11.2. Using a table of derivatives. Introduction. Prerequisites. Learning Style. Learning Outcomes
|
|
|
- Blanche Eaton
- 7 years ago
- Views:
Transcription
1 Using a table of derivatives 11.2 Introduction In Block 1 you were introduced to the idea of a derivative and you calculated some derivatives from first principles. Rather than always calculate the derivative of a function from first principles it is common practise to use a table of derivatives. This block provides such a table and shows you how to use it. Prerequisites Before starting this Block you should... Learning Outcomes After completing this Block you should be able to... use a table of derivatives understand the meaning of the term derivative understand what is meant by the notation dy dx Learning Style To achieve what is expected of you... allocate sufficient study time briefly revise the prerequisite material attempt every guided exercise and most of the other exercises
2 1. Table of derivatives Table 1 lists some of the common functions used in engineering and their corresponding derivatives. Remember, that in each case the function in the right-hand column tell us the rate of change, or the gradient of the graph, of the function on the left at a particular value of x. Table 1 Common functions and their derivatives (In this table k, n and c are constants) Function Derivative constant 0 x 1 kx k x n kx n e x e kx nx n 1 knx n 1 e x ke kx ln x 1/x ln kx 1/x sin x cos x sin kx k cos kx sin(kx + c) k cos(kx + c) cos x sin x cos kx k sin kx cos(kx + c) k sin(kx + c) tan x sec 2 x tan kx k sec 2 kx tan(kx + c) k sec 2 (kx + c) Key Point Particularly important is the rule for differentiating powers of functions. If y = x n then dy dx = nxn 1 For example, if y = x 3 then dy dx =3x2. Example Use Table 1 to find dy dx when y is given by (a) 7x (b) 14 (c) 5x2 (d) 4x 7 Engineering Mathematics: Open Learning Unit Level 1 2
3 Solution (a) We note that 7x is of the form kx where k = 7. Using Table 1 we then have dy =7. dx (b) Noting that 14 is a constant we see that dy =0. dx (c) We see that 5x 2 is of the form kx n, with k = 5 and n = 2. The derivative, knx n 1, is then 10x 1, or more simply, 10x. Soify =5x 2, then dy =10x. dx (d) We see that 4x 7 is of the form kx n, with k = 4 and n = 7. Hence the derivative, dy, is given dx by 28x 6. Try each part of this exercise Find dy dx when y is (a) x (b) 5 x 3 (c) 7 x Part (a) Write x as x 1 2, and use the result for x n with n = 1 2. Part (b) Write 5 x 3 as 5x 3 and use the result for differentiating kx n. Part (c) Write 7 x as 7x 1. Try each part of this exercise Use Table 1 to find dz given (a) z = dt et (b) z = e 8t (c) z = e 3t Although Table 1 is written using x as the independent variable, the Table can be used for any variable. Part (a) From Table 1, if y = e x, then dy = dx ex. Hence if z = e t then dz = dt et. Part (b) Use the result for e kx in Table 1 Part (c) Use the result for e kx in Table 1. Try each part of this exercise Find the derivative, dy dx (a) sin 2x (b) cos x 2 (c) tan 5x Part (a) Use the result for sin kx in Table 1. Part (b) Note that cos x is the same as cos 1 x. Use the result for cos kx in Table 1, and take 2 2 k = 1. 2 Part (c) Use the result for tan kx in Table 1, and take k =5. 3 Engineering Mathematics: Open Learning Unit Level 1
4 More exercises for you to try 1. Find the derivative of the following functions: (a) 9x 2 (b) 5 (c) 6x 3 (d) 13x 4 (e) ln 5t 2. Find dz when z is given by dt (a) 5 (b) t t 3 (c) 5t 2 (d) 3t Find the derivative of each of the following functions (a) sin 5x (b) cos 4t (c) tan 3r (d) e 2v (e) 1 e 3t 4. Find the derivative of the following (a) cos 2x 3 (b) sin( 2x) (c) tan πx (d) e x 2 (e) ln 2 3 x 2. Extending the Table of derivatives We now quote two simple rules which enable us to extend the range of functions which we can differentiate. The first rule is for differentiating sums, or differences, of functions. The reader should note that all of the rules quoted below can be obtained from first principles using the approach outlined in Block 1. Key Point The derivative of f(x)+g(x) is df dx + dg dx The derivative of f(x) g(x) is df dx dg dx This rule says that to find the derivative of the sum (or difference) of two functions, we simply calculate the sum (or difference) of the derivatives of each function. Example Find the derivative of y = x 6 + x 4. Solution We simply calculate the sum of the derivatives of each separate function: dy dx =6x5 +4x 3 The second rule tells us how to differentiate a multiple of a function. Engineering Mathematics: Open Learning Unit Level 1 4
5 Key Point If k is a constant the derivative of kf(x) isk df dx This rule tells us that if a function is multiplied by a constant, k, then the derivative is also multiplied by the same constant, k. Example Find the derivative of y =8e 2x. Solution Here we are interested in differentiating a multiple of the function e 2x. We simply differentiate e 2x and multiply the result by 8. Thus dy dx =8 2e2x =16e 2x Example Find the derivative of each of the following functions. (a) y = 6 sin 2x (b) y = 6 sin 2x +3x 2 (c) y = 6 sin 2x +3x 2 5e 3x Solution (a) From Table 1, the derivative of sin 2x is 2 cos 2x. Hence the derivative of 6 sin 2x is 6(2 cos 2x), that is, 12 cos 2x. dy y = 6 sin 2x, = 6(2 cos 2x) = 12 cos 2x dx (b) The function is the sum of two terms: 6 sin 2x and 3x 2. Part (a) has already been differentiated to 6 sin 2x, so we consider the derivative of 3x 2. The derivative of x 2 is 2x and so the derivative of 3x 2 is 3(2x), that is, 6x. These derivatives are now summed: y = 6 sin 2x +3x 2 dy = 12 cos 2x +6x dx (c) We differentiate each part of the function in turn. y = 6 sin 2x +3x 2 5e 3x dy dx = 6(2 cos 2x) + 3(2x) 5(3e3x ) = 12 cos 2x +6x 15e 3x 5 Engineering Mathematics: Open Learning Unit Level 1
6 Try each part of this exercise Find dy dx where y =7x5 3e 5x. Part (a) The derivative of x 5 is 5x 4. Hence obtain the derivative of 7x 5. Part (b) Obtain the derivative of e 5x. Part (c) Hence obtain the derivative of 3e 5x. Part (d) So obtain the derivative of y =7x 5 3e 5x. Try each part of this exercise Find dy dx where y = 4 cos x x3. Part (a) Obtain the derivative of cos x 2. Part (b) The derivative of 17 is 0. Now obtain the derivative of 9x 3. Part (c) So now obtain the derivative of y = 4 cos x x3. More exercises for you to try 1. Find dy when y is given by: dx (a) 3x 7 +8x 3 (b) 3x 4 +2x (c) + 14 x 2 x 2. Find the derivative of each of the following functions: (a) z(t) = 5 sin t + sin 5t (b) h(v) = 3 cos 2v 6 sin v 2 3x (d) 3+2x 4 (e) (2 + 3x) 2 (c) m(n) =4e 2n + 2 e 2n + n2 2 (d) H(t) = e3t tan 2t (e) S(r) =(r2 +1) 2 4e 2r 3. Differentiate the following functions. (a) A(t) = (3 + e t ) 2 (b) B(s) =πe 2s sin πs s (c) V (r) =(1+ 1 r )2 +(r +1) 2 (d) M(θ) = 6 sin 2θ 2 cos θ 4 +2θ2 (e) H(t) = 4 tan 3t + 3 sin 2t 2 cos 4t Engineering Mathematics: Open Learning Unit Level 1 6
7 3. Evaluating a derivative The need to find the rate of change of a function at a particular point occurs often. We do this by finding the derivative of the function, and then evaluating the derivative at that point. When taking derivatives of trigonometric functions, any angles must be measured in radians. Consider a function, y(x). We use the notation dy (a) dx ory (a) to denote the derivative of y evaluated at x = a. Soy (0.5) means the value of the derivative of y when x =0.5. Example Find the value of the derivative of y = x 3 where x = 2. Interpret your result. Solution We have y = x 3 and so dy dx =3x2. When x =2, dy = dx 3(2)2 = 12, that is, dy (2) = 12 (Equivalently, dx y (2) = 12). The derivative is positive when x = 2 and so y is increasing at this point. When x =2,y is increasing at a rate of 12 vertical units per horizontal unit. More exercises for you to try 1. Calculate the derivative of y = x 2 + sin x when x = Calculate the rate of change of i(t) = 4 sin 2t +3t when (a) t = π 3 (b) t = Calculate the rate of change of F (t) =5sint 3 cos 2t when (a) t = 0 (b) t =1.3 7 Engineering Mathematics: Open Learning Unit Level 1
8 End of Block 11.2 Engineering Mathematics: Open Learning Unit Level 1 8
9 dy dx = nxn 1 = 1 2 x = 1 2 x 1 2. This may be written as 1 2 x. 9 Engineering Mathematics: Open Learning Unit Level 1
10 5( 3)x 3 1 = 15x 4 Engineering Mathematics: Open Learning Unit Level 1 10
11 7( 1)x 1 1 = 7x 2 = 7 x 2 11 Engineering Mathematics: Open Learning Unit Level 1
12 8e 8t Engineering Mathematics: Open Learning Unit Level 1 12
13 3e 3t 13 Engineering Mathematics: Open Learning Unit Level 1
14 (a) Using the result for sin kx, and taking k = 2, we see that when y = cos x 2 dy dx = 2 cos 2x Engineering Mathematics: Open Learning Unit Level 1 14
15 1 2 sin x 2 15 Engineering Mathematics: Open Learning Unit Level 1
16 5 sec 2 5x Engineering Mathematics: Open Learning Unit Level 1 16
17 1. (a) 18x (b) 0 (c) 18x 2 (d) 52x 3 (e) 1 t 2. (a) 15t 4 (b) 3 2 t 1 2 (c) 10t 3 (d) 9 4 t (a) 5 cos 5x (b) 4 sin 4t (c) 3 sec 2 3r (d) 2e 2v (e) 3e 3t 4. (a) 2 3 sin 2x 3 (b) 2 cos( 2x) (c) π sec2 πx (d) 1 2 e x 2 (e) 1 x 17 Engineering Mathematics: Open Learning Unit Level 1
18 35x 4 Engineering Mathematics: Open Learning Unit Level 1 18
19 5e 5x 19 Engineering Mathematics: Open Learning Unit Level 1
20 3(5e 5x )=15e 5x Engineering Mathematics: Open Learning Unit Level 1 20
21 35x 4 15e 5x 21 Engineering Mathematics: Open Learning Unit Level 1
22 1 2 sin x 2 Engineering Mathematics: Open Learning Unit Level 1 22
23 27x 2 23 Engineering Mathematics: Open Learning Unit Level 1
24 2 sin x 2 27x2 Engineering Mathematics: Open Learning Unit Level 1 24
25 1. (a) 21x 6 +24x 2 (b) 12x 3 +3x 0.5 (c) x 3 x 2 (d) 1 (e) x 2 2. (a) 5 cos t + 5 cos 5t (b) 6 sin 2v 3 cos v (c) 8e 2n 4e 2n + n 2 (d) 3e3t sec 2 2t (e) 4r 3 +4r +8e 2r 3. (a) 6e t +2e 2t (b) 2πe 2s 1 s 2 +2π cos(πs) (c) r + 2 (d) 12 cos 2θ + 1 sin θ +4θ r 2 r (e) 12 sec 2 3t + 6 cos 2t + 8 sin 4t 25 Engineering Mathematics: Open Learning Unit Level 1
26 (a) 1 (b) (a) 5 (b) Engineering Mathematics: Open Learning Unit Level 1 26
14.1. Basic Concepts of Integration. Introduction. Prerequisites. Learning Outcomes. Learning Style
Basic Concepts of Integration 14.1 Introduction When a function f(x) is known we can differentiate it to obtain its derivative df. The reverse dx process is to obtain the function f(x) from knowledge of
Algebra I Notes Relations and Functions Unit 03a
OBJECTIVES: F.IF.A.1 Understand the concept of a function and use function notation. Understand that a function from one set (called the domain) to another set (called the range) assigns to each element
Series FOURIER SERIES. Graham S McDonald. A self-contained Tutorial Module for learning the technique of Fourier series analysis
Series FOURIER SERIES Graham S McDonald A self-contained Tutorial Module for learning the technique of Fourier series analysis Table of contents Begin Tutorial c 004 g.s.mcdonald@salford.ac.uk 1. Theory.
19.6. Finding a Particular Integral. Introduction. Prerequisites. Learning Outcomes. Learning Style
Finding a Particular Integral 19.6 Introduction We stated in Block 19.5 that the general solution of an inhomogeneous equation is the sum of the complementary function and a particular integral. We have
Using a table of derivatives
Using a table of derivatives In this unit we construct a Table of Derivatives of commonly occurring functions. This is done using the knowledge gained in previous units on differentiation from first principles.
Implicit Differentiation
Implicit Differentiation Sometimes functions are given not in the form y = f(x) but in a more complicated form in which it is difficult or impossible to express y explicitly in terms of x. Such functions
The Derivative. Philippe B. Laval Kennesaw State University
The Derivative Philippe B. Laval Kennesaw State University Abstract This handout is a summary of the material students should know regarding the definition and computation of the derivative 1 Definition
The Exponential Form of a Complex Number
The Exponential Form of a Complex Number 10.3 Introduction In this block we introduce a third way of expressing a complex number: the exponential form. We shall discover, through the use of the complex
Chapter 7 Outline Math 236 Spring 2001
Chapter 7 Outline Math 236 Spring 2001 Note 1: Be sure to read the Disclaimer on Chapter Outlines! I cannot be responsible for misfortunes that may happen to you if you do not. Note 2: Section 7.9 will
is identically equal to x 2 +3x +2
Partial fractions 3.6 Introduction It is often helpful to break down a complicated algebraic fraction into a sum of simpler fractions. 4x+7 For example it can be shown that has the same value as 1 + 3
36 CHAPTER 1. LIMITS AND CONTINUITY. Figure 1.17: At which points is f not continuous?
36 CHAPTER 1. LIMITS AND CONTINUITY 1.3 Continuity Before Calculus became clearly de ned, continuity meant that one could draw the graph of a function without having to lift the pen and pencil. While this
1.5. Factorisation. Introduction. Prerequisites. Learning Outcomes. Learning Style
Factorisation 1.5 Introduction In Block 4 we showed the way in which brackets were removed from algebraic expressions. Factorisation, which can be considered as the reverse of this process, is dealt with
Representation of functions as power series
Representation of functions as power series Dr. Philippe B. Laval Kennesaw State University November 9, 008 Abstract This document is a summary of the theory and techniques used to represent functions
3.2. Solving quadratic equations. Introduction. Prerequisites. Learning Outcomes. Learning Style
Solving quadratic equations 3.2 Introduction A quadratic equation is one which can be written in the form ax 2 + bx + c = 0 where a, b and c are numbers and x is the unknown whose value(s) we wish to find.
2 Session Two - Complex Numbers and Vectors
PH2011 Physics 2A Maths Revision - Session 2: Complex Numbers and Vectors 1 2 Session Two - Complex Numbers and Vectors 2.1 What is a Complex Number? The material on complex numbers should be familiar
Math Placement Test Practice Problems
Math Placement Test Practice Problems The following problems cover material that is used on the math placement test to place students into Math 1111 College Algebra, Math 1113 Precalculus, and Math 2211
Microeconomic Theory: Basic Math Concepts
Microeconomic Theory: Basic Math Concepts Matt Van Essen University of Alabama Van Essen (U of A) Basic Math Concepts 1 / 66 Basic Math Concepts In this lecture we will review some basic mathematical concepts
Techniques of Integration
CHPTER 7 Techniques of Integration 7.. Substitution Integration, unlike differentiation, is more of an art-form than a collection of algorithms. Many problems in applied mathematics involve the integration
2.2 Derivative as a Function
2.2 Derivative as a Function Recall that we defined the derivative as f (a) = lim h 0 f(a + h) f(a) h But since a is really just an arbitrary number that represents an x-value, why don t we just use x
19.7. Applications of Differential Equations. Introduction. Prerequisites. Learning Outcomes. Learning Style
Applications of Differential Equations 19.7 Introduction Blocks 19.2 to 19.6 have introduced several techniques for solving commonly-occurring firstorder and second-order ordinary differential equations.
Section 4.4. Using the Fundamental Theorem. Difference Equations to Differential Equations
Difference Equations to Differential Equations Section 4.4 Using the Fundamental Theorem As we saw in Section 4.3, using the Fundamental Theorem of Integral Calculus reduces the problem of evaluating a
3 Contour integrals and Cauchy s Theorem
3 ontour integrals and auchy s Theorem 3. Line integrals of complex functions Our goal here will be to discuss integration of complex functions = u + iv, with particular regard to analytic functions. Of
1. (from Stewart, page 586) Solve the initial value problem.
. (from Stewart, page 586) Solve the initial value problem.. (from Stewart, page 586) (a) Solve y = y. du dt = t + sec t u (b) Solve y = y, y(0) = 0., u(0) = 5. (c) Solve y = y, y(0) = if possible. 3.
y cos 3 x dx y cos 2 x cos x dx y 1 sin 2 x cos x dx
Trigonometric Integrals In this section we use trigonometric identities to integrate certain combinations of trigonometric functions. We start with powers of sine and cosine. EXAMPLE Evaluate cos 3 x dx.
9.4. The Scalar Product. Introduction. Prerequisites. Learning Style. Learning Outcomes
The Scalar Product 9.4 Introduction There are two kinds of multiplication involving vectors. The first is known as the scalar product or dot product. This is so-called because when the scalar product of
2 Integrating Both Sides
2 Integrating Both Sides So far, the only general method we have for solving differential equations involves equations of the form y = f(x), where f(x) is any function of x. The solution to such an equation
SOLVING TRIGONOMETRIC INEQUALITIES (CONCEPT, METHODS, AND STEPS) By Nghi H. Nguyen
SOLVING TRIGONOMETRIC INEQUALITIES (CONCEPT, METHODS, AND STEPS) By Nghi H. Nguyen DEFINITION. A trig inequality is an inequality in standard form: R(x) > 0 (or < 0) that contains one or a few trig functions
8.1. Cramer s Rule for Solving Simultaneous Linear Equations. Introduction. Prerequisites. Learning Outcomes. Learning Style
Cramer s Rule for Solving Simultaneous Linear Equations 8.1 Introduction The need to solve systems of linear equations arises frequently in engineering. The analysis of electric circuits and the control
Solutions to Homework 10
Solutions to Homework 1 Section 7., exercise # 1 (b,d): (b) Compute the value of R f dv, where f(x, y) = y/x and R = [1, 3] [, 4]. Solution: Since f is continuous over R, f is integrable over R. Let x
Lecture 3 : The Natural Exponential Function: f(x) = exp(x) = e x. y = exp(x) if and only if x = ln(y)
Lecture 3 : The Natural Exponential Function: f(x) = exp(x) = Last day, we saw that the function f(x) = ln x is one-to-one, with domain (, ) and range (, ). We can conclude that f(x) has an inverse function
MATHEMATICS FOR ENGINEERING INTEGRATION TUTORIAL 3 - NUMERICAL INTEGRATION METHODS
MATHEMATICS FOR ENGINEERING INTEGRATION TUTORIAL - NUMERICAL INTEGRATION METHODS This tutorial is essential pre-requisite material for anyone studying mechanical engineering. This tutorial uses the principle
Evaluating trigonometric functions
MATH 1110 009-09-06 Evaluating trigonometric functions Remark. Throughout this document, remember the angle measurement convention, which states that if the measurement of an angle appears without units,
1 Lecture: Integration of rational functions by decomposition
Lecture: Integration of rational functions by decomposition into partial fractions Recognize and integrate basic rational functions, except when the denominator is a power of an irreducible quadratic.
1.7. Partial Fractions. 1.7.1. Rational Functions and Partial Fractions. A rational function is a quotient of two polynomials: R(x) = P (x) Q(x).
.7. PRTIL FRCTIONS 3.7. Partial Fractions.7.. Rational Functions and Partial Fractions. rational function is a quotient of two polynomials: R(x) = P (x) Q(x). Here we discuss how to integrate rational
3.1. Solving linear equations. Introduction. Prerequisites. Learning Outcomes. Learning Style
Solving linear equations 3.1 Introduction Many problems in engineering reduce to the solution of an equation or a set of equations. An equation is a type of mathematical expression which contains one or
MA4001 Engineering Mathematics 1 Lecture 10 Limits and Continuity
MA4001 Engineering Mathematics 1 Lecture 10 Limits and Dr. Sarah Mitchell Autumn 2014 Infinite limits If f(x) grows arbitrarily large as x a we say that f(x) has an infinite limit. Example: f(x) = 1 x
THE COMPLEX EXPONENTIAL FUNCTION
Math 307 THE COMPLEX EXPONENTIAL FUNCTION (These notes assume you are already familiar with the basic properties of complex numbers.) We make the following definition e iθ = cos θ + i sin θ. (1) This formula
10.3. The Exponential Form of a Complex Number. Introduction. Prerequisites. Learning Outcomes
The Exponential Form of a Complex Number 10.3 Introduction In this Section we introduce a third way of expressing a complex number: the exponential form. We shall discover, through the use of the complex
15.3. Calculating centres of mass. Introduction. Prerequisites. Learning Outcomes. Learning Style
Calculating centres of mass 15.3 Introduction In this block we show how the idea of integration as the limit of a sum can be used to find the centre of mass of an object such as a thin plate, like a sheet
Taylor and Maclaurin Series
Taylor and Maclaurin Series In the preceding section we were able to find power series representations for a certain restricted class of functions. Here we investigate more general problems: Which functions
Solving simultaneous equations using the inverse matrix
Solving simultaneous equations using the inverse matrix 8.2 Introduction The power of matrix algebra is seen in the representation of a system of simultaneous linear equations as a matrix equation. Matrix
1. Introduction sine, cosine, tangent, cotangent, secant, and cosecant periodic
1. Introduction There are six trigonometric functions: sine, cosine, tangent, cotangent, secant, and cosecant; abbreviated as sin, cos, tan, cot, sec, and csc respectively. These are functions of a single
INTEGRATING FACTOR METHOD
Differential Equations INTEGRATING FACTOR METHOD Graham S McDonald A Tutorial Module for learning to solve 1st order linear differential equations Table of contents Begin Tutorial c 2004 g.s.mcdonald@salford.ac.uk
a cos x + b sin x = R cos(x α)
a cos x + b sin x = R cos(x α) In this unit we explore how the sum of two trigonometric functions, e.g. cos x + 4 sin x, can be expressed as a single trigonometric function. Having the ability to do this
correct-choice plot f(x) and draw an approximate tangent line at x = a and use geometry to estimate its slope comment The choices were:
Topic 1 2.1 mode MultipleSelection text How can we approximate the slope of the tangent line to f(x) at a point x = a? This is a Multiple selection question, so you need to check all of the answers that
Section 10.4 Vectors
Section 10.4 Vectors A vector is represented by using a ray, or arrow, that starts at an initial point and ends at a terminal point. Your textbook will always use a bold letter to indicate a vector (such
Homework #2 Solutions
MAT Spring Problems Section.:, 8,, 4, 8 Section.5:,,, 4,, 6 Extra Problem # Homework # Solutions... Sketch likely solution curves through the given slope field for dy dx = x + y...8. Sketch likely solution
F Matrix Calculus F 1
F Matrix Calculus F 1 Appendix F: MATRIX CALCULUS TABLE OF CONTENTS Page F1 Introduction F 3 F2 The Derivatives of Vector Functions F 3 F21 Derivative of Vector with Respect to Vector F 3 F22 Derivative
Recognizing Types of First Order Differential Equations E. L. Lady
Recognizing Types of First Order Differential Equations E. L. Lady Every first order differential equation to be considered here can be written can be written in the form P (x, y)+q(x, y)y =0. This means
The Vector or Cross Product
The Vector or ross Product 1 ppendix The Vector or ross Product We saw in ppendix that the dot product of two vectors is a scalar quantity that is a maximum when the two vectors are parallel and is zero
Period of Trigonometric Functions
Period of Trigonometric Functions In previous lessons we have learned how to translate any primary trigonometric function horizontally or vertically, and how to Stretch Vertically (change Amplitude). In
National 5 Mathematics Course Assessment Specification (C747 75)
National 5 Mathematics Course Assessment Specification (C747 75) Valid from August 013 First edition: April 01 Revised: June 013, version 1.1 This specification may be reproduced in whole or in part for
MAC 1114. Learning Objectives. Module 10. Polar Form of Complex Numbers. There are two major topics in this module:
MAC 1114 Module 10 Polar Form of Complex Numbers Learning Objectives Upon completing this module, you should be able to: 1. Identify and simplify imaginary and complex numbers. 2. Add and subtract complex
Trigonometric Functions and Triangles
Trigonometric Functions and Triangles Dr. Philippe B. Laval Kennesaw STate University August 27, 2010 Abstract This handout defines the trigonometric function of angles and discusses the relationship between
Review Solutions MAT V1102. 1. (a) If u = 4 x, then du = dx. Hence, substitution implies 1. dx = du = 2 u + C = 2 4 x + C.
Review Solutions MAT V. (a) If u 4 x, then du dx. Hence, substitution implies dx du u + C 4 x + C. 4 x u (b) If u e t + e t, then du (e t e t )dt. Thus, by substitution, we have e t e t dt e t + e t u
Practice Final Math 122 Spring 12 Instructor: Jeff Lang
Practice Final Math Spring Instructor: Jeff Lang. Find the limit of the sequence a n = ln (n 5) ln (3n + 8). A) ln ( ) 3 B) ln C) ln ( ) 3 D) does not exist. Find the limit of the sequence a n = (ln n)6
6.1. The Exponential Function. Introduction. Prerequisites. Learning Outcomes. Learning Style
The Exponential Function 6.1 Introduction In this block we revisit the use of exponents. We consider how the expression a x is defined when a is a positive number and x is irrational. Previously we have
Inner Product Spaces
Math 571 Inner Product Spaces 1. Preliminaries An inner product space is a vector space V along with a function, called an inner product which associates each pair of vectors u, v with a scalar u, v, and
DERIVATIVES AS MATRICES; CHAIN RULE
DERIVATIVES AS MATRICES; CHAIN RULE 1. Derivatives of Real-valued Functions Let s first consider functions f : R 2 R. Recall that if the partial derivatives of f exist at the point (x 0, y 0 ), then we
ANALYTICAL METHODS FOR ENGINEERS
UNIT 1: Unit code: QCF Level: 4 Credit value: 15 ANALYTICAL METHODS FOR ENGINEERS A/601/1401 OUTCOME - TRIGONOMETRIC METHODS TUTORIAL 1 SINUSOIDAL FUNCTION Be able to analyse and model engineering situations
1 TRIGONOMETRY. 1.0 Introduction. 1.1 Sum and product formulae. Objectives
TRIGONOMETRY Chapter Trigonometry Objectives After studying this chapter you should be able to handle with confidence a wide range of trigonometric identities; be able to express linear combinations of
SECOND-ORDER LINEAR DIFFERENTIAL EQUATIONS
SECOND-ORDER LINEAR DIFFERENTIAL EQUATIONS A second-order linear differential equation has the form 1 Px d y dx dy Qx dx Rxy Gx where P, Q, R, and G are continuous functions. Equations of this type arise
Math 432 HW 2.5 Solutions
Math 432 HW 2.5 Solutions Assigned: 1-10, 12, 13, and 14. Selected for Grading: 1 (for five points), 6 (also for five), 9, 12 Solutions: 1. (2y 3 + 2y 2 ) dx + (3y 2 x + 2xy) dy = 0. M/ y = 6y 2 + 4y N/
Trigonometry Review with the Unit Circle: All the trig. you ll ever need to know in Calculus
Trigonometry Review with the Unit Circle: All the trig. you ll ever need to know in Calculus Objectives: This is your review of trigonometry: angles, six trig. functions, identities and formulas, graphs:
Calculus. Contents. Paul Sutcliffe. Office: CM212a.
Calculus Paul Sutcliffe Office: CM212a. www.maths.dur.ac.uk/~dma0pms/calc/calc.html Books One and several variables calculus, Salas, Hille & Etgen. Calculus, Spivak. Mathematical methods in the physical
Introduction to Differential Calculus. Christopher Thomas
Mathematics Learning Centre Introduction to Differential Calculus Christopher Thomas c 1997 University of Sydney Acknowledgements Some parts of this booklet appeared in a similar form in the booklet Review
Introduction to Complex Numbers in Physics/Engineering
Introduction to Complex Numbers in Physics/Engineering ference: Mary L. Boas, Mathematical Methods in the Physical Sciences Chapter 2 & 14 George Arfken, Mathematical Methods for Physicists Chapter 6 The
SOLUTIONS. f x = 6x 2 6xy 24x, f y = 3x 2 6y. To find the critical points, we solve
SOLUTIONS Problem. Find the critical points of the function f(x, y = 2x 3 3x 2 y 2x 2 3y 2 and determine their type i.e. local min/local max/saddle point. Are there any global min/max? Partial derivatives
LS.6 Solution Matrices
LS.6 Solution Matrices In the literature, solutions to linear systems often are expressed using square matrices rather than vectors. You need to get used to the terminology. As before, we state the definitions
Calculus 1: Sample Questions, Final Exam, Solutions
Calculus : Sample Questions, Final Exam, Solutions. Short answer. Put your answer in the blank. NO PARTIAL CREDIT! (a) (b) (c) (d) (e) e 3 e Evaluate dx. Your answer should be in the x form of an integer.
Grade 12 Pre-Calculus Mathematics (40S) A Course for Independent Study
Grade 1 Pre-Calculus Mathematics (40S) A Course for Independent Study GRADE 1 PRE-CALCULUS MATHEMATICS (40S) A Course for Independent Study 007 Manitoba Education, Citizenship and Youth Manitoba Education,
Intermediate Value Theorem, Rolle s Theorem and Mean Value Theorem
Intermediate Value Theorem, Rolle s Theorem and Mean Value Theorem February 21, 214 In many problems, you are asked to show that something exists, but are not required to give a specific example or formula
Review of Fundamental Mathematics
Review of Fundamental Mathematics As explained in the Preface and in Chapter 1 of your textbook, managerial economics applies microeconomic theory to business decision making. The decision-making tools
Lies My Calculator and Computer Told Me
Lies My Calculator and Computer Told Me 2 LIES MY CALCULATOR AND COMPUTER TOLD ME Lies My Calculator and Computer Told Me See Section.4 for a discussion of graphing calculators and computers with graphing
100. In general, we can define this as if b x = a then x = log b
Exponents and Logarithms Review 1. Solving exponential equations: Solve : a)8 x = 4! x! 3 b)3 x+1 + 9 x = 18 c)3x 3 = 1 3. Recall: Terminology of Logarithms If 10 x = 100 then of course, x =. However,
MATH 132: CALCULUS II SYLLABUS
MATH 32: CALCULUS II SYLLABUS Prerequisites: Successful completion of Math 3 (or its equivalent elsewhere). Math 27 is normally not a sufficient prerequisite for Math 32. Required Text: Calculus: Early
PRE-CALCULUS GRADE 12
PRE-CALCULUS GRADE 12 [C] Communication Trigonometry General Outcome: Develop trigonometric reasoning. A1. Demonstrate an understanding of angles in standard position, expressed in degrees and radians.
Phasors. Phasors. by Prof. Dr. Osman SEVAİOĞLU Electrical and Electronics Engineering Department. ^ V cos (wt + θ) ^ V sin (wt + θ)
V cos (wt θ) V sin (wt θ) by Prof. Dr. Osman SEVAİOĞLU Electrical and Electronics Engineering Department EE 209 Fundamentals of Electrical and Electronics Engineering, Prof. Dr. O. SEVAİOĞLU, Page 1 Vector
1.4. Removing Brackets. Introduction. Prerequisites. Learning Outcomes. Learning Style
Removing Brackets 1. Introduction In order to simplify an expression which contains brackets it is often necessary to rewrite the expression in an equivalent form but without any brackets. This process
RADIUS OF CURVATURE AND EVOLUTE OF THE FUNCTION y=f(x)
RADIUS OF CURVATURE AND EVOLUTE OF THE FUNCTION y=f( In introductory calculus one learns about the curvature of a function y=f( also about the path (evolute that the center of curvature traces out as x
Factoring Special Polynomials
6.6 Factoring Special Polynomials 6.6 OBJECTIVES 1. Factor the difference of two squares 2. Factor the sum or difference of two cubes In this section, we will look at several special polynomials. These
To give it a definition, an implicit function of x and y is simply any relationship that takes the form:
2 Implicit function theorems and applications 21 Implicit functions The implicit function theorem is one of the most useful single tools you ll meet this year After a while, it will be second nature to
Nonhomogeneous Linear Equations
Nonhomogeneous Linear Equations In this section we learn how to solve second-order nonhomogeneous linear differential equations with constant coefficients, that is, equations of the form ay by cy G x where
Section 3.7. Rolle s Theorem and the Mean Value Theorem. Difference Equations to Differential Equations
Difference Equations to Differential Equations Section.7 Rolle s Theorem and the Mean Value Theorem The two theorems which are at the heart of this section draw connections between the instantaneous rate
Chapter 11. Techniques of Integration
Chapter Techniques of Integration Chapter 6 introduced the integral. There it was defined numerically, as the limit of approximating Riemann sums. Evaluating integrals by applying this basic definition
SOLVING TRIGONOMETRIC EQUATIONS
Mathematics Revision Guides Solving Trigonometric Equations Page 1 of 17 M.K. HOME TUITION Mathematics Revision Guides Level: AS / A Level AQA : C2 Edexcel: C2 OCR: C2 OCR MEI: C2 SOLVING TRIGONOMETRIC
1.7. formulae and transposition. Introduction. Prerequisites. Learning Outcomes. Learning Style
formulae and transposition 1.7 Introduction formulae are used frequently in almost all aspects of engineering in order to relate a physical quantity to one or more others. Many well-known physical laws
GeoGebra. 10 lessons. Gerrit Stols
GeoGebra in 10 lessons Gerrit Stols Acknowledgements GeoGebra is dynamic mathematics open source (free) software for learning and teaching mathematics in schools. It was developed by Markus Hohenwarter
Triangle Trigonometry and Circles
Math Objectives Students will understand that trigonometric functions of an angle do not depend on the size of the triangle within which the angle is contained, but rather on the ratios of the sides of
3. KINEMATICS IN TWO DIMENSIONS; VECTORS.
3. KINEMATICS IN TWO DIMENSIONS; VECTORS. Key words: Motion in Two Dimensions, Scalars, Vectors, Addition of Vectors by Graphical Methods, Tail to Tip Method, Parallelogram Method, Negative Vector, Vector
Second Order Linear Nonhomogeneous Differential Equations; Method of Undetermined Coefficients. y + p(t) y + q(t) y = g(t), g(t) 0.
Second Order Linear Nonhomogeneous Differential Equations; Method of Undetermined Coefficients We will now turn our attention to nonhomogeneous second order linear equations, equations with the standard
AP Calculus AB 2006 Scoring Guidelines
AP Calculus AB 006 Scoring Guidelines The College Board: Connecting Students to College Success The College Board is a not-for-profit membership association whose mission is to connect students to college
Integration ALGEBRAIC FRACTIONS. Graham S McDonald and Silvia C Dalla
Integration ALGEBRAIC FRACTIONS Graham S McDonald and Silvia C Dalla A self-contained Tutorial Module for practising the integration of algebraic fractions Table of contents Begin Tutorial c 2004 g.s.mcdonald@salford.ac.uk
Analysis of Stresses and Strains
Chapter 7 Analysis of Stresses and Strains 7.1 Introduction axial load = P / A torsional load in circular shaft = T / I p bending moment and shear force in beam = M y / I = V Q / I b in this chapter, we
Section 5-9 Inverse Trigonometric Functions
46 5 TRIGONOMETRIC FUNCTIONS Section 5-9 Inverse Trigonometric Functions Inverse Sine Function Inverse Cosine Function Inverse Tangent Function Summar Inverse Cotangent, Secant, and Cosecant Functions
1. First-order Ordinary Differential Equations
Advanced Engineering Mathematics 1. First-order ODEs 1 1. First-order Ordinary Differential Equations 1.1 Basic concept and ideas 1.2 Geometrical meaning of direction fields 1.3 Separable differential
An Introduction to Calculus. Jackie Nicholas
Mathematics Learning Centre An Introduction to Calculus Jackie Nicholas c 2004 University of Sydney Mathematics Learning Centre, University of Sydney 1 Some rules of differentiation and how to use them
Week 13 Trigonometric Form of Complex Numbers
Week Trigonometric Form of Complex Numbers Overview In this week of the course, which is the last week if you are not going to take calculus, we will look at how Trigonometry can sometimes help in working
MATH 304 Linear Algebra Lecture 20: Inner product spaces. Orthogonal sets.
MATH 304 Linear Algebra Lecture 20: Inner product spaces. Orthogonal sets. Norm The notion of norm generalizes the notion of length of a vector in R n. Definition. Let V be a vector space. A function α
www.mathsbox.org.uk ab = c a If the coefficients a,b and c are real then either α and β are real or α and β are complex conjugates
Further Pure Summary Notes. Roots of Quadratic Equations For a quadratic equation ax + bx + c = 0 with roots α and β Sum of the roots Product of roots a + b = b a ab = c a If the coefficients a,b and c