values. Procedure:. Choose variables and/or draw a labeled figure..
Test # 1 Review Math MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
Test # 1 Review Math 13 Name MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Find the slope of the curve at the given point P and an equation of the
PACKET Unit 4 Honors ICM Functions and Limits 1
PACKET Unit 4 Honors ICM Functions and Limits 1 Day 1 Homework For each of the rational functions find: a. domain b. -intercept(s) c. y-intercept Graph #8 and #10 with at least 5 EXACT points. 1. f 6.
Section 3.2 The Derivative as a Function Graphing the Derivative
Math 80 www.timetodare.com Derivatives Section 3. The Derivative as a Function Graphing the Derivative ( ) In the previous section we defined the slope of the tangent to a curve with equation y= f ( )
4 The Cartesian Coordinate System- Pictures of Equations
The Cartesian Coordinate Sstem- Pictures of Equations Concepts: The Cartesian Coordinate Sstem Graphs of Equations in Two Variables -intercepts and -intercepts Distance in Two Dimensions and the Pthagorean
2.5. Infinite Limits and Vertical Asymptotes. Infinite Limits
. Infinite Limits and Vertical Asmptotes. Infinite Limits and Vertical Asmptotes In this section we etend the concept of it to infinite its, which are not its as before, but rather an entirel new use of
Eigenvectors and Eigenvalues 1
Ma 2015 page 1 Eigenvectors and Eigenvalues 1 In this handout, we will eplore eigenvectors and eigenvalues. We will begin with an eploration, then provide some direct eplanation and worked eamples, and
1.3 LIMITS AT INFINITY; END BEHAVIOR OF A FUNCTION
. Limits at Infinit; End Behavior of a Function 89. LIMITS AT INFINITY; END BEHAVIOR OF A FUNCTION Up to now we have been concerned with its that describe the behavior of a function f) as approaches some
Math Review Packet #5 Algebra II (Part 2) Notes
SCIE 0, Spring 0 Miller Math Review Packet #5 Algebra II (Part ) Notes Quadratic Functions (cont.) So far, we have onl looked at quadratic functions in which the term is squared. A more general form of
A11.1 Areas under curves
Applications 11.1 Areas under curves A11.1 Areas under curves Before ou start You should be able to: calculate the value of given the value of in algebraic equations of curves calculate the area of a trapezium.
Limits and Continuous Functions. 2.2 Introduction to Limits. We first interpret limits loosely. We write. lim f(x) = L
2 Limits and Continuous Functions 2.2 Introduction to Limits We first interpret limits loosel. We write lim f() = L and sa the limit of f() as approaches c, equals L if we can make the values of f() arbitraril
Finding Limits Graphically and Numerically. An Introduction to Limits
60_00.qd //0 :05 PM Page 8 8 CHAPTER Limits and Their Properties Section. Finding Limits Graphicall and Numericall Estimate a it using a numerical or graphical approach. Learn different was that a it can
x c x c This suggests the following definition.
110 Chapter 1 / Limits and Continuit 1.5 CONTINUITY A thrown baseball cannot vanish at some point and reappear someplace else to continue its motion. Thus, we perceive the path of the ball as an unbroken
Unit 3 NOTES Honors Common Core Math 2 1. Day 1: Properties of Exponents
Unit NOTES Honors Common Core Math Da : Properties of Eponents Warm-Up: Before we begin toda s lesson, how much do ou remember about eponents? Use epanded form to write the rules for the eponents. OBJECTIVE
1.7 Inverse Functions
71_0107.qd 1/7/0 10: AM Page 17 Section 1.7 Inverse Functions 17 1.7 Inverse Functions Inverse Functions Recall from Section 1. that a function can be represented b a set of ordered pairs. For instance,
AP Calculus AB/IB Math SL2 Unit 1: Limits and Continuity. Name:
AP Calculus AB/IB Math SL Unit : Limits and Continuity Name: Block: Date:. A bungee jumper dives from a tower at time t = 0. Her height h (in feet) at time t (in seconds) is given by the graph below. In
2.1 How Do We Measure Speed? Student Notes HH6ed
2.1 How Do We Measure Speed? Student Notes HH6ed Part I: Using a table of values for a position function The table below represents the position of an object as a function of time. Use the table to answer
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 1.2: A Catalog of Functions
Section 1.: A Catalog of Functions As we discussed in the last section, in the sciences, we often tr to find an equation which models some given phenomenon in the real world - for eample, temperature as
5.6 RATIOnAl FUnCTIOnS. Using Arrow notation. learning ObjeCTIveS
CHAPTER PolNomiAl ANd rational functions learning ObjeCTIveS In this section, ou will: Use arrow notation. Solve applied problems involving rational functions. Find the domains of rational functions. Identif
Math Worksheet 1 SHOW ALL OF YOUR WORK! f(x) = (x a) 2 + b. = x 2 + 6x + ( 6 2 )2 ( 6 2 )2 + 7 = (x 2 + 6x + 9) = (x + 3) 2 2
Names Date. Consider the function Math 0550 Worksheet SHOW ALL OF YOUR WORK! f() = + 6 + 7 (a) Complete the square and write the function in the form f() = ( a) + b. f() = + 6 + 7 = + 6 + ( 6 ) ( 6 ) +
Problems for Chapter 3.
Problems for Chapter 3. Let A denote a nonempty set of reals. The complement of A, denoted by A, or A C is the set of all points not in A. We say that belongs to the interior of A, Int A, if there eists
4.3 Worksheet - Derivatives of Inverse Functions
AP Calculus 3.8 Worksheet 4.3 Worksheet - Derivatives of Inverse Functions All work must be shown in this course for full credit. Unsupported answers ma receive NO credit.. What are the following derivatives
Introducing Instantaneous Rate of Change
Introducing Instantaneous Rate of Change The diagram shows a door with an automatic closer. At time t = 0 seconds someone pushes the door. It swings open, slows down, stops, starts closing, then closes
Grade 12 (MCV4UE) AP Calculus Page 1 of 5 Derivative of a Function & Differentiability
Grade 2 (MCV4UE) AP Calculus Page of 5 The Derivative at a Point f ( a h) f ( a) Recall, lim provides the slope of h0 h the tangent to the graph y f ( at the point, f ( a), and the instantaneous rate of
Limits. or Use Only in Pilot Program F The Idea of Limits 2.2 Definitions of Limits 2.3 Techniques for Computing.
Limits or Use Onl in Pilot Program F 03 04. he Idea of Limits. Definitions of Limits.3 echniques for Computing Limits.4 Infinite Limits.5 Limits at Infinit.6 Continuit.7 Precise Definitions of Limits Biologists
AP Calculus AB Ch. 2 Derivatives (Part I) Intro to Derivatives: Definition of the Derivative and the Tangent Line 9/15/14
AP Calculus AB Ch. Derivatives (Part I) Name Intro to Derivatives: Deinition o the Derivative an the Tangent Line 9/15/1 A linear unction has the same slope at all o its points, but non-linear equations
1.6 CONTINUITY OF TRIGONOMETRIC, EXPONENTIAL, AND INVERSE FUNCTIONS
.6 Continuit of Trigonometric, Eponential, and Inverse Functions.6 CONTINUITY OF TRIGONOMETRIC, EXPONENTIAL, AND INVERSE FUNCTIONS In this section we will discuss the continuit properties of trigonometric
Overview. Graphing More Accurately First and second derivatives Critical points Extrema
Overview Graphing More Accuratel First and second derivatives Critical points Etrema 6.1 Overall shape of the graph of a function Up with hope, down with dope, Increasing functions have positive slope
KEY IDEAS. Chapter 1 Function Transformations. 1.1 Horizontal and Vertical Translations Pre-Calculus 12 Student Workbook MHR 1
Chapter Function Transformations. Horizontal and Vertical Translations A translation can move the graph of a function up or down (vertical translation) and right or left (horizontal translation). A translation
NO CALCULATORS: 1. Find A) 1 B) 0 C) D) 2. Find the points of discontinuity of the function y of discontinuity.
AP CALCULUS BC NO CALCULATORS: MIDTERM REVIEW. Find lim 7 7 9. B) C) D). Find the points of discontinuit of the function of discontinuit. 9. For each discontinuit identif the tpe A. Removable discontinuit
4.3 Exercises. local maximum or minimum. The second derivative is. e 1 x 2x 1. f x x 2 e 1 x 1 x 2 e 1 x 2x x 4
SECTION 4.3 HOW DERIVATIVES AFFECT THE SHAPE OF A GRAPH 297 local maimum or minimum. The second derivative is f 2 e 2 e 2 4 e 2 4 Since e and 4, we have f when and when 2 f. So the curve is concave downward
MATH 114 Calculus Notes on Chapter 2 (Limits) (pages 60-? in Stewart)
Still under construction. MATH 114 Calculus Notes on Chapter 2 (Limits) (pages 60-? in Stewart) As seen in A Preview of Calculus, the concept of it underlies the various branches of calculus. Hence we
Review: Limits of Functions - 10/7/16
Review: Limits of Functions - 10/7/16 1 Right and Left Hand Limits Definition 1.0.1 We write lim a f() = L to mean that the function f() approaches L as approaches a from the left. We call this the left
LIMITS AND DERIVATIVES
2 LIMITS AND DERIVATIVES LIMITS AND DERIVATIVES 1. Equation In Section 2.7, we considered the derivative of a function f at a fixed number a: f '( a) lim h 0 f ( a h) f ( a) h In this section, we change
8.1 Exponents and Roots
Section 8. Eponents and Roots 75 8. Eponents and Roots Before defining the net famil of functions, the eponential functions, we will need to discuss eponent notation in detail. As we shall see, eponents
2.1 The Tangent and Velocity Problems
2.1 The Tangent and Velocity Problems Tangents What is a tangent? Tangent lines and Secant lines Estimating slopes from discrete data: Example: 1. A tank holds 1000 gallons of water, which drains from
Chapter 3.5: Rational Functions
Chapter.5: Rational Functions A rational number is a ratio of two integers. A rational function is a quotient of two polynomials. All rational numbers are, therefore, rational functions as well. Let s
8 Differential Calculus 1 Introduction
8 Differential Calculus Introduction The ideas that are the basis for calculus have been with us for a ver long time. Between 5 BC and 5 BC, Greek mathematicians were working on problems that would find
FIRST- AND SECOND-ORDER IVPS The problem given in (1) is also called an nth-order initial-value problem. For example, Solve: Solve:
.2 INITIAL-VALUE PROBLEMS 3.2 INITIAL-VALUE PROBLEMS REVIEW MATERIAL Normal form of a DE Solution of a DE Famil of solutions INTRODUCTION We are often interested in problems in which we seek a solution
11.1 Double Riemann Sums and Double Integrals over Rectangles
Chapter 11 Multiple Integrals 11.1 ouble Riemann Sums and ouble Integrals over Rectangles Motivating Questions In this section, we strive to understand the ideas generated b the following important questions:
UNIT 3. Recall From Unit 2 Rational Functions
UNIT 3 Recall From Unit Rational Functions f() is a rational function if where p() and q() are and. Rational functions often approach for values of. Rational Functions are not graphs There various types
Properties of Limits
33460_003qd //04 :3 PM Page 59 SECTION 3 Evaluating Limits Analticall 59 Section 3 Evaluating Limits Analticall Evaluate a it using properties of its Develop and use a strateg for finding its Evaluate
Announcements. Topics: Homework:
Topics: Announcements - section 2.6 (limits at infinity [skip Precise Definitions (middle of pg. 134 end of section)]) - sections 2.1 and 2.7 (rates of change, the derivative) - section 2.8 (the derivative
3.5 Continuity of a Function One Sided Continuity Intermediate Value Theorem... 23
Chapter 3 Limit and Continuity Contents 3. Definition of Limit 3 3.2 Basic Limit Theorems 8 3.3 One sided Limit 4 3.4 Infinite Limit, Limit at infinity and Asymptotes 5 3.4. Infinite Limit and Vertical
Laurie s Notes. Overview of Section 3.5
Overview of Section.5 Introduction Sstems of linear equations were solved in Algebra using substitution, elimination, and graphing. These same techniques are applied to nonlinear sstems in this lesson.
Derivatives of Multivariable Functions
Chapter 10 Derivatives of Multivariable Functions 10.1 Limits Motivating Questions What do we mean b the limit of a function f of two variables at a point (a, b)? What techniques can we use to show that
2.1 Limits, Rates of Change, and Tangent Lines. Preliminary Questions 1. Average velocity is defined as a ratio of which two quantities?
LIMITS. Limits, Rates of Change, and Tangent Lines Preinar Questions. Average velocit is defined as a ratio of which two quantities? Average velocit is defined as the ratio of distance traveled to time
3.3 Limits and Infinity
Calculus Maimus. Limits Infinity Infinity is not a concrete number, but an abstract idea. It s not a destination, but a really long, never-ending journey. It s one of those mind-warping ideas that is difficult
7.7. Inverse Trigonometric Functions. Defining the Inverses
7.7 Inverse Trigonometric Functions 57 7.7 Inverse Trigonometric Functions Inverse trigonometric functions arise when we want to calculate angles from side measurements in triangles. The also provide useful
2.1 Limits, Rates of Change and Slopes of Tangent Lines
2.1 Limits, Rates of Change and Slopes of Tangent Lines (1) Average rate of change of y f x over an interval x 0,x 1 : f x 1 f x 0 x 1 x 0 Instantaneous rate of change of f x at x x 0 : f x lim 1 f x 0
Slide 1. Slide 2. Slide 3 Remark is a new function derived from called derivative. 2.2 The derivative as a Function
Slide 1 2.2 The derivative as a Function Slide 2 Recall: The derivative of a function number : at a fixed Definition (Derivative of ) For any number, the derivative of is Slide 3 Remark is a new function
Review of elements of Calculus (functions in one variable)
Review of elements of Calculus (functions in one variable) Mainly adapted from the lectures of prof Greg Kelly Hanford High School, Richland Washington http://online.math.uh.edu/houstonact/ https://sites.google.com/site/gkellymath/home/calculuspowerpoints
Chapter One. Chapter One
Chapter One Chapter One CHAPTER ONE Hughes Hallett et al c 005, John Wile & Sons ConcepTests and Answers and Comments for Section.. Which of the following functions has its domain identical with its range?
1.1 Limits & Continuity
1.1 Limits & Continuity What do you see below? We are building the House of Calculus, one side at a time... and we need a solid FOUNDATION. Page 1 of 11 Eample 1: (Calculator) For f ( ) (a) fill in the
CHAPTER 3 Applications of Differentiation
CHAPTER Applications of Differentiation Section. Etrema on an Interval.............. 0 Section. Rolle s Theorem and the Mean Value Theorem. 07 Section. Increasing and Decreasing Functions and the First
Math 123 Summary of Important Algebra & Trigonometry Concepts Chapter 1 & Appendix D, Stewart, Calculus Early Transcendentals
Math Summar of Important Algebra & Trigonometr Concepts Chapter & Appendi D, Stewart, Calculus Earl Transcendentals Function a rule that assigns to each element in a set D eactl one element, called f (
Rolle s Theorem. THEOREM 3 Rolle s Theorem. x x. then there is at least one number c in (a, b) at which ƒ scd = 0.
4.2 The Mean Value Theorem 255 4.2 The Mean Value Theorem f '(c) 0 f() We know that constant functions have zero derivatives, ut could there e a complicated function, with man terms, the derivatives of
Exponential, Logistic, and Logarithmic Functions
CHAPTER 3 Eponential, Logistic, and Logarithmic Functions 3.1 Eponential and Logistic Functions 3.2 Eponential and Logistic Modeling 3.3 Logarithmic Functions and Their Graphs 3.4 Properties of Logarithmic
All work must be shown in this course for full credit. Unsupported answers may receive NO credit.
AP Calculus.4 Worksheet All work must be shown in this course for full credit. Unsupported answers may receive NO credit.. What is a difference quotient?. How do you find the slope of a curve (aka slope
Roberto s Notes on Differential Calculus Chapter 1: Limits and continuity Section 7. Discontinuities. is the tool to use,
Roberto s Notes on Differential Calculus Chapter 1: Limits and continuity Section 7 Discontinuities What you need to know already: The concept and definition of continuity. What you can learn here: The
2.1 Rates of Change and Limits AP Calculus
.1 Rates of Change and Limits AP Calculus.1 RATES OF CHANGE AND LIMITS Limits Limits are what separate Calculus from pre calculus. Using a it is also the foundational principle behind the two most important
6. Graph each of the following functions. What do you notice? What happens when x = 2 on the graph of b?
Pre Calculus Worksheet 1. Da 1 1. The relation described b the set of points {(-,5,0,5,,8,,9 ) ( ) ( ) ( )} is NOT a function. Eplain wh. For questions - 4, use the graph at the right.. Eplain wh the graph
is the intuition: the derivative tells us the change in output y (from f(b)) in response to a change of input x at x = b.
Uses of differentials to estimate errors. Recall the derivative notation df d is the intuition: the derivative tells us the change in output y (from f(b)) in response to a change of input at = b. Eamples.