Lecture 5: Inverse Trigonometric Functions
|
|
|
- Brittney French
- 6 years ago
- Views:
Transcription
1 Lecture 5: Inverse Trigonometric Functions 5 The inverse sine function The function f(x = sin(x is not one-to-one on (,, but is on [ π, π Moreover, f still has range [, when restricte to this interval Hence it is reasonable to restrict f to [ π, π to obtain an inverse for the sine function Definition We call the inverse of the sine function, when restricte to [ π, π, the arcsine function, with value at a point x enote by arcsin(x or sin (x In other wors, y = sin (x if an only if sin(y = x an π y π Note that the arcsine function has omain [, an range [ π, π sin (0 = 0, sin ( = π, an sin ( = π, ( sin = π 6 Note that sin(sin (x = x for any x an sin (sin(x = x for any π x However, the latter equality oes not hol for other values of x For example, π sin (sin ( ( 7π = sin = π 6 6 Since for x in [, we have sin(sin (x = x, it follows that x sin(sin (x = x x Hence cos(sin (x x sin (x =, 5-
2 an so Lecture 5: Inverse Trigonometric Functions 5- x sin (x = Now if y = sin (x, then π y π Since cos(y 0 for π y π, we have Thus we have the following proposition cos(sin (x an sin(y = x Thus cos (y = sin (y = x cos(sin (x = cos(y = x Proposition x sin (x = x for all < x < x sin (3x = 6x 9x Note that f(x = sin (x is an increasing function on [,, with graph as shown below Graphs of y = sin(x an y = sin (x We now have the integration formula x = x sin (x + c
3 Lecture 5: Inverse Trigonometric Functions 5-3 Using the substitution u = x, we have 0 x = x 0 u u = sin (u 0 = ( sin sin (0 = π 5 The inverse tangent function Note that the function f(x = tan(x is one-to-one if its omain is restricte to ( π, π Moreover, f still has range (, when restricte to this interval Hence it is reasonable to restrict f to ( π, π to obtain an inverse for the tangent function Definition We call the inverse of the tangent function, when restricte to ( π, π, the arctangent function, with value at a point x enote by arctan(x or tan (x In other wors, y = tan (x if an only if tan(y = x an π < y < π Note that the arctangent function has omain (, an range ( π, π an tan (0 = 0, tan ( = π, tan ( 3 = π 3 Since tan(tan (x = x for all x in (,, we have x tan(tan (x = x x Hence sec (tan (x x tan (x =,
4 an so Lecture 5: Inverse Trigonometric Functions 5- x tan (x = Now if y = tan (x, then π < y < π Thus we have the following proposition sec (tan (x an tan(y = x Thus sec (y = + tan (y = + x Proposition x tan (x = + x If f(x = tan ( x, then f (x = ( ( x + = 8 + x Note that f(x = tan (x is an increasing function on (, with an Hence the lines y = π an y = π lim x tan (x = π lim x tan (x = π are horizontal asymptotes for the graph of f Moreover, f (x = ( + x x (x = ( + x, from which it follows that the graph of f is concave upwar on (, 0 an concave ownwar on (0, The graph is shown below Graphs of y = tan(x an y = tan (x
5 Lecture 5: Inverse Trigonometric Functions 5-5 We now have the integration formula + x x = tan (x + c + x x = Using the substitution u = x, we have + ( x x = + u u = tan (u + c = ( x tan + c 53 The inverse secant function Note that the function f(x = sec(x is one-to-one if its omain is restricte to [ 0, π [ π, 3π Moreover, f still has range (, [, when restricte to this omain Hence it is reasonable to restrict f to this omain to obtain an inverse for the secant function Definition We call the inverse of the secant function, when restricte to [ 0, π [ π, 3π, the arcsecant function, with value at a point x enote by arcsec(x or sec (x In other wors, y = sec (x if an only if sec(y = x an either 0 y < π or π y < 3π Note that the arcsecant function has omain (, [, an range [ 0, π [ π, 3π sec ( = 0, sec ( = π,
6 Lecture 5: Inverse Trigonometric Functions 5-6 an sec ( = π 3 Since sec(sec (x = x, we have Hence an so Now if y = sec (x, then x sec(sec (x = x x sec(sec (x tan(sec (x x sec (x =, x sec (x = x tan(sec (x tan (y = sec (y = x Since tan(y 0 for y in the range of the arcsecant function, we have tan(sec (x = x Thus we have the following proposition Proposition For all x or x, x sec (x = x x The previous proposition gives us the integration formula x x x = sec (x + c The graph of f(x = sec (x is shown below Notice how the vertical asymptotes of g(x = sec(x are transforme into horizontal asymptotes for f Graphs of y = sec(x an y = sec (x
7 Lecture 5: Inverse Trigonometric Functions The remaining inverse trigonometric functions The remaining inverse trigonometric functions are not use as frequently as the three we have consiere above They are efine as follows: y = cos (x if an only if cos(y = x an 0 y π, y = cot (x if an only if cot(y = x an 0 < y < π, an y = csc (x if an only if csc(y = x an 0 < y π or π < y 3π One may show that an x cos (x =, x x cot (x = + x, x csc (x = x x Note that these are just the negations of the erivatives of the inverses of the corresponing co-functions In particular, they o not give us any new integration formulas Moreover, this implies that these functions iffer by only a constant from the inverses of their corresponing co-functions
Inverse Trig Functions
6.6i Inverse Trigonometric Functions Inverse Sine Function Does g(x) = sin(x) have an inverse? What restriction would we need to make so that at least a piece of this function has an inverse? Given f (x)
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
June 9 Math 1113 sec 002 Summer 2014
June 9 Math 1113 sec 002 Summer 2014 Section 6.5: Inverse Trigonometric Functions Definition: (Inverse Sine) For x in the interval [ 1, 1] the inverse sine of x is denoted by either and is defined by the
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
MTH 112: Elementary Functions
1/19 MTH 11: Elementary Functions Section 6.6 6.6:Inverse Trigonometric functions /19 Inverse Trig functions 1 1 functions satisfy the horizontal line test: Any horizontal line crosses the graph of a 1
Inverse Trigonometric Functions. September 5, 2018
Inverse Trigonometric Functions September 5, 08 / 7 Restricted Sine Function. The trigonometric function sin x is not a one-to-one functions..0 0.5 Π 6, 5Π 6, Π Π Π Π 0.5 We still want an inverse, so what
MTH 112: Elementary Functions
MTH 11: Elementary Functions F. Patricia Medina Department of Mathematics. Oregon State University Section 6.6 Inverse Trig functions 1 1 functions satisfy the horizontal line test: Any horizontal line
INVERSE FUNCTIONS DERIVATIVES. terms on one side and everything else on the other. (3) Factor out dy. for the following functions: 1.
INVERSE FUNCTIONS DERIVATIVES Recall the steps for computing y implicitly: (1) Take of both sies, treating y like a function. (2) Expan, a, subtract to get the y terms on one sie an everything else on
Section Inverse Trigonometry. In this section, we will define inverse since, cosine and tangent functions. x is NOT one-to-one.
Section 5.4 - Inverse Trigonometry In this section, we will define inverse since, cosine and tangent functions. RECALL Facts about inverse functions: A function f ) is one-to-one if no two different inputs
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
NOTES ON INVERSE TRIGONOMETRIC FUNCTIONS
NOTES ON INVERSE TRIGONOMETRIC FUNCTIONS MATH 5 (S). Definitions of Inverse Trigonometric Functions () y = sin or y = arcsin is the inverse function of y = sin on [, ]. The omain of y = sin = arcsin is
16 Inverse Trigonometric Functions
6 Inverse Trigonometric Functions Concepts: Restricting the Domain of the Trigonometric Functions The Inverse Sine Function The Inverse Cosine Function The Inverse Tangent Function Using the Inverse Trigonometric
L Hôpital s Rule was discovered by Bernoulli but written for the first time in a text by L Hôpital.
7.5. Ineterminate Forms an L Hôpital s Rule L Hôpital s Rule was iscovere by Bernoulli but written for the first time in a text by L Hôpital. Ineterminate Forms 0/0 an / f(x) If f(x 0 ) = g(x 0 ) = 0,
6.6 Inverse Trigonometric Functions
6.6 6.6 Inverse Trigonometric Functions We recall the following definitions from trigonometry. If we restrict the sine function, say fx) sinx, π x π then we obtain a one-to-one function. π/, /) π/ π/ Since
Find the indicated derivative. 1) Find y(4) if y = 3 sin x. A) y(4) = 3 cos x B) y(4) = 3 sin x C) y(4) = - 3 cos x D) y(4) = - 3 sin x
Assignment 5 Name Find the indicated derivative. ) Find y(4) if y = sin x. ) A) y(4) = cos x B) y(4) = sin x y(4) = - cos x y(4) = - sin x ) y = (csc x + cot x)(csc x - cot x) ) A) y = 0 B) y = y = - csc
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
There are some trigonometric identities given on the last page.
MA 114 Calculus II Fall 2015 Exam 4 December 15, 2015 Name: Section: Last 4 digits of student ID #: No books or notes may be used. Turn off all your electronic devices and do not wear ear-plugs during
Linear and quadratic approximation
Linear an quaratic approximation November 11, 2013 Definition: Suppose f is a function that is ifferentiable on an interval I containing the point a. The linear approximation to f at a is the linear function
1 Applications of the Chain Rule
November 7, 08 MAT86 Week 6 Justin Ko Applications of the Chain Rule We go over several eamples of applications of the chain rule to compute erivatives of more complicate functions. Chain Rule: If z =
TOTAL NAME DATE PERIOD AP CALCULUS AB UNIT 4 ADVANCED DIFFERENTIATION TECHNIQUES DATE TOPIC ASSIGNMENT /6 10/8 10/9 10/10 X X X X 10/11 10/12
NAME DATE PERIOD AP CALCULUS AB UNIT ADVANCED DIFFERENTIATION TECHNIQUES DATE TOPIC ASSIGNMENT 0 0 0/6 0/8 0/9 0/0 X X X X 0/ 0/ 0/5 0/6 QUIZ X X X 0/7 0/8 0/9 0/ 0/ 0/ 0/5 UNIT EXAM X X X TOTAL AP Calculus
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
Function and Relation Library
1 of 7 11/6/2013 7:56 AM Function and Relation Library Trigonometric Functions: Angle Definitions Legs of A Triangle Definitions Sine Cosine Tangent Secant Cosecant Cotangent Trig functions are functions
I.e., the range of f(x) = arctan(x) is all real numbers y such that π 2 < y < π 2
Inverse Trigonometric Functions: The inverse sine function, denoted by fx = arcsinx or fx = sin 1 x is defined by: y = sin 1 x if and only if siny = x and π y π I.e., the range of fx = arcsinx is all real
5, tan = 4. csc = Simplify: 3. Simplify: 4. Factor and simplify: cos x sin x cos x
Precalculus Final Review 1. Given the following values, evaluate (if possible) the other four trigonometric functions using the fundamental trigonometric identities or triangles csc = - 3 5, tan = 4 3.
DIFFERENTIATION RULES
3 DIFFERENTIATION RULES DIFFERENTIATION RULES The functions that we have met so far can be described by expressing one variable explicitly in terms of another variable. y For example,, or y = x sin x,
The function is a periodic function. That means that the functions repeats its values in regular intervals, which we call the period.
Section 5.4 - Inverse Trigonometric Functions The Inverse Sine Function Consider the graph of the sine function f ( x) sin( x). The function is a periodic function. That means that the functions repeats
SANDERSON HIGH SCHOOL AP CALCULUS AB/BC SUMMER REVIEW PACKET
SANDERSON HIGH SCHOOL AP CALCULUS AB/BC SUMMER REVIEW PACKET 017-018 Name: 1. This packet is to be handed in on Monday August 8, 017.. All work must be shown on separate paper attached to the packet. 3.
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
Implicit Differentiation and Inverse Trigonometric Functions
Implicit Differentiation an Inverse Trigonometric Functions MATH 161 Calculus I J. Robert Buchanan Department of Mathematics Summer 2018 Explicit vs. Implicit Functions 0.5 1 y 0.0 y 2 0.5 3 4 1.0 0.5
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)
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
12) y = -2 sin 1 2 x - 2
Review -Test 1 - Unit 1 and - Math 41 Name SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. Find and simplify the difference quotient f(x + h) - f(x),
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
CK- 12 Algebra II with Trigonometry Concepts 1
14.1 Graphing Sine and Cosine 1. A.,1 B. (, 1) C. 3,0 D. 11 1, 6 E. (, 1) F. G. H. 11, 4 7, 1 11, 3. 3. 5 9,,,,,,, 4 4 4 4 3 5 3, and, 3 3 CK- 1 Algebra II with Trigonometry Concepts 1 4.ans-1401-01 5.
TRIGONOMETRIC FUNCTIONS. Copyright Cengage Learning. All rights reserved.
12 TRIGONOMETRIC FUNCTIONS Copyright Cengage Learning. All rights reserved. 12.2 The Trigonometric Functions Copyright Cengage Learning. All rights reserved. The Trigonometric Functions and Their Graphs
2 Trigonometric functions
Theodore Voronov. Mathematics 1G1. Autumn 014 Trigonometric functions Trigonometry provides methods to relate angles and lengths but the functions we define have many other applications in mathematics..1
SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. and θ is in quadrant IV. 1)
Chapter 5-6 Review Math 116 Name SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. Use the fundamental identities to find the value of the trigonometric
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,
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,
4.3 Inverse Trigonometric Properties
www.ck1.org Chapter. Inverse Trigonometric Functions. Inverse Trigonometric Properties Learning Objectives Relate the concept of inverse functions to trigonometric functions. Reduce the composite function
Formulas to remember
Complex numbers Let z = x + iy be a complex number The conjugate z = x iy Formulas to remember The real part Re(z) = x = z+z The imaginary part Im(z) = y = z z i The norm z = zz = x + y The reciprocal
Chapter 1. Functions 1.3. Trigonometric Functions
1.3 Trigonometric Functions 1 Chapter 1. Functions 1.3. Trigonometric Functions Definition. The number of radians in the central angle A CB within a circle of radius r is defined as the number of radius
TRIGONOMETRY OUTCOMES
TRIGONOMETRY OUTCOMES C10. Solve problems involving limits of trigonometric functions. C11. Apply derivatives of trigonometric functions. C12. Solve problems involving inverse trigonometric functions.
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
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.)
Mth Review Problems for Test 2 Stewart 8e Chapter 3. For Test #2 study these problems, the examples in your notes, and the homework.
For Test # study these problems, the examples in your notes, and the homework. Derivative Rules D [u n ] = nu n 1 du D [ln u] = du u D [log b u] = du u ln b D [e u ] = e u du D [a u ] = a u ln a du D [sin
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
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
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
Chapter 3: Transcendental Functions
Chapter 3: Transcendental Functions Spring 2018 Department of Mathematics Hong Kong Baptist University 1 / 32 Except for the power functions, the other basic elementary functions are also called the transcendental
FUNCTIONS AND MODELS
1 FUNCTIONS AND MODELS FUNCTIONS AND MODELS 1.6 Inverse Functions and Logarithms In this section, we will learn about: Inverse functions and logarithms. INVERSE FUNCTIONS The table gives data from an experiment
Section The Chain Rule and Implicit Differentiation with Application on Derivative of Logarithm Functions
Section 3.4-3.6 The Chain Rule an Implicit Differentiation with Application on Derivative of Logarithm Functions Ruipeng Shen September 3r, 5th Ruipeng Shen MATH 1ZA3 September 3r, 5th 1 / 3 The Chain
THEOREM: THE CONSTANT RULE
MATH /MYERS/ALL FORMULAS ON THIS REVIEW MUST BE MEMORIZED! DERIVATIVE REVIEW THEOREM: THE CONSTANT RULE The erivative of a constant function is zero. That is, if c is a real number, then c 0 Eample 1:
Math 1 Lecture 20. Dartmouth College. Wednesday
Math 1 Lecture 20 Dartmouth College Wenesay 10-26-16 Contents Reminers/Announcements Last Time Derivatives of Trigonometric Functions Reminers/Announcements WebWork ue Friay x-hour problem session rop
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,
Authors Gregory Hartman, Ph.D. Brian Heinold, Ph.D. Troy Siemers, Ph.D. Dimplekumar Chalishajar, Ph.D. Editor Jennifer Bowen, Ph.D.
Section 7 Derivatives of Inverse Functions AP E XC I Version 0 Authors Gregory Hartman, PhD Department of Applie Mathema cs Virginia Military Ins tute Brian Heinol, PhD Department of Mathema cs an Computer
Chapter 5 Notes. 5.1 Using Fundamental Identities
Chapter 5 Notes 5.1 Using Fundamental Identities 1. Simplify each expression to its lowest terms. Write the answer to part as the product of factors. (a) sin x csc x cot x ( 1+ sinσ + cosσ ) (c) 1 tanx
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
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
For a semi-circle with radius r, its circumfrence is πr, so the radian measure of a semi-circle (a straight line) is
Radian Measure Given any circle with radius r, if θ is a central angle of the circle and s is the length of the arc sustained by θ, we define the radian measure of θ by: θ = s r For a semi-circle with
TO EARN ANY CREDIT, YOU MUST SHOW STEPS LEADING TO THE ANSWER
Prof. Israel N. Nwaguru MATH 11 CHAPTER,,, AND - REVIEW WORKOUT EACH PROBLEM NEATLY AND ORDERLY ON SEPARATE SHEET THEN CHOSE THE BEST ANSWER TO EARN ANY CREDIT, YOU MUST SHOW STEPS LEADING TO THE ANSWER
Inverse Relations. 5 are inverses because their input and output are switched. For instance: f x x. x 5. f 4
Inverse Functions Inverse Relations The inverse of a relation is the set of ordered pairs obtained by switching the input with the output of each ordered pair in the original relation. (The domain of the
( ) a (graphical) transformation of y = f ( x )? x 0,2π. f ( 1 b) = a if and only if f ( a ) = b. f 1 1 f
Warm-Up: Solve sinx = 2 for x 0,2π 5 (a) graphically (approximate to three decimal places) y (b) algebraically BY HAND EXACTLY (do NOT approximate except to verify your solutions) x x 0,2π, xscl = π 6,y,,
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.
Topics and Concepts. 1. Limits
Topics and Concepts 1. Limits (a) Evaluating its (Know: it exists if and only if the it from the left is the same as the it from the right) (b) Infinite its (give rise to vertical asymptotes) (c) Limits
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
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
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
Formulas From Calculus
Formulas You Shoul Memorize (an I o mean Memorize!) S 997 Pat Rossi Formulas From Calculus. [sin ()] = cos () 2. [cos ()] = sin () 3. [tan ()] = sec2 () 4. [cot ()] = csc2 () 5. [sec ()] = sec () tan ()
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)
4-3 Trigonometric Functions on the Unit Circle
Find the exact value of each trigonometric function, if defined. If not defined, write undefined. 9. sin The terminal side of in standard position lies on the positive y-axis. Choose a point P(0, 1) on
Chapter 5 Logarithmic, Exponential, and Other Transcendental Functions
Chapter 5 Logarithmic, Exponential, an Other Transcenental Functions 5.1 The Natural Logarithmic Function: Differentiation 5.2 The Natural Logarithmic Function: Integration 5.3 Inverse Functions 5.4 Exponential
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?
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
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
9. The x axis is a horizontal line so a 1 1 function can touch the x axis in at most one place.
O Answers: Chapter 7 Contemporary Calculus PROBLEM ANSWERS Chapter Seven Section 7.0. f is one to one ( ), y is, g is not, h is not.. f is not, y is, g is, h is not. 5. I think SS numbers are supposeo
Differentiability, Computing Derivatives, Trig Review
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
DRAFT - Math 101 Lecture Note - Dr. Said Algarni
3 Differentiation Rules 3.1 The Derivative of Polynomial and Exponential Functions In this section we learn how to differentiate constant functions, power functions, polynomials, and exponential functions.
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
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
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
1 Lecture 20: Implicit differentiation
Lecture 20: Implicit ifferentiation. Outline The technique of implicit ifferentiation Tangent lines to a circle Derivatives of inverse functions by implicit ifferentiation Examples.2 Implicit ifferentiation
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
WORKBOOK. MATH 32. CALCULUS AND ANALYTIC GEOMETRY II.
WORKBOOK. MATH 32. CALCULUS AND ANALYTIC GEOMETRY II. DEPARTMENT OF MATHEMATICS AND COMPUTER SCIENCE Contributors: U. N. Iyer, P. Laul, I. Petrovic. (Many problems have been irectly taken from Single Variable
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
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
Differential and Integral Calculus
School of science an engineering El Akhawayn University Monay, March 31 st, 2008 Outline 1 Definition of hyperbolic functions: The hyperbolic cosine an the hyperbolic sine of the real number x are enote
Trigonometric Identities Exam Questions
Trigonometric Identities Exam Questions Name: ANSWERS January 01 January 017 Multiple Choice 1. Simplify the following expression: cos x 1 cot x a. sin x b. cos x c. cot x d. sec x. Identify a non-permissible
Practice Questions From Calculus II. 0. State the following calculus rules (these are many of the key rules from Test 1 topics).
Math 132. Practice Questions From Calculus II I. Topics Covered in Test I 0. State the following calculus rules (these are many of the key rules from Test 1 topics). (Trapezoidal Rule) b a f(x) dx (Fundamental
Name Date Period. MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
AB Fall Final Exam Review 200-20 Name Date Period MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Solve the problem. ) The position of a particle
AP Calculus Summer Packet
AP Calculus Summer Packet Writing The Equation Of A Line Example: Find the equation of a line that passes through ( 1, 2) and (5, 7). ü Things to remember: Slope formula, point-slope form, slopeintercept
Calculus I Exam 1 Review Fall 2016
Problem 1: Decide whether the following statements are true or false: (a) If f, g are differentiable, then d d x (f g) = f g. (b) If a function is continuous, then it is differentiable. (c) If a function
7.3 Inverse Trigonometric Functions
58 transcendental functions 73 Inverse Trigonometric Functions We now turn our attention to the inverse trigonometric functions, their properties and their graphs, focusing on properties and techniques
Inverse Trigonometric Functions
Inverse Trigonometric Functions Lori Jordan, (LoriJ) Brenda Meery, (BrendaM) Say Thanks to the Authors Click http://www.ck1.org/saythanks (No sign in required) To access a customizable version of this
Lecture 25: The Sine and Cosine Functions. tan(x) 1+y
Lecture 5: The Sine Cosine Functions 5. Denitions We begin b dening functions s : c : ; i! R ; i! R b Note that 8 >< q tan(x) ; if x s(x) + tan (x) ; >: ; if x 8 >< q ; if x c(x) + tan (x) ; >: 0; if x.
Core Mathematics 3 Differentiation
http://kumarmaths.weebly.com/ Core Mathematics Differentiation C differentiation Page Differentiation C Specifications. By the end of this unit you should be able to : Use chain rule to find the derivative
Announcements. Topics: Homework: - sections 4.5 and * Read these sections and study solved examples in your textbook!
Announcements Topics: - sections 4.5 and 5.1-5.5 * Read these sections and study solved examples in your textbook! Homework: - review lecture notes thoroughly - work on practice problems from the textbook
SESSION 6 Trig. Equations and Identities. Math 30-1 R 3. (Revisit, Review and Revive)
SESSION 6 Trig. Equations and Identities Math 30-1 R 3 (Revisit, Review and Revive) 1 P a g e 2 P a g e Mathematics 30-1 Learning Outcomes Specific Outcome 5: Solve, algebraically and graphically, first
Section 4.8 Anti Derivative and Indefinite Integrals 2 Lectures. Dr. Abdulla Eid. College of Science. MATHS 101: Calculus I
Section 4.8 Anti Derivative and Indefinite Integrals 2 Lectures College of Science MATHS 101: Calculus I (University of Bahrain) 1 / 28 Indefinite Integral Given a function f, if F is a function such that
Name Date Period. Calculater Permitted MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
PreAP Precalculus Spring Final Exam Review Name Date Period Calculater Permitted MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Simplify the expression.