Inverse Trigonometric Functions. September 5, 2018

Size: px
Start display at page:

Download "Inverse Trigonometric Functions. September 5, 2018"

Transcription

1 Inverse Trigonometric Functions September 5, 08 / 7

2 Restricted Sine Function. The trigonometric function sin x is not a one-to-one functions Π 6, 5Π 6, Π Π Π Π 0.5 We still want an inverse, so what to do? We must restrict the domain of the sin function. We want to choose some interval (as large as we can find) so that [ sin(x) is one-to-one on that interval. From the graph we see that π, π ] looks likes a good choice. To be super cautious note d sin(x) [ ] = cos(x) 0 on dx sin(x) is increasing on π, π ]. [ π, π and 0 only at the end points so / 7

3 The restricted sine function Define the restricted sine function by sin x π x π f(x) = undefined otherwise We have Domain(f) = [ π, π ] and Range(f) = [, ] and graph Π Π 4 4 Π Π / 7

4 Inverse Sine Function (arcsin x = sin x). The restricted sine function is one-to-one and hence has an inverse, shown in red in the diagram below..5, Π.0 Π, 0.5 Π Π 4 4 Π Π Π, 4 Π, This inverse function, f (x), is denoted by.5 f (x) = sin x or arcsin x. sin is terrible notation since you will confuse it with the cosecant function. 4 / 7

5 Properties of arcsin(x). Domain(arcsin) = [, ] and Range(arcsin) = [ π, π ]. Since f (x) = y if and only if f(y) = x, we have: arcsin x = y if and only if sin(y) = x and π y π. Since f(f (x)) = x f (f(x)) = x we have: sin(arcsin(x)) = x for x [, ] [ arcsin(sin(x)) = x for x π, π ]. From the graph: arcsin(x) is an odd function and so arcsin( x) = arcsin(x). 5 / 7

6 Evaluating arcsin(x). Example Evaluate arcsin We see that the point Therefore arcsin ( ) (, π 4 ( ) = π 4. using the graph above. ) is on the graph of y = arcsin(x). 6 / 7

7 ( ) ( ) 3 3 Example Evaluate arcsin and arcsin. ( ) 3 arcsin = y is the same statement as: y is an angle between π and π 3 with sin y =. Consulting our unit circle, we see that y = π 3. ( ) ( ) 3 3 arcsin = arcsin = π 3 7 / 7

8 More Examples For arcsin(x) Example Evaluate arcsin(sin π). It is tempting to write π until you realize the answer is between π and π. We have sin π = 0, hence arcsin(sin π) = arcsin(0) = 0. Example Evaluate cos(arcsin( 3/)). ( ) 3 We saw above that arcsin = π 3. ( ( )) 3 ( π ) Therefore cos arcsin = cos = 3. 8 / 7

9 Preparation for the method of Trigonometric Substitution Example Give a formula in terms of x for tan ( arcsin(x) ). We draw a right angled triangle with θ = arcsin(x). x θ - x From this we see that tan ( arcsin(x) ) = tan(θ) = x x. 9 / 7

10 Derivative of arcsin(x). d dx arcsin(x) = x, < x <. Proof We have arcsin(x) = y if and only if sin(y) = x and π/ y π/. Using implicit differentiation, we get cos(y) dy dx = or dy dx = cos(y). Now we know that cos (y) + sin (y) =, hence we have that cos (y) + x = and cos(y) = ± x For y between π and π, cos(y) 0 so cos(y) = x and d dx arcsin(x) = x 0 / 7

11 Derivative of arcsin(x): Example Example Find the derivative d dx arcsin( cos(x) ). We have d dx arcsin( cos(x) ) = = d ( cos(x) ) cos x dx sin x cos x cos x = sin x cos x cos x / 7

12 Inverse Cosine Function Inverse Cosine Function We can define the function cos x = arccos(x) similarly. The details are given at the end of your lecture notes. Domain(arccos) = [, ] and Range(arccos) = [0, π]. arccos(x) = y if and only if cos(y) = x and 0 y π. cos(arccos(x)) = x for x [, ] arccos(cos(x)) = x for x [ 0, π ]. / 7

13 Derivative of Inverse Cosine Function Inverse Cosine Function Also by implicit differentiation, we can show that (see end of the lecture notes) d dx arccos(x) = d dx arcsin(x) = x Note that since both have the same derivative, we must have arccos(x) = arcsin(x) + C for some constant C. Letting x = 0, we get that arcsin(x) + arccos(x) = π. This means that the arccos is not an odd function. In fact arccos( x) = π arccos(x). 3 / 7

14 Restricted Tangent Function The tangent function is not a one to one function. The restricted tangent function is given by tan x π < x < π h(x) = undefined otherwise We see from the graph of the restricted tangent function (or from its derivative) that the function is one-to-one and hence has an inverse, which we denote by h (x) = tan x or arctan x. 4 / 7

15 6 4 Π 4, Π Π 4 Π 4 4 Π Π 4, Π Π / 7

16 Properties of arctan(x). Domain(arctan) = (, ) and Range(arctan) = ( π, π ). Since h (x) = y if and only if h(y) = x, we have: arctan(x) = y if and only if tan(y) = x and π < y < π. Since h(h (x)) = x and h (h(x)) = x, we have: tan(arctan(x)) = x for x (, ) ( arctan(tan(x)) = x for x π, π ) 6 / 7

17 Π 4 From the graph, we have: arctan( x) = arctan(x). Also, since lim tan x = and lim tan x =, x ( π ) x ( π )+ we have lim x arctan(x) = π and lim x arctan(x) = π Π Π 4, Π / 7

18 Evaluating arctan(x) ( Example Find arctan() and arctan 3 ). arctan() is the unique angle, θ, between π and π with tan(θ) = sin(θ) cos(θ) =. By inspecting the unit circle, we see that θ = π 4. ) ( arctan is the unique angle, θ, between π 3 and π with tan(θ) = sin(θ) cos(θ) =. By inspecting the unit circle, we see that 3 θ = π 6. 8 / 7

19 Example ( ( Find cos arctan 3 )). ( ( )) ( π ) cos arctan 3 = cos = / 7

20 Derivative of arctan(x). Using implicit differentiation, we get d dx arctan(x) = x, < x <. + tan(y) = x; y sec (y) = so y ( + tan (y) ) = so y = + x. We can use the chain rule in conjunction with the above derivative. Example Find the domain and derivative of arctan ( ln(x) ) Domain = Domain of ln x = (0, ) d dx arctan( ln(x) ) = + (ln x) x = x( + (ln x) ). 0 / 7

21 Integration Formulas Reversing the derivative formulas above, we get dx = arcsin(x) + C, x dx = arctan(x) + C, + x Example / 0 + 4x dx We use substitution. Let u = x, then du = dx, u(0) = 0, u(/) =. / 0 + 4x dx = 0 + u du = arctan(u) 0 = ( ) arctan() arctan(0) = ( π ) 4 0 = π 8. / 7

22 Integration Example 9 x dx dx = 9 x dx = 3 x 3 9 dx x 9 Let u = x, then dx = 3du 3 dx = 9 x 3 3 ( x ) du = arcsin(u) + C = arcsin + C u 3 / 7

23 arcsec(x) (on 09/07) Here is a graph of the secant function. There are vertical asymptotes at π + kπ, k any integer. There is disagreement in the literature as to how to restrict it. The [ book uses 0, π ) [ π, 3π ) so we will too, but some authors prefer [ 0, π ) ( π ], π. 3 / 7

24 We are using the red curves, but some authors prefer the red in the first quadrant union the blue in the fourth. Our domain is [ 0, π ) [ π, 3π ) and the range is (, ] [, ). (The other choice has the same range.) 4 / 7

25 The red curve is the graph y = arcsec(x). The domain [ is(, ] [, ) and the range is 0, π ) [ π, 3π ). There are two limits (or horizontal asymptotes) lim x arcsec(x) = 3π and lim x arcsec(x) = π 5 / 7

26 d dx arcsec(x) = x x y = arcsec(x); sec y = x; y sec(y) tan(y) =. Now sec (y) = + tan (y) so tan (y) = sec y and hence tan(y) = ± sec y = ± x. Hence y = ± x x. From the graph, when x < 0 the curve is decreasing so the derivative is negative and the sign is +. Similarly, when x > 0 the curve is increasing so the derivative is positive and the sign is +. 6 / 7

27 The other choice for the arcsec gives a function which is always increasing but has derivative x. Since we will only use the x arcsec to integrate, we prefer the definition we chose. There are also inverse trig functions arccot and arccsc. These do not allow us to integrate anything new so we will not discuss them. 7 / 7

I.e., the range of f(x) = arctan(x) is all real numbers y such that π 2 < y < π 2

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

More information

MTH 112: Elementary Functions

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

More information

Inverse Trig 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)

More information

MTH 112: Elementary Functions

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

More information

SET 1. (1) Solve for x: (a) e 2x = 5 3x

SET 1. (1) Solve for x: (a) e 2x = 5 3x () Solve for x: (a) e x = 5 3x SET We take natural log on both sides: ln(e x ) = ln(5 3x ) x = 3 x ln(5) Now we take log base on both sides: log ( x ) = log (3 x ln 5) x = log (3 x ) + log (ln(5)) x x

More information

6.6 Inverse Trigonometric Functions

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

More information

June 9 Math 1113 sec 002 Summer 2014

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

More information

2 Trigonometric functions

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

More information

Lecture 5: Inverse Trigonometric Functions

Lecture 5: Inverse Trigonometric Functions 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

More information

Math 1 Lecture 22. Dartmouth College. Monday

Math 1 Lecture 22. Dartmouth College. Monday Math 1 Lecture 22 Dartmouth College Monday 10-31-16 Contents Reminders/Announcements Last Time Implicit Differentiation Derivatives of Inverse Functions Derivatives of Inverse Trigonometric Functions Examish

More information

SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. and θ is in quadrant IV. 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

More information

Chapter 3: Transcendental Functions

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

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

Section 3.5: Implicit Differentiation

Section 3.5: Implicit Differentiation Section 3.5: Implicit Differentiation In the previous sections, we considered the problem of finding the slopes of the tangent line to a given function y = f(x). The idea of a tangent line however is not

More information

9. The x axis is a horizontal line so a 1 1 function can touch the x axis in at most one place.

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

More information

For a semi-circle with radius r, its circumfrence is πr, so the radian measure of a semi-circle (a straight line) is

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

More information

Chapter 7: Techniques of Integration

Chapter 7: Techniques of Integration Chapter 7: Techniques of Integration MATH 206-01: Calculus II Department of Mathematics University of Louisville last corrected September 14, 2013 1 / 43 Chapter 7: Techniques of Integration 7.1. Integration

More information

Week 1: need to know. November 14, / 20

Week 1: need to know. November 14, / 20 Week 1: need to know How to find domains and ranges, operations on functions (addition, subtraction, multiplication, division, composition), behaviors of functions (even/odd/ increasing/decreasing), library

More information

There are some trigonometric identities given on the last page.

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

More information

6.1 The Inverse Sine, Cosine, and Tangent Functions Objectives

6.1 The Inverse Sine, Cosine, and Tangent Functions Objectives Objectives 1. Find the Exact Value of an Inverse Sine, Cosine, or Tangent Function. 2. Find an Approximate Value of an Inverse Sine Function. 3. Use Properties of Inverse Functions to Find Exact Values

More information

Math Section 4.3 Unit Circle Trigonometry

Math Section 4.3 Unit Circle Trigonometry Math 10 - Section 4. Unit Circle Trigonometry An angle is in standard position if its vertex is at the origin and its initial side is along the positive x axis. Positive angles are measured counterclockwise

More information

2 Recollection of elementary functions. II

2 Recollection of elementary functions. II Recollection of elementary functions. II Last updated: October 5, 08. In this section we continue recollection of elementary functions. In particular, we consider exponential, trigonometric and hyperbolic

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

Section Inverse Trigonometry. In this section, we will define inverse since, cosine and tangent functions. x is NOT one-to-one.

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

More information

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. 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

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

THE INVERSE TRIGONOMETRIC FUNCTIONS

THE INVERSE TRIGONOMETRIC FUNCTIONS THE INVERSE TRIGONOMETRIC FUNCTIONS Question 1 (**+) Solve the following trigonometric equation ( x ) π + 3arccos + 1 = 0. 1 x = Question (***) It is given that arcsin x = arccos y. Show, by a clear method,

More information

7.3 Inverse Trigonometric Functions

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

More information

More with Angles Reference Angles

More with Angles Reference Angles More with Angles Reference Angles A reference angle is the angle formed by the terminal side of an angle θ, and the (closest) x axis. A reference angle, θ', is always 0 o

More information

Exercises. 880 Foundations of Trigonometry

Exercises. 880 Foundations of Trigonometry 880 Foundations of Trigonometry 0.. Exercises For a link to all of the additional resources available for this section, click OSttS Chapter 0 materials. In Exercises - 0, find the exact value. For help

More information

SANDERSON HIGH SCHOOL AP CALCULUS AB/BC SUMMER REVIEW PACKET

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.

More information

Trigonometric substitutions (8.3).

Trigonometric substitutions (8.3). Review for Eam 2. 5 or 6 problems. No multiple choice questions. No notes, no books, no calculators. Problems similar to homeworks. Eam covers: 7.4, 7.6, 7.7, 8-IT, 8., 8.2. Solving differential equations

More information

Interpreting Derivatives, Local Linearity, Newton s

Interpreting Derivatives, Local Linearity, Newton s Unit #4 : Method Interpreting Derivatives, Local Linearity, Newton s Goals: Review inverse trigonometric functions and their derivatives. Create and use linearization/tangent line formulas. Investigate

More information

Math 106 Calculus 1 Topics for first exam

Math 106 Calculus 1 Topics for first exam Math 06 Calculus Topics for first exam Precalculus = what comes before its. Lines and their slopes: slope= rise over run = (change in y-value)/(corresponding change in x value) y y 0 slope-intercept: y

More information

Inverse Relations. 5 are inverses because their input and output are switched. For instance: f x x. x 5. f 4

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

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

A List of Definitions and Theorems

A List of Definitions and Theorems Metropolitan Community College Definition 1. Two angles are called complements if the sum of their measures is 90. Two angles are called supplements if the sum of their measures is 180. Definition 2. One

More information

Chapter 5 Notes. 5.1 Using Fundamental Identities

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

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

Trigonometry Trigonometry comes from the Greek word meaning measurement of triangles Angles are typically labeled with Greek letters

Trigonometry Trigonometry comes from the Greek word meaning measurement of triangles Angles are typically labeled with Greek letters Trigonometry Trigonometry comes from the Greek word meaning measurement of triangles Angles are typically labeled with Greek letters α( alpha), β ( beta), θ ( theta) as well as upper case letters A,B,

More information

1. The graph of a function f is given above. Answer the question: a. Find the value(s) of x where f is not differentiable. Ans: x = 4, x = 3, x = 2,

1. The graph of a function f is given above. Answer the question: a. Find the value(s) of x where f is not differentiable. Ans: x = 4, x = 3, x = 2, 1. The graph of a function f is given above. Answer the question: a. Find the value(s) of x where f is not differentiable. x = 4, x = 3, x = 2, x = 1, x = 1, x = 2, x = 3, x = 4, x = 5 b. Find the value(s)

More information

TRIGONOMETRY OUTCOMES

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.

More information

CALCULUS: Graphical,Numerical,Algebraic by Finney,Demana,Watts and Kennedy Chapter 3: Derivatives 3.3: Derivative of a function pg.

CALCULUS: Graphical,Numerical,Algebraic by Finney,Demana,Watts and Kennedy Chapter 3: Derivatives 3.3: Derivative of a function pg. CALCULUS: Graphical,Numerical,Algebraic b Finne,Demana,Watts and Kenned Chapter : Derivatives.: Derivative of a function pg. 116-16 What ou'll Learn About How to find the derivative of: Functions with

More information

Practice Differentiation Math 120 Calculus I Fall 2015

Practice Differentiation Math 120 Calculus I Fall 2015 . x. Hint.. (4x 9) 4x + 9. Hint. Practice Differentiation Math 0 Calculus I Fall 0 The rules of differentiation are straightforward, but knowing when to use them and in what order takes practice. Although

More information

These items need to be included in the notebook. Follow the order listed.

These items need to be included in the notebook. Follow the order listed. * Use the provided sheets. * This notebook should be your best written work. Quality counts in this project. Proper notation and terminology is important. We will follow the order used in class. Anyone

More information

Function and Relation Library

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

More information

Math Analysis Chapter 5 Notes: Analytic Trigonometric

Math Analysis Chapter 5 Notes: Analytic Trigonometric Math Analysis Chapter 5 Notes: Analytic Trigonometric Day 9: Section 5.1-Verifying Trigonometric Identities Fundamental Trig Identities Reciprocal Identities: 1 1 1 sin u = cos u = tan u = cscu secu cot

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

(Section 4.7: Inverse Trig Functions) 4.82 PART F: EVALUATING INVERSE TRIG FUNCTIONS. Think:

(Section 4.7: Inverse Trig Functions) 4.82 PART F: EVALUATING INVERSE TRIG FUNCTIONS. Think: PART F: EVALUATING INVERSE TRIG FUNCTIONS Think: (Section 4.7: Inverse Trig Functions) 4.82 A trig function such as sin takes in angles (i.e., real numbers in its domain) as inputs and spits out outputs

More information

Section 6.2 Trigonometric Functions: Unit Circle Approach

Section 6.2 Trigonometric Functions: Unit Circle Approach Section. Trigonometric Functions: Unit Circle Approach The unit circle is a circle of radius centered at the origin. If we have an angle in standard position superimposed on the unit circle, the terminal

More information

b n x n + b n 1 x n b 1 x + b 0

b n x n + b n 1 x n b 1 x + b 0 Math Partial Fractions Stewart 7.4 Integrating basic rational functions. For a function f(x), we have examined several algebraic methods for finding its indefinite integral (antiderivative) F (x) = f(x)

More information

3. Use absolute value notation to write an inequality that represents the statement: x is within 3 units of 2 on the real line.

3. Use absolute value notation to write an inequality that represents the statement: x is within 3 units of 2 on the real line. PreCalculus Review Review Questions 1 The following transformations are applied in the given order) to the graph of y = x I Vertical Stretch by a factor of II Horizontal shift to the right by units III

More information

Ch 5 and 6 Exam Review

Ch 5 and 6 Exam Review Ch 5 and 6 Exam Review Note: These are only a sample of the type of exerices that may appear on the exam. Anything covered in class or in homework may appear on the exam. Use the fundamental identities

More information

Math Section 4.3 Unit Circle Trigonometry

Math Section 4.3 Unit Circle Trigonometry Math 10 - Section 4. Unit Circle Trigonometry An angle is in standard position if its vertex is at the origin and its initial side is along the positive x axis. Positive angles are measured counterclockwise

More information

y= sin3 x+sin6x x 1 1 cos(2x + 4 ) = cos x + 2 = C(x) (M2) Therefore, C(x) is periodic with period 2.

y= sin3 x+sin6x x 1 1 cos(2x + 4 ) = cos x + 2 = C(x) (M2) Therefore, C(x) is periodic with period 2. . (a).5 0.5 y sin x+sin6x 0.5.5 (A) (C) (b) Period (C) []. (a) y x 0 x O x Notes: Award for end points Award for a maximum of.5 Award for a local maximum of 0.5 Award for a minimum of 0.75 Award for the

More information

DEPARTMENT OF MATHEMATICS

DEPARTMENT OF MATHEMATICS DEPARTMENT OF MATHEMATICS A2 level Mathematics Core 3 course workbook 2015-2016 Name: Welcome to Core 3 (C3) Mathematics. We hope that you will use this workbook to give you an organised set of notes for

More information

Pre-Calculus 40 Final Outline/Review:

Pre-Calculus 40 Final Outline/Review: 2016-2017 Pre-Calculus 40 Final Outline/Review: Non-Calculator Section: 16 multiple choice (32 pts) and 6 open ended (24 pts). Calculator Section: 8 multiple choice (16 pts) and 11 open ended (36 pts).

More information

Problem Sets for MATH1110 Fall 2017

Problem Sets for MATH1110 Fall 2017 Problem Sets for MATH1110 Fall 2017 Matt Hin November 27, 2017 Contents A Lectures L1-1 A.1 Lecture 1-22 August 2017................................... L1-1 A.2 Lecture 2-24 August 2017...................................

More information

x n cos 2x dx. dx = nx n 1 and v = 1 2 sin(2x). Andreas Fring (City University London) AS1051 Lecture Autumn / 36

x n cos 2x dx. dx = nx n 1 and v = 1 2 sin(2x). Andreas Fring (City University London) AS1051 Lecture Autumn / 36 We saw in Example 5.4. that we sometimes need to apply integration by parts several times in the course of a single calculation. Example 5.4.4: For n let S n = x n cos x dx. Find an expression for S n

More information

1 Lecture 39: The substitution rule.

1 Lecture 39: The substitution rule. Lecture 39: The substitution rule. Recall the chain rule and restate as the substitution rule. u-substitution, bookkeeping for integrals. Definite integrals, changing limits. Symmetry-integrating even

More information

DuVal High School Summer Review Packet AP Calculus

DuVal High School Summer Review Packet AP Calculus DuVal High School Summer Review Packet AP Calculus Welcome to AP Calculus AB. This packet contains background skills you need to know for your AP Calculus. My suggestion is, you read the information and

More information

Recall we measure angles in terms of degrees or radians: 360 = 2π radians

Recall we measure angles in terms of degrees or radians: 360 = 2π radians Review: trigonometry 8 Review of trigonometry 8.1 Definition of sine and cosine Recall we measure angles in terms of degrees or radians: 360 = 2π radians 8.2 Basic facts: 1. Te point (cos(t), sin(t)) lies

More information

MATH 127 SAMPLE FINAL EXAM I II III TOTAL

MATH 127 SAMPLE FINAL EXAM I II III TOTAL MATH 17 SAMPLE FINAL EXAM Name: Section: Do not write on this page below this line Part I II III TOTAL Score Part I. Multiple choice answer exercises with exactly one correct answer. Each correct answer

More information

Chapter 1: Limits and Continuity

Chapter 1: Limits and Continuity Chapter 1: Limits and Continuity Winter 2015 Department of Mathematics Hong Kong Baptist University 1/69 1.1 Examples where limits arise Calculus has two basic procedures: differentiation and integration.

More information

Using the Definitions of the Trigonometric Functions

Using the Definitions of the Trigonometric Functions 1.4 Using the Definitions of the Trigonometric Functions Reciprocal Identities Signs and Ranges of Function Values Pythagorean Identities Quotient Identities February 1, 2013 Mrs. Poland Objectives Objective

More information

Core 3 (A2) Practice Examination Questions

Core 3 (A2) Practice Examination Questions Core 3 (A) Practice Examination Questions Trigonometry Mr A Slack Trigonometric Identities and Equations I know what secant; cosecant and cotangent graphs look like and can identify appropriate restricted

More information

Final Exam. Math 3 December 7, 2010

Final Exam. Math 3 December 7, 2010 Final Exam Math 3 December 7, 200 Name: On this final examination for Math 3 in Fall 200, I will work individually, neither giving nor receiving help, guided by the Dartmouth Academic Honor Principle.

More information

AP Calculus AB Summer Math Packet

AP Calculus AB Summer Math Packet Name Date Section AP Calculus AB Summer Math Packet This assignment is to be done at you leisure during the summer. It is meant to help you practice mathematical skills necessary to be successful in Calculus

More information

CK- 12 Algebra II with Trigonometry Concepts 1

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.

More information

AP Calculus Summer Packet

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

More information

A-Level Mathematics TRIGONOMETRY. G. David Boswell - R2S Explore 2019

A-Level Mathematics TRIGONOMETRY. G. David Boswell - R2S Explore 2019 A-Level Mathematics TRIGONOMETRY G. David Boswell - R2S Explore 2019 1. Graphs the functions sin kx, cos kx, tan kx, where k R; In these forms, the value of k determines the periodicity of the trig functions.

More information

3.4 The Chain Rule. F (x) = f (g(x))g (x) Alternate way of thinking about it: If y = f(u) and u = g(x) where both are differentiable functions, then

3.4 The Chain Rule. F (x) = f (g(x))g (x) Alternate way of thinking about it: If y = f(u) and u = g(x) where both are differentiable functions, then 3.4 The Chain Rule To find the derivative of a function that is the composition of two functions for which we already know the derivatives, we can use the Chain Rule. The Chain Rule: Suppose F (x) = f(g(x)).

More information

Announcements. Topics: Homework: - sections 4.5 and * Read these sections and study solved examples in your textbook!

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

More information

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

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

More information

Albertson AP Calculus AB AP CALCULUS AB SUMMER PACKET DUE DATE: The beginning of class on the last class day of the first week of school.

Albertson AP Calculus AB AP CALCULUS AB SUMMER PACKET DUE DATE: The beginning of class on the last class day of the first week of school. Albertson AP Calculus AB Name AP CALCULUS AB SUMMER PACKET 2015 DUE DATE: The beginning of class on the last class day of the first week of school. This assignment is to be done at you leisure during the

More information

which can also be written as: 21 Basic differentiation - Chain rule: (f g) (b) = f (g(b)) g (b) dg dx for f(y) and g(x), = df dy

which can also be written as: 21 Basic differentiation - Chain rule: (f g) (b) = f (g(b)) g (b) dg dx for f(y) and g(x), = df dy 2 Basic differentiation - Chain rule: Chain rule: Suppose f and g are differentiable functions so that we can form their composition f g. The derivative of f g at input b can be computed in terms of the

More information

Summary: Primer on Integral Calculus:

Summary: Primer on Integral Calculus: Physics 2460 Electricity and Magnetism I, Fall 2006, Primer on Integration: Part I 1 Summary: Primer on Integral Calculus: Part I 1. Integrating over a single variable: Area under a curve Properties of

More information

Lesson 28 Working with Special Triangles

Lesson 28 Working with Special Triangles Lesson 28 Working with Special Triangles Pre-Calculus 3/3/14 Pre-Calculus 1 Review Where We ve Been We have a new understanding of angles as we have now placed angles in a circle on a coordinate plane

More information

NAME: DATE: CLASS: AP CALCULUS AB SUMMER MATH 2018

NAME: DATE: CLASS: AP CALCULUS AB SUMMER MATH 2018 NAME: DATE: CLASS: AP CALCULUS AB SUMMER MATH 2018 A] Refer to your pre-calculus notebook, the internet, or the sheets/links provided for assistance. B] Do not wait until the last minute to complete this

More information

MAT 1332: Calculus for Life Sciences. A course based on the book Modeling the dynamics of life by F.R. Adler

MAT 1332: Calculus for Life Sciences. A course based on the book Modeling the dynamics of life by F.R. Adler MT 33: Calculus for Life Sciences course based on the book Modeling the dynamics of life by F.R. dler Supplementary material University of Ottawa Frithjof Lutscher, with Robert Smith? January 3, MT 33:

More information

Topics and Concepts. 1. Limits

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

More information

Test one Review Cal 2

Test one Review Cal 2 Name: Class: Date: ID: A Test one Review Cal 2 Short Answer. Write the following expression as a logarithm of a single quantity. lnx 2ln x 2 ˆ 6 2. Write the following expression as a logarithm of a single

More information

4.3 Inverse Trigonometric Properties

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

More information

DIFFERENTIATION RULES

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,

More information

Hello Future Calculus Level One Student,

Hello Future Calculus Level One Student, Hello Future Calculus Level One Student, This assignment must be completed and handed in on the first day of class. This assignment will serve as the main review for a test on this material. The test will

More information

Chapter 5 Logarithmic, Exponential, and Other Transcendental Functions

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

More information

Chapter 1. Functions 1.3. Trigonometric Functions

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

More information

Derivatives of Trig and Inverse Trig Functions

Derivatives of Trig and Inverse Trig Functions Derivatives of Trig and Inverse Trig Functions Math 102 Section 102 Mingfeng Qiu Nov. 28, 2018 Office hours I m planning to have additional office hours next week. Next Monday (Dec 3), which time works

More information

4 The Trigonometric Functions

4 The Trigonometric Functions Mathematics Learning Centre, University of Sydney 8 The Trigonometric Functions The definitions in the previous section apply to between 0 and, since the angles in a right angle triangle can never be greater

More information

Practice Questions From Calculus II. 0. State the following calculus rules (these are many of the key rules from Test 1 topics).

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

More information

FUNCTIONS AND MODELS

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

More information

4181H Problem Set 10 Selected Solutions. Chapter 15. = lim. (by L Hôpital s Rule) (0) = lim. x 2, = lim x 3 ( sin x)x + cos x cos x = lim

4181H Problem Set 10 Selected Solutions. Chapter 15. = lim. (by L Hôpital s Rule) (0) = lim. x 2, = lim x 3 ( sin x)x + cos x cos x = lim 48H Problem Set 0 Selected Solutions Chapter 5 # 3(a From the definition of the derivative we have sin f f( f(0 (0 sin 0 0 0 0 = cos (by L Hôpital s Rule = sin (by L Hôpital s Rule = 0 (b From the definition

More information

Trigonometry Outline

Trigonometry Outline Trigonometr Outline Introduction Knowledge of the content of this outline is essential to perform well in calculus. The reader is urged to stud each of the three parts of the outline. Part I contains the

More information

Mth Review Problems for Test 2 Stewart 8e Chapter 3. For Test #2 study these problems, the examples in your notes, and the homework.

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

More information

MATH 151, FALL SEMESTER 2011 COMMON EXAMINATION 3 - VERSION B - SOLUTIONS

MATH 151, FALL SEMESTER 2011 COMMON EXAMINATION 3 - VERSION B - SOLUTIONS Name (print): Signature: MATH 5, FALL SEMESTER 0 COMMON EXAMINATION - VERSION B - SOLUTIONS Instructor s name: Section No: Part Multiple Choice ( questions, points each, No Calculators) Write your name,

More information

Chapter 5 Analytic Trigonometry

Chapter 5 Analytic Trigonometry Chapter 5 Analytic Trigonometry Overview: 5.1 Using Fundamental Identities 5.2 Verifying Trigonometric Identities 5.3 Solving Trig Equations 5.4 Sum and Difference Formulas 5.5 Multiple-Angle and Product-to-sum

More information

Precalculus Review. Functions to KNOW! 1. Polynomial Functions. Types: General form Generic Graph and unique properties. Constants. Linear.

Precalculus Review. Functions to KNOW! 1. Polynomial Functions. Types: General form Generic Graph and unique properties. Constants. Linear. Precalculus Review Functions to KNOW! 1. Polynomial Functions Types: General form Generic Graph and unique properties Constants Linear Quadratic Cubic Generalizations for Polynomial Functions - The domain

More information

Formulas to remember

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

More information

4-3 Trigonometric Functions on the Unit Circle

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

More information

MATH 150 TOPIC 16 TRIGONOMETRIC EQUATIONS

MATH 150 TOPIC 16 TRIGONOMETRIC EQUATIONS Math 150 T16-Trigonometric Equations Page 1 MATH 150 TOPIC 16 TRIGONOMETRIC EQUATIONS In calculus, you will often have to find the zeros or x-intercepts of a function. That is, you will have to determine

More information