6. Differentiating the exponential and logarithm functions
|
|
- Briana Brooks
- 7 years ago
- Views:
Transcription
1 1 6. Differentiating te exponential and logaritm functions We wis to find and use derivatives for functions of te form f(x) = a x, were a is a constant. By far te most convenient suc function for tis purpose is te exponential function wit base te special number e. Definition. Te number e, wic is approximately , is te number suc tat lim e = 1 Te number e is called Euler's number, after te great matematician Leonard Euler ( ). Te major reason for te use of e is te following teorem, wic says tat e x is its own derivative: Teorem. Dx [e x ] = e x. Proof. For any function f(x), recall tat f '(x) = lim f(x+) - f(x) Hence Dx [e x ] = lim e (x+) - e x = lim e x e - e x = lim e x (e - 1) = e x (1) [by te definition of e] = e x. We often call te function f(x) = e x te exponential function, since it involves taking an exponent. Example. Find Dx [5 e x ] Answer: 5 e x Derivatives get substantially more complicated wen we remember te cain rule:
2 2 Recall tat te composition g f of te functions f and g is te function (g f)(x) = g(f(x)). Tis means, "do te function f to x, ten do g to te result." Cain Rule. Suppose f and g ave derivatives. Ten (g f)'(x) = g'(f(x)) f '(x). Tis means tat we can use te cain rule for te exponential function as well: Teorem. (Exponential cain rule) Dx [e f(x) ] = e f(x) Dx[f(x)]. Proof. Tis is te cain rule were g(y) = e y. Example. Find Dx [e 6x ] Dx [e 6x ] = e 6x Dx [6x ] = 6 e 6x Example. Find Dx [e x2 ] Dx [e x2 ] = e x2 Dx [x 2 ] = 2x e x2 Example. Find Dx [5 e 3x ] Dx [5 e 3x ] = 5 e 3x Dx [3x ] = 15 e 3x Example. Find Dx [2x e 4x ] Dx [2x e 4x ] = (2x) Dx [ e 4x ] + e 4x Dx [2x ] [by te Product Rule] = (2x) e 4x Dx[4x] + e 4x 2 = (2x) e 4x (4) + 2 e 4x = 8x e 4x + 2 e 4x Example. If f(t) = e 4t e 2t find f '(t). Use te quotient rule (1 + e 2t ) Dt[e 4t ] - e 4t Dt[1 + e 2t ] f '(t) = (1 + e 2t ) 4 e 4t - e 4t 2 e 2t = e 4t + 4 e 2t e 4t -2 e 4t e 2t =
3 3 4 e 4t + 4 e 6t -2 e 6t = e 4t + 2 e 6t = Closely related to te exponential function is te logaritm function. Recall te natural logaritm of x, written ln(x), is te number suc tat e ln(x) = x. Tus ln(x) is te power to wic e is raised to yield te number x. It makes sense only if x>0. Teorem. Dx [ln(x)] = 1/x if x > 0. Wy is it true tat Dx [ln(x)] = 1/x if x > 0? Te argument is a clever one. Let g(x) = e [ln(x)]. Ten by te Exponential Cain Rule we ave Dx[g(x)] = e [ln(x)] Dx [ln(x)] But g(x) = e [ln(x)] = x, so Dx[g(x)] = 1. Hence 1= e [ln(x)] Dx [ln(x)] 1= x Dx [ln(x)] Dx [ln(x)] = 1/x Example. If g(x) = 3 ln(x) find g '(2). g '(x) = 3 Dx [ln(x)] = 3 /x. Hence g '(2) = 3/2 = 1.5. Teorem. (Logaritm cain rule) Dx [ln (f(x))] Dx [f(x)] f '(x) = = f(x) f(x) Proof. Write g(x) = ln(x). Ten Dx [ln (f(x))] = Dx [g (f(x))] = g '(f(x)) f '(x) [by te Cain Rule] = [1 / f(x)] f '(x) = f '(x) / f(x) Example. Find Dx [ln ()] Dx [ln ()]
4 4 = Dx [] = 2x Example. Find f '(x) if f(x) = 2x ln () f '(x) = Dx [2 x ln ()] = (2x) Dx [ ln ()] +[ ln ()] Dx [2 x] = (2x) Dx [ ] [ ln ()] (2) = (2x) (2x) ln () = 4x ln () Example. Find f '(x) if f(x) = ln() x+1 By te quotient rule we ave f '(x) = (x+1) Dx [ln()] - [ln()] Dx [x+1] (x+1) 2 f '(x) = (x+1) [1/()] Dx [] - [ln()] (1) (x+1) 2 f '(x) = (x+1) [1/()] (2x) - ln() (x+1) 2 f '(x) = (x+1)(2x) /() - ln() (x+1) 2 Simplify by multiplying numerator and denominator by (x 2 +1).
5 5 f '(x) = (x+1)(2x) - () ln() (x+1) 2 () Example. Find f '(x) if f(x) = e 3x ln(2x + 1). Use te product rule: f ' (x) = e 3x Dx[ln(2x + 1)] + ln(2x + 1) Dx[ e 3x ] f ' (x) = e 3x [1/(2x + 1)]Dx[2x+1] + ln(2x + 1) e 3x (3) f ' (x) = e 3x [2/(2x + 1)]+ 3 e 3x ln(2x + 1) f ' (x) = 2 e 3x /(2x + 1)+ 3 e 3x ln(2x + 1) We sall often ave to solve equations involving exponents or logaritms. Tese will arise wen we need to find critical numbers, or were a function is increasing of decreasing, or finding maxima or minima. Tese typically involve te identities ln (e c ) = c e ln(b) = b Example. Solve for x if e 2x = 5. Give your answer (a) exactly (b) to 5 decimal places e 2x = 5 ln[e 2x ] = ln(5) 2 x = ln(5) x = [ln(5)]/2 Hence (a) as answer [ln(5)]/2. To find te answer to (b), use your calculator: to 5 decimal places means Example. Solve for x if ln(3x) - 4 = 0. Give your answer (a) exactly (b) to 5 decimal places Solution ln(3x) - 4 = 0 ln(3x) = 4 e [ln(3x)] = e 4 3x = e 4 (a) x = (e 4 )/3 (b) x = Example. Solve for x if e 2x - 5 x e 2x = 0. e 2x - 5 x e 2x = 0 Factor: e 2x (1-5 x) = 0. If a product is 0, one of te factors is 0. Hence eiter e 2x = 0 or 1-5x = 0 But e 2x > 0 since it is a positive number raised to a power, so it is never 0. Hence 1-5x = 0
6 6 5x = 1 x = 1/5 Example. Solve for x if x 2 e 3x + 12 e 3x = 7 x e 3x Rewrite as an expression = 0: x 2 e 3x - 7 x e 3x + 12 e 3x = 0 Factor e 3x (x 2-7 x + 12 ) = 0 Since e 3x >0, we must ave x 2-7 x + 12 = 0 (x-4)(x-3) = 0 x - 4 = 0 or x-3 = 0 x = 4 or x = 3 Example. Solve for x if 3 e -2x - 5 e -6x = 0. Give te answer to 4 decimal places. 3 e -2x - 5 e -6x = 0. 3 e -2x = 5 e -6x 3 e -2x / e -6x = 5 3 e -2x -(-6x) = 5 3 e 4x = 5 e 4x = 5/3 ln e 4x = ln(5/3) 4x = ln(5/3) x = (1/4) ln(5/3) = We can use tese new functions to answer questions about relative extrema and absolute extrema. Example. Let f(x) = x e -2x. (a) Tell were f is increasing, and were f is decreasing. (b) Locate and classify all critical numbers. (c) Find te absolute maximum and absolute minimum for 0 x 4, and were tey occur. f '(x) = x Dx [e -2x ] + e -2x Dx[x] = x e -2x Dx [-2x] + e -2x (1) = x e -2x (-2) + e -2x = -2x e -2x + e -2x To find te critical numbers we solve f '(x) = 0: -2x e -2x + e -2x = 0 Multiply by e 2x : -2x + 1 = 0 2x = 1 x = 1/2 We now find te cart for f '(x) by finding te value at points not equal to a critical number:
7 7 (+) 1/2 (-) (a) f is increasing for x < 1/2. f is decreasing for x > 1/2. (b) Te only critical number is 1/2 relative maximum (c) For 0 x 4 we form a cart including te endpoints and critical numbers in te interval x f(x) 1/ Te absolute maximum is at x = 1/2. Te absolute minimum is 0 at x = 0. Example. Let f(x) = ln (x) for x > 0. 2x (a) Tell were f is increasing, and were f is decreasing. (b) Locate and classify all critical numbers. (c) Find te absolute maximum and absolute minimum for 1 x 10, and were tey occur. Give answers to 3 decimal places We must solve f '(x) = 0. By te quotient rule 2x Dx[ ln(x)] - (ln (x)) Dx[2x] f ' (x) = (2x) 2 2x [1/x] - (ln (x)) (2) f ' (x) = x ln (x) f ' (x) = x 2 Hence f '(x) = 0 wen 2-2 ln(x) = 0 2 ln(x) = 2 ln(x) = 1 e ln(x) = e 1 x = e We find te cart for f '(x) as follows, remembering tat x>0. Note f(0) is meaningless. 0 (+) e (-) Hence (a) f is increasing for 0<x<e and f is decreasing for x>3. (b) Te only critical number is x = e, wic is a relative maximum.
8 8 (c) We must consider te endpoints x = 1 and x = 10 togeter wit critical numbers in te interval. Since 1<e<10, we must consider x= e. Hence our cart of values is x f(x) 1 ln (1) / (2(1)) = 0 10 ln (10) / 20 = e ln (e) / (2e) =.184 Te absolute maximum is.184 at x = e = Te absolute minimum is 0 at x = 1. Example. Te concentration C(t) of a drug in te blood t ours after a pill is swallowed is given by C(t) = 7 e -0.2t - 7 e -0.5t mg/l. (a) Wen is te concentration at a maximum? (b) Wat is te maximum concentration? (c) For wic t is te concentration increasing? (d) For wic t is te concentration decreasing? Give answers to 2 decimal places. We differentiate C(t): C '(t) = 7 e -0.2t (-0.2) - 7 e -0.5t (-0.5) = -1.4 e -0.2t +3.5 e -0.5t We need te critical numbers, so we solve C '(t) = e -0.2t +3.5 e -0.5t = 0 Divide by e -0.2t : e -0.5t / e -0.2t = e -0.5t+0.2t = e -0.3t = e -0.3t = 1.4 e -0.3t = 1.4/3.5 = 0.4 ln ( e -0.3t ) = ln( 0.4) -0.3 t = ln (0.4) t = (ln (0.4))/(-0.3) t = 3.05 ours Note te cart for C ' is ten (+) 3.05 (-) Hence te answers are (a) t = 3.05 ours. (b) C(3.05) = 2.28 mg/l (c) 0<t< 3.05 ours (d) t > 3.05 ours
Math 113 HW #5 Solutions
Mat 3 HW #5 Solutions. Exercise.5.6. Suppose f is continuous on [, 5] and te only solutions of te equation f(x) = 6 are x = and x =. If f() = 8, explain wy f(3) > 6. Answer: Suppose we ad tat f(3) 6. Ten
More informationCHAPTER 8: DIFFERENTIAL CALCULUS
CHAPTER 8: DIFFERENTIAL CALCULUS 1. Rules of Differentiation As we ave seen, calculating erivatives from first principles can be laborious an ifficult even for some relatively simple functions. It is clearly
More informationDerivatives Math 120 Calculus I D Joyce, Fall 2013
Derivatives Mat 20 Calculus I D Joyce, Fall 203 Since we ave a good understanding of its, we can develop derivatives very quickly. Recall tat we defined te derivative f x of a function f at x to be te
More informationLecture 10: What is a Function, definition, piecewise defined functions, difference quotient, domain of a function
Lecture 10: Wat is a Function, definition, piecewise defined functions, difference quotient, domain of a function A function arises wen one quantity depends on anoter. Many everyday relationsips between
More informationSection 3.3. Differentiation of Polynomials and Rational Functions. Difference Equations to Differential Equations
Difference Equations to Differential Equations Section 3.3 Differentiation of Polynomials an Rational Functions In tis section we begin te task of iscovering rules for ifferentiating various classes of
More informationf(a + h) f(a) f (a) = lim
Lecture 7 : Derivative AS a Function In te previous section we defined te derivative of a function f at a number a (wen te function f is defined in an open interval containing a) to be f (a) 0 f(a + )
More informationSections 3.1/3.2: Introducing the Derivative/Rules of Differentiation
Sections 3.1/3.2: Introucing te Derivative/Rules of Differentiation 1 Tangent Line Before looking at te erivative, refer back to Section 2.1, looking at average velocity an instantaneous velocity. Here
More informationMath 229 Lecture Notes: Product and Quotient Rules Professor Richard Blecksmith richard@math.niu.edu
Mat 229 Lecture Notes: Prouct an Quotient Rules Professor Ricar Blecksmit ricar@mat.niu.eu 1. Time Out for Notation Upate It is awkwar to say te erivative of x n is nx n 1 Using te prime notation for erivatives,
More informationInstantaneous Rate of Change:
Instantaneous Rate of Cange: Last section we discovered tat te average rate of cange in F(x) can also be interpreted as te slope of a scant line. Te average rate of cange involves te cange in F(x) over
More informationMath 120 Final Exam Practice Problems, Form: A
Math 120 Final Exam Practice Problems, Form: A Name: While every attempt was made to be complete in the types of problems given below, we make no guarantees about the completeness of the problems. Specifically,
More information5.1 Derivatives and Graphs
5.1 Derivatives and Graphs What does f say about f? If f (x) > 0 on an interval, then f is INCREASING on that interval. If f (x) < 0 on an interval, then f is DECREASING on that interval. A function has
More informationSolving Exponential Equations
Solving Exponential Equations Deciding How to Solve Exponential Equations When asked to solve an exponential equation such as x + 6 = or x = 18, the first thing we need to do is to decide which way is
More informationChapter 7 Numerical Differentiation and Integration
45 We ave a abit in writing articles publised in scientiþc journals to make te work as Þnised as possible, to cover up all te tracks, to not worry about te blind alleys or describe ow you ad te wrong idea
More informationHomework # 3 Solutions
Homework # 3 Solutions February, 200 Solution (2.3.5). Noting that and ( + 3 x) x 8 = + 3 x) by Equation (2.3.) x 8 x 8 = + 3 8 by Equations (2.3.7) and (2.3.0) =3 x 8 6x2 + x 3 ) = 2 + 6x 2 + x 3 x 8
More information2.1: The Derivative and the Tangent Line Problem
.1.1.1: Te Derivative and te Tangent Line Problem Wat is te deinition o a tangent line to a curve? To answer te diiculty in writing a clear deinition o a tangent line, we can deine it as te iting position
More informationCHAPTER FIVE. Solutions for Section 5.1. Skill Refresher. Exercises
CHAPTER FIVE 5.1 SOLUTIONS 265 Solutions for Section 5.1 Skill Refresher S1. Since 1,000,000 = 10 6, we have x = 6. S2. Since 0.01 = 10 2, we have t = 2. S3. Since e 3 = ( e 3) 1/2 = e 3/2, we have z =
More informationACT Math Facts & Formulas
Numbers, Sequences, Factors Integers:..., -3, -2, -1, 0, 1, 2, 3,... Rationals: fractions, tat is, anyting expressable as a ratio of integers Reals: integers plus rationals plus special numbers suc as
More informationLecture 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
More informationCHAPTER 7. Di erentiation
CHAPTER 7 Di erentiation 1. Te Derivative at a Point Definition 7.1. Let f be a function defined on a neigborood of x 0. f is di erentiable at x 0, if te following it exists: f 0 fx 0 + ) fx 0 ) x 0 )=.
More information6.4 Logarithmic Equations and Inequalities
6.4 Logarithmic Equations and Inequalities 459 6.4 Logarithmic Equations and Inequalities In Section 6.3 we solved equations and inequalities involving exponential functions using one of two basic strategies.
More informationPractice with Proofs
Practice with Proofs October 6, 2014 Recall the following Definition 0.1. A function f is increasing if for every x, y in the domain of f, x < y = f(x) < f(y) 1. Prove that h(x) = x 3 is increasing, using
More information100. 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,
More information1. Then f has a relative maximum at x = c if f(c) f(x) for all values of x in some
Section 3.1: First Derivative Test Definition. Let f be a function with domain D. 1. Then f has a relative maximum at x = c if f(c) f(x) for all values of x in some open interval containing c. The number
More informationSAT Subject Math Level 1 Facts & Formulas
Numbers, Sequences, Factors Integers:..., -3, -2, -1, 0, 1, 2, 3,... Reals: integers plus fractions, decimals, and irrationals ( 2, 3, π, etc.) Order Of Operations: Aritmetic Sequences: PEMDAS (Parenteses
More information2 Limits and Derivatives
2 Limits and Derivatives 2.7 Tangent Lines, Velocity, and Derivatives A tangent line to a circle is a line tat intersects te circle at exactly one point. We would like to take tis idea of tangent line
More informationcorrect-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
More informationA power series about x = a is the series of the form
POWER SERIES AND THE USES OF POWER SERIES Elizabeth Wood Now we are finally going to start working with a topic that uses all of the information from the previous topics. The topic that we are going to
More information1 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.
More informationCalculus 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.
More informationCompute the derivative by definition: The four step procedure
Compute te derivative by definition: Te four step procedure Given a function f(x), te definition of f (x), te derivative of f(x), is lim 0 f(x + ) f(x), provided te limit exists Te derivative function
More informationFINAL EXAM SECTIONS AND OBJECTIVES FOR COLLEGE ALGEBRA
FINAL EXAM SECTIONS AND OBJECTIVES FOR COLLEGE ALGEBRA 1.1 Solve linear equations and equations that lead to linear equations. a) Solve the equation: 1 (x + 5) 4 = 1 (2x 1) 2 3 b) Solve the equation: 3x
More informationInverse Functions and Logarithms
Section 3. Inverse Functions and Logarithms 1 Kiryl Tsishchanka Inverse Functions and Logarithms DEFINITION: A function f is called a one-to-one function if it never takes on the same value twice; that
More information1.7 Graphs of Functions
64 Relations and Functions 1.7 Graphs of Functions In Section 1.4 we defined a function as a special type of relation; one in which each x-coordinate was matched with only one y-coordinate. We spent most
More informationCartesian Products and Relations
Cartesian Products and Relations Definition (Cartesian product) If A and B are sets, the Cartesian product of A and B is the set A B = {(a, b) :(a A) and (b B)}. The following points are worth special
More informationFINITE DIFFERENCE METHODS
FINITE DIFFERENCE METHODS LONG CHEN Te best known metods, finite difference, consists of replacing eac derivative by a difference quotient in te classic formulation. It is simple to code and economic to
More informationThe EOQ Inventory Formula
Te EOQ Inventory Formula James M. Cargal Matematics Department Troy University Montgomery Campus A basic problem for businesses and manufacturers is, wen ordering supplies, to determine wat quantity of
More informationDifferentiation and Integration
This material is a supplement to Appendix G of Stewart. You should read the appendix, except the last section on complex exponentials, before this material. Differentiation and Integration Suppose we have
More informationThe Derivative as a Function
Section 2.2 Te Derivative as a Function 200 Kiryl Tsiscanka Te Derivative as a Function DEFINITION: Te derivative of a function f at a number a, denoted by f (a), is if tis limit exists. f (a) f(a+) f(a)
More informationThe Method of Partial Fractions Math 121 Calculus II Spring 2015
Rational functions. as The Method of Partial Fractions Math 11 Calculus II Spring 015 Recall that a rational function is a quotient of two polynomials such f(x) g(x) = 3x5 + x 3 + 16x x 60. The method
More informationSolving DEs by Separation of Variables.
Solving DEs by Separation of Variables. Introduction and procedure Separation of variables allows us to solve differential equations of the form The steps to solving such DEs are as follows: dx = gx).
More informationMATH 425, PRACTICE FINAL EXAM SOLUTIONS.
MATH 45, PRACTICE FINAL EXAM SOLUTIONS. Exercise. a Is the operator L defined on smooth functions of x, y by L u := u xx + cosu linear? b Does the answer change if we replace the operator L by the operator
More informationSolutions by: KARATUĞ OZAN BiRCAN. PROBLEM 1 (20 points): Let D be a region, i.e., an open connected set in
KOÇ UNIVERSITY, SPRING 2014 MATH 401, MIDTERM-1, MARCH 3 Instructor: BURAK OZBAGCI TIME: 75 Minutes Solutions by: KARATUĞ OZAN BiRCAN PROBLEM 1 (20 points): Let D be a region, i.e., an open connected set
More informationMicroeconomic 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
More informationAbsolute Value Equations and Inequalities
Key Concepts: Compound Inequalities Absolute Value Equations and Inequalities Intersections and unions Suppose that A and B are two sets of numbers. The intersection of A and B is the set of all numbers
More information2 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
More information2.6. Probability. In general the probability density of a random variable satisfies two conditions:
2.6. PROBABILITY 66 2.6. Probability 2.6.. Continuous Random Variables. A random variable a real-valued function defined on some set of possible outcomes of a random experiment; e.g. the number of points
More information12.6 Logarithmic and Exponential Equations PREPARING FOR THIS SECTION Before getting started, review the following:
Section 1.6 Logarithmic and Exponential Equations 811 1.6 Logarithmic and Exponential Equations PREPARING FOR THIS SECTION Before getting started, review the following: Solve Quadratic Equations (Section
More informationLinear and quadratic Taylor polynomials for functions of several variables.
ams/econ 11b supplementary notes ucsc Linear quadratic Taylor polynomials for functions of several variables. c 010, Yonatan Katznelson Finding the extreme (minimum or maximum) values of a function, is
More informationSimplify the rational expression. Find all numbers that must be excluded from the domain of the simplified rational expression.
MAC 1105 Final Review Simplify the rational expression. Find all numbers that must be excluded from the domain of the simplified rational expression. 1) 8x 2-49x + 6 x - 6 A) 1, x 6 B) 8x - 1, x 6 x -
More informationMethod To Solve Linear, Polynomial, or Absolute Value Inequalities:
Solving Inequalities An inequality is the result of replacing the = sign in an equation with ,, or. For example, 3x 2 < 7 is a linear inequality. We call it linear because if the < were replaced with
More informationx), etc. In general, we have
BASIC CALCULUS REFRESHER. Introduction. Ismor Fischer, Ph.D. Dept. of Statistics UW-Madison This is a very condensed and simplified version of basic calculus, which is a prerequisite for many courses in
More informationf(x) f(a) x a Our intuition tells us that the slope of the tangent line to the curve at the point P is m P Q =
Lecture 6 : Derivatives and Rates of Cange In tis section we return to te problem of finding te equation of a tangent line to a curve, y f(x) If P (a, f(a)) is a point on te curve y f(x) and Q(x, f(x))
More informationDefinition of derivative
Definition of derivative Contents 1. Slope-The Concept 2. Slope of a curve 3. Derivative-The Concept 4. Illustration of Example 5. Definition of Derivative 6. Example 7. Extension of the idea 8. Example
More informationNegative Integer Exponents
7.7 Negative Integer Exponents 7.7 OBJECTIVES. Define the zero exponent 2. Use the definition of a negative exponent to simplify an expression 3. Use the properties of exponents to simplify expressions
More informationThe Mean Value Theorem
The Mean Value Theorem THEOREM (The Extreme Value Theorem): If f is continuous on a closed interval [a, b], then f attains an absolute maximum value f(c) and an absolute minimum value f(d) at some numbers
More informationAn important theme in this book is to give constructive definitions of mathematical objects. Thus, for instance, if you needed to evaluate.
Chapter 10 Series and Approximations An important theme in this book is to give constructive definitions of mathematical objects. Thus, for instance, if you needed to evaluate 1 0 e x2 dx, you could set
More informationALGEBRA REVIEW LEARNING SKILLS CENTER. Exponents & Radicals
ALGEBRA REVIEW LEARNING SKILLS CENTER The "Review Series in Algebra" is taught at the beginning of each quarter by the staff of the Learning Skills Center at UC Davis. This workshop is intended to be an
More informationLogarithmic and Exponential Equations
11.5 Logarithmic and Exponential Equations 11.5 OBJECTIVES 1. Solve a logarithmic equation 2. Solve an exponential equation 3. Solve an application involving an exponential equation Much of the importance
More informationSection 1. Logarithms
Worksheet 2.7 Logarithms and Exponentials Section 1 Logarithms The mathematics of logarithms and exponentials occurs naturally in many branches of science. It is very important in solving problems related
More information26 Integers: Multiplication, Division, and Order
26 Integers: Multiplication, Division, and Order Integer multiplication and division are extensions of whole number multiplication and division. In multiplying and dividing integers, the one new issue
More informationSolutions to Linear First Order ODE s
First Order Linear Equations In the previous session we learned that a first order linear inhomogeneous ODE for the unknown function x = x(t), has the standard form x + p(t)x = q(t) () (To be precise we
More informationTangent Lines and Rates of Change
Tangent Lines and Rates of Cange 9-2-2005 Given a function y = f(x), ow do you find te slope of te tangent line to te grap at te point P(a, f(a))? (I m tinking of te tangent line as a line tat just skims
More informationPolynomials. Dr. philippe B. laval Kennesaw State University. April 3, 2005
Polynomials Dr. philippe B. laval Kennesaw State University April 3, 2005 Abstract Handout on polynomials. The following topics are covered: Polynomial Functions End behavior Extrema Polynomial Division
More information2.6 Exponents and Order of Operations
2.6 Exponents and Order of Operations We begin this section with exponents applied to negative numbers. The idea of applying an exponent to a negative number is identical to that of a positive number (repeated
More informationMA4001 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
More information1.6. Analyse Optimum Volume and Surface Area. Maximum Volume for a Given Surface Area. Example 1. Solution
1.6 Analyse Optimum Volume and Surface Area Estimation and oter informal metods of optimizing measures suc as surface area and volume often lead to reasonable solutions suc as te design of te tent in tis
More informationRepresentation 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
More informationTo differentiate logarithmic functions with bases other than e, use
To ifferentiate logarithmic functions with bases other than e, use 1 1 To ifferentiate logarithmic functions with bases other than e, use log b m = ln m ln b 1 To ifferentiate logarithmic functions with
More informationSequences and Series
Sequences and Series Consider the following sum: 2 + 4 + 8 + 6 + + 2 i + The dots at the end indicate that the sum goes on forever. Does this make sense? Can we assign a numerical value to an infinite
More informationVerifying Numerical Convergence Rates
1 Order of accuracy Verifying Numerical Convergence Rates We consider a numerical approximation of an exact value u. Te approximation depends on a small parameter, suc as te grid size or time step, and
More informationChapter 7 - Roots, Radicals, and Complex Numbers
Math 233 - Spring 2009 Chapter 7 - Roots, Radicals, and Complex Numbers 7.1 Roots and Radicals 7.1.1 Notation and Terminology In the expression x the is called the radical sign. The expression under the
More informationSection 4.5 Exponential and Logarithmic Equations
Section 4.5 Exponential and Logarithmic Equations Exponential Equations An exponential equation is one in which the variable occurs in the exponent. EXAMPLE: Solve the equation x = 7. Solution 1: We have
More informationWriting Mathematics Papers
Writing Matematics Papers Tis essay is intended to elp your senior conference paper. It is a somewat astily produced amalgam of advice I ave given to students in my PDCs (Mat 4 and Mat 9), so it s not
More informationUsing 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.
More informationMath 53 Worksheet Solutions- Minmax and Lagrange
Math 5 Worksheet Solutions- Minmax and Lagrange. Find the local maximum and minimum values as well as the saddle point(s) of the function f(x, y) = e y (y x ). Solution. First we calculate the partial
More information5.1 Radical Notation and Rational Exponents
Section 5.1 Radical Notation and Rational Exponents 1 5.1 Radical Notation and Rational Exponents We now review how exponents can be used to describe not only powers (such as 5 2 and 2 3 ), but also roots
More information1 Calculus of Several Variables
1 Calculus of Several Variables Reading: [Simon], Chapter 14, p. 300-31. 1.1 Partial Derivatives Let f : R n R. Then for each x i at each point x 0 = (x 0 1,..., x 0 n) the ith partial derivative is defined
More informationThe 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
More informationM(0) = 1 M(1) = 2 M(h) = M(h 1) + M(h 2) + 1 (h > 1)
Insertion and Deletion in VL Trees Submitted in Partial Fulfillment of te Requirements for Dr. Eric Kaltofen s 66621: nalysis of lgoritms by Robert McCloskey December 14, 1984 1 ackground ccording to Knut
More informationIntroduction 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
More information3 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
More information20. Product rule, Quotient rule
20. Prouct rule, 20.1. Prouct rule Prouct rule, Prouct rule We have seen that the erivative of a sum is the sum of the erivatives: [f(x) + g(x)] = x x [f(x)] + x [(g(x)]. One might expect from this that
More informationPre-Algebra Lecture 6
Pre-Algebra Lecture 6 Today we will discuss Decimals and Percentages. Outline: 1. Decimals 2. Ordering Decimals 3. Rounding Decimals 4. Adding and subtracting Decimals 5. Multiplying and Dividing Decimals
More informationProjective Geometry. Projective Geometry
Euclidean versus Euclidean geometry describes sapes as tey are Properties of objects tat are uncanged by rigid motions» Lengts» Angles» Parallelism Projective geometry describes objects as tey appear Lengts,
More informationIncreasing for all. Convex for all. ( ) Increasing for all (remember that the log function is only defined for ). ( ) Concave for all.
1. Differentiation The first derivative of a function measures by how much changes in reaction to an infinitesimal shift in its argument. The largest the derivative (in absolute value), the faster is evolving.
More informationTaylor 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
More informationRolle s Theorem. q( x) = 1
Lecture 1 :The Mean Value Theorem We know that constant functions have derivative zero. Is it possible for a more complicated function to have derivative zero? In this section we will answer this question
More informationActivity 1: Using base ten blocks to model operations on decimals
Rational Numbers 9: Decimal Form of Rational Numbers Objectives To use base ten blocks to model operations on decimal numbers To review the algorithms for addition, subtraction, multiplication and division
More informationEconomics 121b: Intermediate Microeconomics Problem Set 2 1/20/10
Dirk Bergemann Department of Economics Yale University s by Olga Timoshenko Economics 121b: Intermediate Microeconomics Problem Set 2 1/20/10 This problem set is due on Wednesday, 1/27/10. Preliminary
More informationIn other words the graph of the polynomial should pass through the points
Capter 3 Interpolation Interpolation is te problem of fitting a smoot curve troug a given set of points, generally as te grap of a function. It is useful at least in data analysis (interpolation is a form
More information5.3 Improper Integrals Involving Rational and Exponential Functions
Section 5.3 Improper Integrals Involving Rational and Exponential Functions 99.. 3. 4. dθ +a cos θ =, < a
More information6.1. 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
More information9.2 Summation Notation
9. Summation Notation 66 9. Summation Notation In the previous section, we introduced sequences and now we shall present notation and theorems concerning the sum of terms of a sequence. We begin with a
More information1 3 4 = 8i + 20j 13k. x + w. y + w
) Find the point of intersection of the lines x = t +, y = 3t + 4, z = 4t + 5, and x = 6s + 3, y = 5s +, z = 4s + 9, and then find the plane containing these two lines. Solution. Solve the system of equations
More informationStudent Outcomes. Lesson Notes. Classwork. Discussion (10 minutes)
NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 5 8 Student Outcomes Students know the definition of a number raised to a negative exponent. Students simplify and write equivalent expressions that contain
More information1 Error in Euler s Method
1 Error in Euler s Method Experience with Euler s 1 method raises some interesting questions about numerical approximations for the solutions of differential equations. 1. What determines the amount of
More informationDefinition 8.1 Two inequalities are equivalent if they have the same solution set. Add or Subtract the same value on both sides of the inequality.
8 Inequalities Concepts: Equivalent Inequalities Linear and Nonlinear Inequalities Absolute Value Inequalities (Sections 4.6 and 1.1) 8.1 Equivalent Inequalities Definition 8.1 Two inequalities are equivalent
More informationf(x + h) f(x) h as representing the slope of a secant line. As h goes to 0, the slope of the secant line approaches the slope of the tangent line.
Derivative of f(z) Dr. E. Jacobs Te erivative of a function is efine as a limit: f (x) 0 f(x + ) f(x) We can visualize te expression f(x+) f(x) as representing te slope of a secant line. As goes to 0,
More informationSection 4.1 Rules of Exponents
Section 4.1 Rules of Exponents THE MEANING OF THE EXPONENT The exponent is an abbreviation for repeated multiplication. The repeated number is called a factor. x n means n factors of x. The exponent tells
More informationPressure. Pressure. Atmospheric pressure. Conceptual example 1: Blood pressure. Pressure is force per unit area:
Pressure Pressure is force per unit area: F P = A Pressure Te direction of te force exerted on an object by a fluid is toward te object and perpendicular to its surface. At a microscopic level, te force
More information