Differentiability, Computing Derivatives, Trig Review

Size: px
Start display at page:

Download "Differentiability, Computing Derivatives, Trig Review"

Transcription

1 Unit #3 : Differentiability, Computing Derivatives, Trig Review Goals: Determine when a function is ifferentiable at a point Relate the erivative graph to the the graph of an original function Compute erivative functions of powers, exponentials, logarithms, an trig functions. Compute erivatives using the prouct rule, quotient rule, an chain rule for erivatives. Review trigonometric functions an their erivatives.

2 Secants vs. Derivatives - 1 In the previous unit we introuce the efinition of the erivative. In this unit we will use an compute the erivative more efficiently. As a lea in, though, let us review how we arrive at the erivative concept.

3 Secants vs. Derivatives - 2 Interpretations of Secants vs Derivatives From Section 2.4 The slope of a secant line gives the average rate of change of f(x) over some interval x. the average velocity over an interval, if f(t) represents position. the average acceleration over an interval, if f(t) represents velocity. Give the units of the slope of a secant line on the following graphs. v(t) (m/s) height (m) Sales (K units) t (s) travel (km) Price ($)

4 Secants vs. Derivatives - 3 The erivative gives the limit of the average slope as the interval x approaches zero. a formula for slopes for the tangent lines to f(x). the instantaneous rate of change of f(x). the velocity, if f(t) represents position. the acceleration, if f(t) represents velocity. Give the units of the slope of the erivative on the following graphs. v(t) (m/s) height (m) Sales (K units) t (s) travel (km) Price ($)

5 Differentiability - 1 Differentiability Recall the efinition of the erivative. f (x) = x f = f x = lim f x 0 x = lim f(x + h) f(x) h 0 h A function f is ifferentiable at a given point a if it has a erivative at a, or the limit above exists. There is also a graphical interpretation ifferentiability: if the graph has a unique an finite slope at a point. Since the slope in question is automatically the slope of the tangent line, we coul also say that f is ifferentiable at a if its graph has a (non-vertical) tangent at (a, f(a)). For functions of the form y = f(x), we o not consier points with vertical tangent lines to have a real-value erivative, because a vertical line oes not have a finite slope.

6 Differentiability - 2 Here are the ways in which a function can fail to be ifferentiable at a point a: 1. The function is not continuous at a. 2. The function has a corner (or a cusp) at a. 3. The function has a vertical tangent at (a, f(a)). Sketch an example graph of each possible case.

7 Differentiability Example - 1 Example: Investigate the limits, continuity an ifferentiability of f(x) = x at x = 0 graphically.

8 Differentiability Example - 2 Use the efinition of the erivative to confirm your graphical analysis.

9 Differentiability Example - 3 We have seen at a point that a function can have, or fail to have, the following escriptors: continuous; limit exists; is ifferentiable. Put these properties in ecreasing orer of stringency, an sketch relevant illustrations.

10 Differentiability Example - 4 Reminer: Differentiability is Common You will notice that, espite our concern about some functions not being ifferentiable, most of our stanar functions (polynomials, rationals, exponentials, logarithms, roots) are ifferentiable at most points. Therefore we shoul investigate what all these possible erivative/slope values coul tell us.

11 Interpreting the Derivative - 1 Interpreting the Derivative Where f (x) > 0, or the erivative is positive, f(x) is increasing. Where f (x) < 0, or the erivative is negative, f(x) is ecreasing. Where f (x) = 0, or the tangent line to the graph is horizontal, f(x) has a critical point.

12 Interpreting the Derivative - 2 Example: Consier the graph of f(x) shown below. A B C On what intervals is f (x) > 0? D E F G

13 Interpreting the Derivative - 3 A B C Where oes f (x) takes on its largest negative value? D E F G

14 Graphs of the Derivative - 1 Graphs, an Graphs of their Derivatives Example: Consier the same graph again, an the graph of its erivative. Ientify important features that associate the two. B A C D E F G A B C D E F G

15 Graphs of the Derivative - 2 Question: Consier the graph of f(x) shown: Which of the following graphs is the graph of the erivative of f(x)?

16 Graphs of the Derivative A 2 1 B C D

17 Computing Derivatives - Basic Formulas - 1 Computing Derivatives Note: The stanar formulas for erivatives are covere in the Grae 12 Ontario curriculum. While they will be reviewe here, stuents who are not familiar with them shoul begin both textbook reaing an the assignment problems for this unit as soon as possible. Beyon the graphical interpretation of erivatives, there are all the algebraic rules. All of these rules are base on the efinition of the erivative, f (x) = x f = f x = lim x 0 f x = lim h 0 f(x + h) f(x) h However, by fining common patterns in the erivatives of certain families of functions, we can compute erivatives much more quickly than by using the efinition.

18 Computing Derivatives - Basic Formulas - 2 Sums, Powers, an Differences Constant Functions: Power rule: x k = 0 x xp = px p 1 Sums : x f(x) + g(x) = ( ) x f(x) + ( ) x g(x) Differences: x f(x) g(x) = ( ) x f(x) ( ) x g(x) Constant Multiplier: ( ) x [kf(x)] = k x f(x), so long as k is a constant

19 Computing Derivatives - Basic Formulas - 3 Example: Evaluate the following erivatives: ( x 4 + 3x 2) x x ( 2.6 x πx )

20 Computing Derivatives - Basic Formulas - 4 Question: The erivative of 3x 2 1 x 2 is A. 6x x 3 B. 6x x 3 C. 6x 2 1 x 3 D. x x

21 Computing Derivatives - Basic Formulas - 5 Exponentials an Logs e as a base: Other bases: Natural Log: Other Logs: x ex = e x x ax = a x (ln(a)) x ln(x) = 1 x x log a(x) = 1 1 xln(a)

22 Computing Derivatives - Basic Formulas - 6 Example: Evaluate the following erivatives: ( 4 10 x + 10 x 4) x x (ex + log 10 (x)) (Exponential an log erivatives are relatively straightforwar, until we mix in the prouct, quotient, an chain rules.)

23 Computing Derivatives - Prouct an Quotient Rules - 1 Prouct an Quotient Rules Proucts: x f(x) g(x) = f (x)g(x) + f(x)g (x) Quotients: x f(x) g(x) = f (x)g(x) f(x)g (x) (g(x)) 2 Example: ( 4x 2 e x) x Evaluate the following erivatives:

24 Computing Derivatives - Prouct an Quotient Rules - 2 x (x ln(x)) x ) (5 x2 ln(x)

25 Computing Derivatives - Prouct an Quotient Rules - 3 Question: The erivative of 10x x 3 10 x A. ln(10) x x ( 3x 4 ) is: B. 10x ln(10)x 3 10 x (3x 2 ) x 6 C. 10x 1 ln(10) x3 10 x (3x 2 ) x 6 D. ln(10)10 x x x ( 3x 4 )

26 Computing Derivatives - Chain Rule - 1 Chain Rule Neste Functions: x [f(g(x))] = f (g(x)) g (x) Liebnitz form x f g f(g(x)) = g x

27 Computing Derivatives - Chain Rule - 2 Example: x ex2 Evaluate the following erivatives:

28 Computing Derivatives - Chain Rule - 3 x ln(x4 )

29 Computing Derivatives - Chain Rule - 4 x ( 1 ) 1 + x 3

30 Computing Derivatives - Chain Rule - 5 x ( x x)

31 Computing Derivatives - Chain Rule - 6 Question: A. 1 2 e 1 x The erivative of e x is B. e x ( x ) C. 1 2 e x ( 1 x ) D. 1 2 e x ( x )

32 Trigonometry Review - 1 Trigonometry Review In our earlier iscussion of functions, we skippe over the trigonometric functions. We return to them now to iscuss both their properties an their erivative rules. The trigonometric functions are usually efine for stuents first using triangles (recall the mnemonic evice, SOHCAHTOA ).

33 Trigonometry Review - 2 Use the 45/45 an 60/30 triangles to compute the sine an cosine of these common angles.

34 Trigonometry Review - 3 Extening Trigonometric Domains One ifficulty with limiting ourselves to the triangle ratio efinition of the trig functions is that the possible angles are limite to the range θ [0, π 2 ] raians or θ [0, 90] egrees. To remove this limitation, mathematicians extene the efinition of the trigonometric functions to a wier omain via the unit circle. θ

35 Trigonometry Review - 4 How oes the circle efinition lea to the trigonometric ientity sin 2 (θ) + cos 2 (θ) = 1?

36 Trigonometry Review - 5 Show how the circle an triangle efinitions efine the same values in the first quarant of the unit circle. It is useful to unerstan both efinitions of trig functions (circle an triangle) as sometimes one is more helpful than the other for a particular task.

37 Sine an Cosine as Functions - 1 Sine an Cosine as Oscillating Functions Despite the geometric source of the trigonometric functions, they are use more commonly in biology an many other sciences as because their perioicity an oscillatory shapes. For many cyclic behaviours in nature, trigonometric functions are a natural first choice for moeling.

38 Sine an Cosine as Functions - 2 Question iagrams? The graph of y = cos(x) is shown in which of the following A B C D Show the amplitue an the average on the correct graph.

39 Sine an Cosine as Functions - 3 Perio an Phase How can you fin the perio of the function cos(ax)?

40 Sine an Cosine as Functions - 4 How can you reliably etermine where the function cos(ax + B) starts on the graph? (For a cosine graph, where the start represents a maximum, the starting time or x value is sometimes calle the phase of the function.)

41 Sine an Cosine as Functions - 5 Consier the graph of the function y = cos(π(x 1)). following properties of the function: amplitue What are the perio average phase

42 Sine an Cosine as Functions - 6 Sketch the graph on the axes below. Inclue at least one full perio of the function.

43 Non-Constant Amplitues - 1 More complicate amplitues In the form y = A + B cos(cx + D), the B factor sets the amplitue. In many interesting cases, however, that amplitue nee not be constant. Sketch the graph of y = 5, an the graph of y = 5 cos(x) on the axes below.

44 Non-Constant Amplitues - 2 Sketch the graph of y = x, an the graph of y = x cos(πx) on the axes below. Use only x 0

45 Non-Constant Amplitues - 3 Use your intuition to sketch the graph of y = e x cos(πx) on the axes below.

46 Derivatives of Trigonometric Functions - 1 Derivatives of Trigonometric Functions Having covere the graphs an properties of trigonometric functions, we can now review the erivative formulae for those same functions. The erivation of the formulas for the erivatives of sin an cos are an interesting stuy in both limits an trigonometric ientities. For those who are intereste, many such erivations can be foun on the web 1. However, it is in some ways more useful to erive the formula in a graphical manner. 1 For example,

47 Derivatives of Trigonometric Functions - 2 Below is a graph of sin(x). Use the graph to sketch the graph of its erivative. 1 3 π /2 π π/2 0 π/2 π 3 π/ π /2 π π/2 0 π/2 π 3 π/2 1

48 Derivatives of Trigonometric Functions - 3 From this sketch, we have evience (though not a proof) that Theorem x sin x =

49 Derivatives of Trigonometric Functions - 4 Most stuents will also be familiar with the other erivative rules for trig functions: x cos(x) = sin(x) x tan(x) = sec2 (x) x sec(x) = sec(x) tan(x) x csc(x) = csc(x) cot(x) x cot(x) = csc2 (x)

50 Derivatives of Trigonometric Functions - 5 Prove the secant erivative rule, sec(x) = sec(x) tan(x), using the efinition x sec(x) = 1 an the other erivative rules. cos(x)

51 Derivatives of Trigonometric Functions - 6 Question: Fin the erivative of cos(πx 2 + 1) A. 4 6 sin(πx 2 + 1) (2πx) B. 6 cos(πx 2 + 1) (2πx) C. 6 sin(πx 2 + 1) (2πx) D. 6 sin(πx 2 + 1) (πx 2 + 1) E. 6 sin(2πx)

Differentiability, Computing Derivatives, Trig Review. Goals:

Differentiability, Computing Derivatives, Trig Review. Goals: Secants vs. Derivatives - Unit #3 : Goals: Differentiability, Computing Derivatives, Trig Review Determine when a function is ifferentiable at a point Relate the erivative graph to the the graph of an

More information

Unit #3 : Differentiability, Computing Derivatives, Trig Review

Unit #3 : Differentiability, Computing Derivatives, Trig Review Unit #3 : Differentiability, Computing Derivatives, Trig Review Goals: Determine when a function is differentiable at a point Relate the derivative graph to the the graph of an original function Compute

More information

Unit #3 : Differentiability, Computing Derivatives

Unit #3 : Differentiability, Computing Derivatives Unit #3 : Differentiability, Computing Derivatives Goals: Determine when a function is differentiable at a point Relate the derivative graph to the the graph of an original function Compute derivative

More information

Calculus I Announcements

Calculus I Announcements Slie 1 Calculus I Announcements Office Hours: Amos Eaton 309, Monays 12:50-2:50 Exam 2 is Thursay, October 22n. The stuy guie is now on the course web page. Start stuying now, an make a plan to succee.

More information

Math 115 Section 018 Course Note

Math 115 Section 018 Course Note Course Note 1 General Functions Definition 1.1. A function is a rule that takes certain numbers as inputs an assigns to each a efinite output number. The set of all input numbers is calle the omain of

More information

f(x) f(a) Limit definition of the at a point in slope notation.

f(x) f(a) Limit definition of the at a point in slope notation. Lesson 9: Orinary Derivatives Review Hanout Reference: Brigg s Calculus: Early Transcenentals, Secon Eition Topics: Chapter 3: Derivatives, p. 126-235 Definition. Limit Definition of Derivatives at a point

More information

Solutions to Practice Problems Tuesday, October 28, 2008

Solutions to Practice Problems Tuesday, October 28, 2008 Solutions to Practice Problems Tuesay, October 28, 2008 1. The graph of the function f is shown below. Figure 1: The graph of f(x) What is x 1 + f(x)? What is x 1 f(x)? An oes x 1 f(x) exist? If so, what

More information

Exam 3 Review. Lesson 19: Concavity, Inflection Points, and the Second Derivative Test. Lesson 20: Absolute Extrema on an Interval

Exam 3 Review. Lesson 19: Concavity, Inflection Points, and the Second Derivative Test. Lesson 20: Absolute Extrema on an Interval Exam 3 Review Lessons 17-18: Relative Extrema, Critical Numbers, an First Derivative Test (from exam 2 review neee for curve sketching) Critical Numbers: where the erivative of a function is zero or unefine.

More information

February 21 Math 1190 sec. 63 Spring 2017

February 21 Math 1190 sec. 63 Spring 2017 February 21 Math 1190 sec. 63 Spring 2017 Chapter 2: Derivatives Let s recall the efinitions an erivative rules we have so far: Let s assume that y = f (x) is a function with c in it s omain. The erivative

More information

Differentiation ( , 9.5)

Differentiation ( , 9.5) Chapter 2 Differentiation (8.1 8.3, 9.5) 2.1 Rate of Change (8.2.1 5) Recall that the equation of a straight line can be written as y = mx + c, where m is the slope or graient of the line, an c is the

More information

SYDE 112, LECTURE 1: Review & Antidifferentiation

SYDE 112, LECTURE 1: Review & Antidifferentiation SYDE 112, LECTURE 1: Review & Antiifferentiation 1 Course Information For a etaile breakown of the course content an available resources, see the Course Outline. Other relevant information for this section

More information

1 Definition of the derivative

1 Definition of the derivative Math 20A - Calculus by Jon Rogawski Chapter 3 - Differentiation Prepare by Jason Gais Definition of the erivative Remark.. Recall our iscussion of tangent lines from way back. We now rephrase this in terms

More information

Derivative Methods: (csc(x)) = csc(x) cot(x)

Derivative Methods: (csc(x)) = csc(x) cot(x) EXAM 2 IS TUESDAY IN QUIZ SECTION Allowe:. A Ti-30x IIS Calculator 2. An 8.5 by inch sheet of hanwritten notes (front/back) 3. A pencil or black/blue pen Covers: 3.-3.6, 0.2, 3.9, 3.0, 4. Quick Review

More information

Math 210 Midterm #1 Review

Math 210 Midterm #1 Review Math 20 Miterm # Review This ocument is intene to be a rough outline of what you are expecte to have learne an retaine from this course to be prepare for the first miterm. : Functions Definition: A function

More information

The derivative of a function f(x) is another function, defined in terms of a limiting expression: f(x + δx) f(x)

The derivative of a function f(x) is another function, defined in terms of a limiting expression: f(x + δx) f(x) Y. D. Chong (2016) MH2801: Complex Methos for the Sciences 1. Derivatives The erivative of a function f(x) is another function, efine in terms of a limiting expression: f (x) f (x) lim x δx 0 f(x + δx)

More information

One of the powerful themes in trigonometry is that the entire subject emanates from a very simple idea: locating a point on the unit circle.

One of the powerful themes in trigonometry is that the entire subject emanates from a very simple idea: locating a point on the unit circle. 2.24 Tanz and the Reciprocals Derivatives of Other Trigonometric Functions One of the powerful themes in trigonometry is that the entire subject emanates from a very simple idea: locating a point on the

More information

Define each term or concept.

Define each term or concept. Chapter Differentiation Course Number Section.1 The Derivative an the Tangent Line Problem Objective: In this lesson you learne how to fin the erivative of a function using the limit efinition an unerstan

More information

JUST THE MATHS UNIT NUMBER DIFFERENTIATION 2 (Rates of change) A.J.Hobson

JUST THE MATHS UNIT NUMBER DIFFERENTIATION 2 (Rates of change) A.J.Hobson JUST THE MATHS UNIT NUMBER 10.2 DIFFERENTIATION 2 (Rates of change) by A.J.Hobson 10.2.1 Introuction 10.2.2 Average rates of change 10.2.3 Instantaneous rates of change 10.2.4 Derivatives 10.2.5 Exercises

More information

Math Chapter 2 Essentials of Calculus by James Stewart Prepared by Jason Gaddis

Math Chapter 2 Essentials of Calculus by James Stewart Prepared by Jason Gaddis Math 231 - Chapter 2 Essentials of Calculus by James Stewart Prepare by Jason Gais Chapter 2 - Derivatives 21 - Derivatives an Rates of Change Definition A tangent to a curve is a line that intersects

More information

Make graph of g by adding c to the y-values. on the graph of f by c. multiplying the y-values. even-degree polynomial. graph goes up on both sides

Make graph of g by adding c to the y-values. on the graph of f by c. multiplying the y-values. even-degree polynomial. graph goes up on both sides Reference 1: Transformations of Graphs an En Behavior of Polynomial Graphs Transformations of graphs aitive constant constant on the outsie g(x) = + c Make graph of g by aing c to the y-values on the graph

More information

x f(x) x f(x) approaching 1 approaching 0.5 approaching 1 approaching 0.

x f(x) x f(x) approaching 1 approaching 0.5 approaching 1 approaching 0. Engineering Mathematics 2 26 February 2014 Limits of functions Consier the function 1 f() = 1. The omain of this function is R + \ {1}. The function is not efine at 1. What happens when is close to 1?

More information

x f(x) x f(x) approaching 1 approaching 0.5 approaching 1 approaching 0.

x f(x) x f(x) approaching 1 approaching 0.5 approaching 1 approaching 0. Engineering Mathematics 2 26 February 2014 Limits of functions Consier the function 1 f() = 1. The omain of this function is R + \ {1}. The function is not efine at 1. What happens when is close to 1?

More information

cosh x sinh x So writing t = tan(x/2) we have 6.4 Integration using tan(x/2) = 2 2t 1 + t 2 cos x = 1 t2 We will revisit the double angle identities:

cosh x sinh x So writing t = tan(x/2) we have 6.4 Integration using tan(x/2) = 2 2t 1 + t 2 cos x = 1 t2 We will revisit the double angle identities: 6.4 Integration using tanx/) We will revisit the ouble angle ientities: sin x = sinx/) cosx/) = tanx/) sec x/) = tanx/) + tan x/) cos x = cos x/) sin x/) tan x = = tan x/) sec x/) tanx/) tan x/). = tan

More information

ARAB ACADEMY FOR SCIENCE TECHNOLOGY AND MARITIME TRANSPORT

ARAB ACADEMY FOR SCIENCE TECHNOLOGY AND MARITIME TRANSPORT ARAB ACADEMY FOR SCIENCE TECHNOLOGY AND MARITIME TRANSPORT Course: Math For Engineering Winter 8 Lecture Notes By Dr. Mostafa Elogail Page Lecture [ Functions / Graphs of Rational Functions] Functions

More information

Review of Differentiation and Integration for Ordinary Differential Equations

Review of Differentiation and Integration for Ordinary Differential Equations Schreyer Fall 208 Review of Differentiation an Integration for Orinary Differential Equations In this course you will be expecte to be able to ifferentiate an integrate quickly an accurately. Many stuents

More information

Differentiation Rules Derivatives of Polynomials and Exponential Functions

Differentiation Rules Derivatives of Polynomials and Exponential Functions Derivatives of Polynomials an Exponential Functions Differentiation Rules Derivatives of Polynomials an Exponential Functions Let s start with the simplest of all functions, the constant function f(x)

More information

2.5 SOME APPLICATIONS OF THE CHAIN RULE

2.5 SOME APPLICATIONS OF THE CHAIN RULE 2.5 SOME APPLICATIONS OF THE CHAIN RULE The Chain Rule will help us etermine the erivatives of logarithms an exponential functions a x. We will also use it to answer some applie questions an to fin slopes

More information

Lecture 6: Calculus. In Song Kim. September 7, 2011

Lecture 6: Calculus. In Song Kim. September 7, 2011 Lecture 6: Calculus In Song Kim September 7, 20 Introuction to Differential Calculus In our previous lecture we came up with several ways to analyze functions. We saw previously that the slope of a linear

More information

Chapter 2 Derivatives

Chapter 2 Derivatives Chapter Derivatives Section. An Intuitive Introuction to Derivatives Consier a function: Slope function: Derivative, f ' For each, the slope of f is the height of f ' Where f has a horizontal tangent line,

More information

Using this definition, it is possible to define an angle of any (positive or negative) measurement by recognizing how its terminal side is obtained.

Using this definition, it is possible to define an angle of any (positive or negative) measurement by recognizing how its terminal side is obtained. Angle in Standard Position With the Cartesian plane, we define an angle in Standard Position if it has its vertex on the origin and one of its sides ( called the initial side ) is always on the positive

More information

Final Exam Study Guide and Practice Problems Solutions

Final Exam Study Guide and Practice Problems Solutions Final Exam Stuy Guie an Practice Problems Solutions Note: These problems are just some of the types of problems that might appear on the exam. However, to fully prepare for the exam, in aition to making

More information

d dx But have you ever seen a derivation of these results? We ll prove the first result below. cos h 1

d dx But have you ever seen a derivation of these results? We ll prove the first result below. cos h 1 Lecture 5 Some ifferentiation rules Trigonometric functions (Relevant section from Stewart, Seventh Eition: Section 3.3) You all know that sin = cos cos = sin. () But have you ever seen a erivation of

More information

Outline. MS121: IT Mathematics. Differentiation Rules for Differentiation: Part 1. Outline. Dublin City University 4 The Quotient Rule

Outline. MS121: IT Mathematics. Differentiation Rules for Differentiation: Part 1. Outline. Dublin City University 4 The Quotient Rule MS2: IT Mathematics Differentiation Rules for Differentiation: Part John Carroll School of Mathematical Sciences Dublin City University Pattern Observe You may have notice the following pattern when we

More information

Computing Derivatives

Computing Derivatives Chapter 2 Computing Derivatives 2.1 Elementary erivative rules Motivating Questions In this section, we strive to unerstan the ieas generate by the following important questions: What are alternate notations

More information

2.1 Derivatives and Rates of Change

2.1 Derivatives and Rates of Change 1a 1b 2.1 Derivatives an Rates of Change Tangent Lines Example. Consier y f x x 2 0 2 x-, 0 4 y-, f(x) axes, curve C Consier a smooth curve C. A line tangent to C at a point P both intersects C at P an

More information

Learning Objectives for Math 165

Learning Objectives for Math 165 Learning Objectives for Math 165 Chapter 2 Limits Section 2.1: Average Rate of Change. State the definition of average rate of change Describe what the rate of change does and does not tell us in a given

More information

DERIVATIVES: LAWS OF DIFFERENTIATION MR. VELAZQUEZ AP CALCULUS

DERIVATIVES: LAWS OF DIFFERENTIATION MR. VELAZQUEZ AP CALCULUS DERIVATIVES: LAWS OF DIFFERENTIATION MR. VELAZQUEZ AP CALCULUS THE DERIVATIVE AS A FUNCTION f x = lim h 0 f x + h f(x) h Last class we examine the limit of the ifference quotient at a specific x as h 0,

More information

Chapter 2. Exponential and Log functions. Contents

Chapter 2. Exponential and Log functions. Contents Chapter. Exponential an Log functions This material is in Chapter 6 of Anton Calculus. The basic iea here is mainly to a to the list of functions we know about (for calculus) an the ones we will stu all

More information

Inverse Trig Functions

Inverse Trig Functions Inverse Trig Functions -8-006 If you restrict fx) = sinx to the interval π x π, the function increases: y = sin x - / / This implies that the function is one-to-one, an hence it has an inverse. The inverse

More information

Derivatives and Its Application

Derivatives and Its Application Chapter 4 Derivatives an Its Application Contents 4.1 Definition an Properties of erivatives; basic rules; chain rules 3 4. Derivatives of Inverse Functions; Inverse Trigonometric Functions; Hyperbolic

More information

By writing (1) as y (x 5 1). (x 5 1), we can find the derivative using the Product Rule: y (x 5 1) 2. we know this from (2)

By writing (1) as y (x 5 1). (x 5 1), we can find the derivative using the Product Rule: y (x 5 1) 2. we know this from (2) 3.5 Chain Rule 149 3.5 Chain Rule Introuction As iscusse in Section 3.2, the Power Rule is vali for all real number exponents n. In this section we see that a similar rule hols for the erivative of a power

More information

Question Instructions Read today's Notes and Learning Goals.

Question Instructions Read today's Notes and Learning Goals. 63 Proucts an Quotients (13051836) Question 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 Instructions Rea toay's Notes an Learning Goals. 1. Question Details fa15 62 chain 1 [3420817] Fin all

More information

Implicit Differentiation

Implicit Differentiation Implicit Differentiation Thus far, the functions we have been concerne with have been efine explicitly. A function is efine explicitly if the output is given irectly in terms of the input. For instance,

More information

4.2 First Differentiation Rules; Leibniz Notation

4.2 First Differentiation Rules; Leibniz Notation .. FIRST DIFFERENTIATION RULES; LEIBNIZ NOTATION 307. First Differentiation Rules; Leibniz Notation In this section we erive rules which let us quickly compute the erivative function f (x) for any polynomial

More information

( 3x +1) 2 does not fit the requirement of the power rule that the base be x

( 3x +1) 2 does not fit the requirement of the power rule that the base be x Section 3 4A: The Chain Rule Introuction The Power Rule is state as an x raise to a real number If y = x n where n is a real number then y = n x n-1 What if we wante to fin the erivative of a variable

More information

Math 106 Exam 2 Topics

Math 106 Exam 2 Topics Implicit ifferentiation Math 106 Exam Topics We can ifferentiate functions; what about equations? (e.g., x +y = 1) graph looks like it has tangent lines tangent line? (a,b) Iea: Preten equation efines

More information

Welcome to AP Calculus!!!

Welcome to AP Calculus!!! Welcome to AP Calculus!!! In preparation for next year, you need to complete this summer packet. This packet reviews & expands upon the concepts you studied in Algebra II and Pre-calculus. Make sure you

More information

Module FP2. Further Pure 2. Cambridge University Press Further Pure 2 and 3 Hugh Neill and Douglas Quadling Excerpt More information

Module FP2. Further Pure 2. Cambridge University Press Further Pure 2 and 3 Hugh Neill and Douglas Quadling Excerpt More information 5548993 - Further Pure an 3 Moule FP Further Pure 5548993 - Further Pure an 3 Differentiating inverse trigonometric functions Throughout the course you have graually been increasing the number of functions

More information

AP Calculus AB One Last Mega Review Packet of Stuff. Take the derivative of the following. 1.) 3.) 5.) 7.) Determine the limit of the following.

AP Calculus AB One Last Mega Review Packet of Stuff. Take the derivative of the following. 1.) 3.) 5.) 7.) Determine the limit of the following. AP Calculus AB One Last Mega Review Packet of Stuff Name: Date: Block: Take the erivative of the following. 1.) x (sin (5x)).) x (etan(x) ) 3.) x (sin 1 ( x3 )) 4.) x (x3 5x) 4 5.) x ( ex sin(x) ) 6.)

More information

You should also review L Hôpital s Rule, section 3.6; follow the homework link above for exercises.

You should also review L Hôpital s Rule, section 3.6; follow the homework link above for exercises. BEFORE You Begin Calculus II If it has been awhile since you ha Calculus, I strongly suggest that you refresh both your ifferentiation an integration skills. I woul also like to remin you that in Calculus,

More information

Tangent Lines Sec. 2.1, 2.7, & 2.8 (continued)

Tangent Lines Sec. 2.1, 2.7, & 2.8 (continued) Tangent Lines Sec. 2.1, 2.7, & 2.8 (continued) Prove this Result How Can a Derivative Not Exist? Remember that the derivative at a point (or slope of a tangent line) is a LIMIT, so it doesn t exist whenever

More information

December Exam Summary

December Exam Summary December Exam Summary 1 Lines and Distances 1.1 List of Concepts Distance between two numbers on the real number line or on the Cartesian Plane. Increments. If A = (a 1, a 2 ) and B = (b 1, b 2 ), then

More information

AP CALCULUS SUMMER WORKSHEET

AP CALCULUS SUMMER WORKSHEET AP CALCULUS SUMMER WORKSHEET DUE: First Day of School Aug. 19, 2010 Complete this assignment at your leisure during the summer. It is designed to help you become more comfortable with your graphing calculator,

More information

Chapter 7. Integrals and Transcendental Functions

Chapter 7. Integrals and Transcendental Functions 7. The Logarithm Define as an Integral Chapter 7. Integrals an Transcenental Functions 7.. The Logarithm Define as an Integral Note. In this section, we introuce the natural logarithm function using efinite

More information

Your signature: (1) (Pre-calculus Review Set Problems 80 and 124.)

Your signature: (1) (Pre-calculus Review Set Problems 80 and 124.) (1) (Pre-calculus Review Set Problems 80 an 14.) (a) Determine if each of the following statements is True or False. If it is true, explain why. If it is false, give a counterexample. (i) If a an b are

More information

UNIT 3: DERIVATIVES STUDY GUIDE

UNIT 3: DERIVATIVES STUDY GUIDE Calculus I UNIT 3: Derivatives REVIEW Name: Date: UNIT 3: DERIVATIVES STUDY GUIDE Section 1: Section 2: Limit Definition (Derivative as the Slope of the Tangent Line) Calculating Rates of Change (Average

More information

Math 106 Exam 2 Topics. du dx

Math 106 Exam 2 Topics. du dx The Chain Rule Math 106 Exam 2 Topics Composition (g f)(x 0 ) = g(f(x 0 )) ; (note: we on t know what g(x 0 ) is.) (g f) ought to have something to o with g (x) an f (x) in particular, (g f) (x 0 ) shoul

More information

Calculus in the AP Physics C Course The Derivative

Calculus in the AP Physics C Course The Derivative Limits an Derivatives Calculus in the AP Physics C Course The Derivative In physics, the ieas of the rate change of a quantity (along with the slope of a tangent line) an the area uner a curve are essential.

More information

Table of Contents Derivatives of Logarithms

Table of Contents Derivatives of Logarithms Derivatives of Logarithms- Table of Contents Derivatives of Logarithms Arithmetic Properties of Logarithms Derivatives of Logarithms Example Example 2 Example 3 Example 4 Logarithmic Differentiation Example

More information

CHAPTER 3 DERIVATIVES (continued)

CHAPTER 3 DERIVATIVES (continued) CHAPTER 3 DERIVATIVES (continue) 3.3. RULES FOR DIFFERENTIATION A. The erivative of a constant is zero: [c] = 0 B. The Power Rule: [n ] = n (n-1) C. The Constant Multiple Rule: [c *f()] = c * f () D. The

More information

AP CALCULUS SUMMER WORKSHEET

AP CALCULUS SUMMER WORKSHEET AP CALCULUS SUMMER WORKSHEET DUE: First Day of School, 2011 Complete this assignment at your leisure during the summer. I strongly recommend you complete a little each week. It is designed to help you

More information

2.1 Limits, Rates of Change and Slopes of Tangent Lines

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

More information

MA4001 Engineering Mathematics 1 Lecture 14 Derivatives of Trigonometric Functions Critical Points

MA4001 Engineering Mathematics 1 Lecture 14 Derivatives of Trigonometric Functions Critical Points MA4001 Engineering Mathematics 1 Lecture 14 Derivatives of Trigonometric Functions Critical Points Dr. Sarah Mitchell Autumn 2014 An important limit To calculate the limits of basic trigonometric functions

More information

8/28/2017 Assignment Previewer

8/28/2017 Assignment Previewer Proucts an Quotients (10862446) Due: Mon Sep 25 2017 07:31 AM MDT Question 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 Instructions Rea toay's Notes an Learning Goals. 1. Question Details

More information

2.5 The Chain Rule Brian E. Veitch

2.5 The Chain Rule Brian E. Veitch 2.5 The Chain Rule This is our last ifferentiation rule for this course. It s also one of the most use. The best way to memorize this (along with the other rules) is just by practicing until you can o

More information

2.1 The derivative. Rates of change. m sec = y f (a + h) f (a)

2.1 The derivative. Rates of change. m sec = y f (a + h) f (a) 2.1 The derivative Rates of change 1 The slope of a secant line is m sec = y f (b) f (a) = x b a and represents the average rate of change over [a, b]. Letting b = a + h, we can express the slope of the

More information

Section 2.1 The Derivative and the Tangent Line Problem

Section 2.1 The Derivative and the Tangent Line Problem Chapter 2 Differentiation Course Number Section 2.1 The Derivative an the Tangent Line Problem Objective: In this lesson you learne how to fin the erivative of a function using the limit efinition an unerstan

More information

MA1021 Calculus I B Term, Sign:

MA1021 Calculus I B Term, Sign: MA1021 Calculus I B Term, 2014 Final Exam Print Name: Sign: Write up your solutions neatly and show all your work. 1. (28 pts) Compute each of the following derivatives: You do not have to simplify your

More information

Chapter 3: Derivatives

Chapter 3: Derivatives Name: Date: Period: AP Calc AB Mr. Mellina Chapter 3: Derivatives Sections: v 2.4 Rates of Change & Tangent Lines v 3.1 Derivative of a Function v 3.2 Differentiability v 3.3 Rules for Differentiation

More information

Calculus & Analytic Geometry I

Calculus & Analytic Geometry I Functions Form the Foundation What is a function? A function is a rule that assigns to each element x (called the input or independent variable) in a set D exactly one element f(x) (called the ouput or

More information

Fall 2016: Calculus I Final

Fall 2016: Calculus I Final Answer the questions in the spaces provie on the question sheets. If you run out of room for an answer, continue on the back of the page. NO calculators or other electronic evices, books or notes are allowe

More information

Differentiation Rules and Formulas

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

More information

2.2 The derivative as a Function

2.2 The derivative as a Function 2.2 The derivative as a Function Recall: The derivative of a function f at a fixed number a: f a f a+h f(a) = lim h 0 h Definition (Derivative of f) For any number x, the derivative of f is f x f x+h f(x)

More information

Using the definition of the derivative of a function is quite tedious. f (x + h) f (x)

Using 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 information

Math Implicit Differentiation. We have discovered (and proved) formulas for finding derivatives of functions like

Math Implicit Differentiation. We have discovered (and proved) formulas for finding derivatives of functions like Math 400 3.5 Implicit Differentiation Name We have iscovere (an prove) formulas for fining erivatives of functions like f x x 3x 4x. 3 This amounts to fining y for 3 y x 3x 4x. Notice that in this case,

More information

Higher. Further Calculus 149

Higher. Further Calculus 149 hsn.uk.net Higher Mathematics UNIT 3 OUTCOME 2 Further Calculus Contents Further Calculus 49 Differentiating sinx an cosx 49 2 Integrating sinx an cosx 50 3 The Chain Rule 5 4 Special Cases of the Chain

More information

Some functions and their derivatives

Some functions and their derivatives Chapter Some functions an their erivatives. Derivative of x n for integer n Recall, from eqn (.6), for y = f (x), Also recall that, for integer n, Hence, if y = x n then y x = lim δx 0 (a + b) n = a n

More information

cosh x sinh x So writing t = tan(x/2) we have 6.4 Integration using tan(x/2) 2t 1 + t 2 cos x = 1 t2 sin x =

cosh x sinh x So writing t = tan(x/2) we have 6.4 Integration using tan(x/2) 2t 1 + t 2 cos x = 1 t2 sin x = 6.4 Integration using tan/ We will revisit the ouble angle ientities: sin = sin/ cos/ = tan/ sec / = tan/ + tan / cos = cos / sin / tan = = tan / sec / tan/ tan /. = tan / + tan / So writing t = tan/ we

More information

Flash Card Construction Instructions

Flash Card Construction Instructions Flash Car Construction Instructions *** THESE CARDS ARE FOR CALCULUS HONORS, AP CALCULUS AB AND AP CALCULUS BC. AP CALCULUS BC WILL HAVE ADDITIONAL CARDS FOR THE COURSE (IN A SEPARATE FILE). The left column

More information

1 Limits Finding limits graphically. 1.3 Finding limits analytically. Examples 1. f(x) = x3 1. f(x) = f(x) =

1 Limits Finding limits graphically. 1.3 Finding limits analytically. Examples 1. f(x) = x3 1. f(x) = f(x) = Theorem 13 (i) If p(x) is a polynomial, then p(x) = p(c) 1 Limits 11 12 Fining its graphically Examples 1 f(x) = x3 1, x 1 x 1 The behavior of f(x) as x approximates 1 x 1 f(x) = 3 x 2 f(x) = x+1 1 f(x)

More information

Unit #6 - Families of Functions, Taylor Polynomials, l Hopital s Rule

Unit #6 - Families of Functions, Taylor Polynomials, l Hopital s Rule Unit # - Families of Functions, Taylor Polynomials, l Hopital s Rule Some problems an solutions selecte or aapte from Hughes-Hallett Calculus. Critical Points. Consier the function f) = 54 +. b) a) Fin

More information

Math 251 Notes. Part I.

Math 251 Notes. Part I. Math 251 Notes. Part I. F. Patricia Meina May 6, 2013 Growth Moel.Consumer price inex. [Problem 20, page 172] The U.S. consumer price inex (CPI) measures the cost of living base on a value of 100 in the

More information

Solutions to Math 41 Second Exam November 4, 2010

Solutions to Math 41 Second Exam November 4, 2010 Solutions to Math 41 Secon Exam November 4, 2010 1. (13 points) Differentiate, using the metho of your choice. (a) p(t) = ln(sec t + tan t) + log 2 (2 + t) (4 points) Using the rule for the erivative of

More information

Calculus I. George Voutsadakis 1. LSSU Math 151. Lake Superior State University. 1 Mathematics and Computer Science

Calculus I. George Voutsadakis 1. LSSU Math 151. Lake Superior State University. 1 Mathematics and Computer Science Calculus I George Voutsadakis 1 1 Mathematics and Computer Science Lake Superior State University LSSU Math 151 George Voutsadakis (LSSU) Calculus I November 2014 1 / 67 Outline 1 Limits Limits, Rates

More information

Math 1A Midterm 2 Fall 2015 Riverside City College (Use this as a Review)

Math 1A Midterm 2 Fall 2015 Riverside City College (Use this as a Review) Name Date Miterm Score Overall Grae Math A Miterm 2 Fall 205 Riversie City College (Use this as a Review) Instructions: All work is to be shown, legible, simplifie an answers are to be boxe in the space

More information

f(g(x)) g (x) dx = f(u) du.

f(g(x)) g (x) dx = f(u) du. 1. Techniques of Integration Section 8-IT 1.1. Basic integration formulas. Integration is more difficult than derivation. The derivative of every rational function or trigonometric function is another

More information

Lecture 4 : General Logarithms and Exponentials. a x = e x ln a, a > 0.

Lecture 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 information

Sec 4.1 Limits, Informally. When we calculated f (x), we first started with the difference quotient. f(x + h) f(x) h

Sec 4.1 Limits, Informally. When we calculated f (x), we first started with the difference quotient. f(x + h) f(x) h 1 Sec 4.1 Limits, Informally When we calculated f (x), we first started with the difference quotient f(x + h) f(x) h and made h small. In other words, f (x) is the number f(x+h) f(x) approaches as h gets

More information

Lectures - Week 10 Introduction to Ordinary Differential Equations (ODES) First Order Linear ODEs

Lectures - Week 10 Introduction to Ordinary Differential Equations (ODES) First Order Linear ODEs Lectures - Week 10 Introuction to Orinary Differential Equations (ODES) First Orer Linear ODEs When stuying ODEs we are consiering functions of one inepenent variable, e.g., f(x), where x is the inepenent

More information

Hyperbolic Functions. Notice: this material must not be used as a substitute for attending. the lectures

Hyperbolic Functions. Notice: this material must not be used as a substitute for attending. the lectures Hyperbolic Functions Notice: this material must not be use as a substitute for attening the lectures 0. Hyperbolic functions sinh an cosh The hyperbolic functions sinh (pronounce shine ) an cosh are efine

More information

Math 1271 Solutions for Fall 2005 Final Exam

Math 1271 Solutions for Fall 2005 Final Exam Math 7 Solutions for Fall 5 Final Eam ) Since the equation + y = e y cannot be rearrange algebraically in orer to write y as an eplicit function of, we must instea ifferentiate this relation implicitly

More information

function independent dependent domain range graph of the function The Vertical Line Test

function independent dependent domain range graph of the function The Vertical Line Test Functions A quantity y is a function of another quantity x if there is some rule (an algebraic equation, a graph, a table, or as an English description) by which a unique value is assigned to y by a corresponding

More information

The Chain Rule. Composition Review. Intuition. = 2(1.5) = 3 times faster than (X)avier.

The Chain Rule. Composition Review. Intuition. = 2(1.5) = 3 times faster than (X)avier. The Chain Rule In the previous section we ha to use a trig ientity to etermine the erivative of. h(x) = sin(2x). We can view h(x) as the composition of two functions. Let g(x) = 2x an f (x) = sin x. Then

More information

Chapter Primer on Differentiation

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

More information

3.2 Differentiability

3.2 Differentiability Section 3 Differentiability 09 3 Differentiability What you will learn about How f (a) Might Fail to Eist Differentiability Implies Local Linearity Numerical Derivatives on a Calculator Differentiability

More information

Section 7.2. The Calculus of Complex Functions

Section 7.2. The Calculus of Complex Functions Section 7.2 The Calculus of Complex Functions In this section we will iscuss limits, continuity, ifferentiation, Taylor series in the context of functions which take on complex values. Moreover, we will

More information

Chapter 1 Overview: Review of Derivatives

Chapter 1 Overview: Review of Derivatives Chapter Overview: Review of Derivatives The purpose of this chapter is to review the how of ifferentiation. We will review all the erivative rules learne last year in PreCalculus. In the net several chapters,

More information

MATH2231-Differentiation (2)

MATH2231-Differentiation (2) -Differentiation () The Beginnings of Calculus The prime occasion from which arose my iscovery of the metho of the Characteristic Triangle, an other things of the same sort, happene at a time when I ha

More information

Inverse Trig Functions

Inverse Trig Functions Inverse Trig Functions -7-08 If you restrict fx) = sinx to the interval π x π, the function increases: y = sin x - / / This implies that the function is one-to-one, an hence it has an inverse. The inverse

More information

Slide 1. Slide 2. Slide 3 Remark is a new function derived from called derivative. 2.2 The derivative as a Function

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

More information