...Teaching Strategy Unit Exponential and Logarithmic equations In this unit I am going to be teaching both Exponential and Logarithmic equations. The different strategies that I have chosen to use will help the students be able to define both kinds of equations, describe their similarities, and describe the relationship between one another. The strategies that I plan on using in this unit are Semantic Question Map, Venn diagram, Semantic Map, Circle graph, and Bio Pyramid. Each strategy has been changed in order to fit with the topic. In this unit Exponential equations will be taught before Logarithmic equations, and connections will need to be made between the two. Semantic Question Map: To open the unit up with exponential equations I plan on using the semantic question map in order to give students and idea of what questions they need to be thinking about as we begin the unit. The questions for example will ask students “what should you do to the exponents if their bases are being multiplied and they are the same?” Students will be required to answer these questions each day at the end of the lesson as we answer each one. This strategy will also be used when we begin to study Logarithmic equations. Circle Graph: This strategy is going to be used at the end of the section on Exponential equations and end of Logarithmic equations. The Circle graph is simply a small circle within a larger circle. The smaller circle will either contain exponential equations, or logarithmic...
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...This watermark does not appear in the registered version - http://www.clicktoconvert.com 1 UNIT - I Lesson 1 - Set theory and Set Operations Contents: 1.1 Aims and Objectives 1.2 Sets and elements 1.3 Further set concepts 1.4 Venn Diagrams 1.5 Operations on Sets 1.6 Set Intersection 1.7 Let – us Sum Up 1.8 Lesson – End Activities 1.9 References 1.1 Aims and Objectives This Lesson introduces some basic concepts in Set Theory, describing sets, elements, Venn diagrams and the union and intersection of sets. 1.2 Sets and elements Sets of objects, numbers, departments, job descriptions, etc. are things that we all deal with every day of our lives. Mathematical Set Theory just puts a structure around this concept so that sets can be used or manipulated in a logical way. The type of notation used is a reasonable and simple one. For example, suppose a company manufactured 5 different products a, b, c, d, and e. Mathematically, we might identify the whole set of products as P, say, and write: P = (a,b,c,d,e) which is translated as 'the set of company products, P, consists of the members (or elements) a, b, c, d and e. The elements of a set are usually put within braces (curly brackets) and the elements separated by commas, as shown for set P above. A mathematical set is a collection of distinct objects, normally referred to as elements or members. Sets are usually denoted by a capital letter and the elements by small letters. Example 1 (Illustrations of sets) This...
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...History of Logarithms Logarithms were invented independently by John Napier, a Scotsman, and by Joost Burgi, a Swiss. Napier's logarithms were published in 1614; Burgi's logarithms were published in 1620. The objective of both men was to simplify mathematical calculations. This approach originally arose out of a desire to simplify multiplication and division to the level of addition and subtraction. Of course, in this era of the cheap hand calculator, this is not necessary anymore but it still serves as a useful way to introduce logarithms. Napier's approach was algebraic and Burgi's approach was geometric. The invention of the common system of logarithms is due to the combined effort of Napier and Henry Biggs in 1624. Natural logarithms first arose as more or less accidental variations of Napier's original logarithms. Their real significance was not recognized until later. The earliest natural logarithms occur in 1618. It can’t be said too often: a logarithm is nothing more than an exponent. The basic concept of logarithms can be expressed as a shortcut…….. Multiplication is a shortcut for Addition: 3 x 5 means 5 + 5 + 5 Exponents are a shortcut for Multiplication: 4^3 means 4 x 4 x 4 Logarithms are a shortcut for Exponents: 10^2 = 100. The present definition of the logarithm is the exponent or power to which a stated number, called the base, is raised to yield a specific number. The logarithm of 100 to the base 10 is 2. This is written: log10 (100) = 2. Before pocket calculators...
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...questionbase.50megs.com A-Level Revision Notes SMP 16-19 Mathematics – Revision Notes Unit 3 – Functions Algebra Of Functions 1. Functions can be combined whereby fg(x) = f(g(x)) = g(x) followed by f(x). 2. The set of values for which a function is defined is the domain (i.e. x values), and the set of values that the function can return is the range (i.e. y values). 3. Many-to-one functions have more than one value in the domain giving one value in the range. It is impossible to have many-to-one functions. 4. The inverse of a function is denoted by f –1(x), and is only a function if f(x) is one-to-one. 5. The graphs of a function and its inverse function have reflection symmetry in the line y = x. 6. Parameters are values in a function that can vary, but for any given function mapping x onto y they will act as constants (e.g. a, b, and c in y = ax2 + bx + c). -p 7. The image of y = f(x) under a translation of is y = f ( x + p ) + q . q 8. The image of y = f(x) after reflection in the y-axis is y = f(–x). 9. The image of y = f(x) after reflection in the x-axis is y = –f(x). 10. If f(–x) = f(x) then f is an even function (i.e. is symmetric about the y-axis). 11. If f(–x) = –f(x) then f is an odd function (i.e. has rotational symmetry about the origin). Circular Functions 1. The sine and cosine functions are periodic – they repeat themselves after a period. − c 2. y = sin( x + c )° + d is obtained by a translation of . d 3. 4. 5. 6. 7. 8. 9. y = a sin...
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...Review Notes for IB Standard Level Math © 2015-2016, Steve Muench steve.muench@gmail.com @stevemuench Please feel free to share the link to these notes http://bit.ly/ib-sl-maths-review-notes or my worked solutions to the November 2014 exam http://bit.ly/ib-sl-maths-nov-2014 or my worked solutions to the May 2015 (Timezone 2) exam http://bit.ly/ib-sl-maths-may-2015-tz2 or my worked solutions to the November 2015 exam https://bit.ly/ib-sl-maths-nov-2015 with any student you believe might benefit from them. If you downloaded these notes from a source other than the bit.ly link above, please check there to make sure you are reading the latest version. It may contain additional content and important corrections! April 8, 2016 1 Contents 1 Algebra 1.1 Rules of Basic Operations . . . . . . . . . . . . . . . . . . . . . 1.2 Rules of Roots . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3 Rules of Exponents . . . . . . . . . . . . . . . . . . . . . . . . . 1.4 Allowed and Disallowed Calculator Functions During the Exam 1.5 Sequences and Series . . . . . . . . . . . . . . . . . . . . . . . . 1.6 Arithmetic Sequences and Series . . . . . . . . . . . . . . . . . 1.7 Sum of Finite Arithmetic Series (u1 + · · · + un ) . . . . . . . . . 1.8 Partial Sum of Finite Arithmetic Series (uj + · · · + un ) . . . . . 1.9 Geometric Sequences and Series . . . . . . . . . . . . . . . . . . 1.10 Sum of Finite Geometric Series . . . . . . . . . . . . . . . . . . ...
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...LSP 121 Homework 5: Logs and Richter Scale, Decibels Open the file logs.xls (found on the QRC website under Excel Files). You will use this file for both parts below. Click on the worksheet tabs at the bottom to access the other file. 1. Richter scale The Richter scale is used to measure the intensity of earthquakes. It is a logarithmic relationship with the following formula: R = log(I) I is the intensity of the earthquake and R is the number on the Richter scale. (Remember that if there is no base written with the log it is base 10). Don’t be scared off by logs. Think of them as a short-cut way of writing large numbers. For example, let’s say the intensity of an earthquake is 100,000,000. Can you imagine if the papers published that big number? What would people think? Most people cannot handle well numbers with large amounts of zeros (unless those people work for the government ;-). Instead of writing 100,000,000, let’s just write how many zeros there are. In this case, there are 8 zeros. That’s exactly what log10 is asking. 10 to what power equals 100,000,000? 10 to the 8th power. So the log(100,000,000) equals 8. Thus, the Richter value for an earthquake with intensity 100,000,000 is 8. Let’s say it another way. Let’s remove the log. Converting the above formula from log form to exponent form would give us: 10R = I (Note: 10 was raised to the R power, which cancels the log function.) Which version you use depends on which variable you...
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...Instructor information Wyatt C. Christian-Carpenter Office: Evans 111D Office Number: 870-230-5043 Google Number: 828-539-0402 Email: CARPENW@hsu.edu Office Hours MWF: 9 – 10 a.m. & 11 a.m. – 12 p.m.; TR: 12:30 – 1:30 p.m. & 2:45 – 3:45 p.m. Meeting Times and Location MWF: 10 – 10:50 a.m., EV205 MWF: 1 – 1:50 p.m., EV 205 TR: 11 a.m. – 12:15 p.m., EV 205 TR 1:30 – 2:45 p.m., EV 207 Text and Required Supplies A Graphical Approach to College Algebra, 6th Edition by John Hornsby, Margaret Lial, Gary Rockswold ©2014 Prentice Hall. Description | | ISBN-10 | ISBN-13 | Approximate Cost | MyMathLab access code | Required | 032119991X | 9780321199911 | $75–100 | Hardcopy or Kindle | Optional | 0321920309 | 9780321920300 | $145–196 | Hardcopy bundled with MML | Optional | 978-0321909817 | 032190981X | $200–290 | The MyMathLab code can be purchased from the Arkadelphia bookstores or online. MWF MyMathLab CourseID: carpenter58666 TR MyMathLab CourseID: carpenter61414 A graphing calculator is required. Any TI newer than a TI-83 is highly recommended, for example, the TI-83+, TI-84+, or TI-nspire. The mathematics department strongly recommends the TI-Nspire CAS if you will take Calculus 1 or above. Course Prerequisite(s) A score of 20 on the ACT Mathematics Section, or equivalent score, or a grade of “C” or better in Intermediate Algebra from an accredited institution is required. However, it is recommended that your ACT score be at least 22. If...
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...MAPOA INSTITUTE OF TEGHNOLOGY Depottment of Mathemotics VISION The Mapua Institute ofrechnology shall be a global center ofexcellence in education by providing instructions that are cur:rent in content and state-of-the-art in delivery; by engaging in cutting-edge, high impact research; and by aggressively taking on presen!day global "oni..nr. MISSION The Mapua Institute of rechnology disseminates, generates, preserves and applies knowledge in various helds of study. 'using The Institute, the most effective and efficient means, provides its students with highly relevant professional and advanced education in preparation for and furtherance ofglobal practice. The Institute engages in research with high socio-economic impact and reports on the results of such inquiries. The Institute brings to bear humanity's vast store ofknowledge on the problems ofindustry and community in order to make the Philippines and the world a better place. BASIC STUDIES EDUCATIONAL OBJECTIVES MISSION a b c d 2. 3. 4. To provide students with a solid foundation in mathematics, physics, general chemistry and engineering drawing and to apply knowledge to engineering, architecture and other related disciplines. To complement the technical trairung of the students with proficiency in oral, written, and graphics communication. To instill in the students human values and cultural rehnement tbrough the humanities and social sciences. To inculcate high ethical standards in the students...
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...Unit 7 Test 5/1/15, 4:09 PM MAT 120.17, Spring 2015 Assessments Unit 8 Test Results Unit 7 Test - Grade Report Score: 100% (19 of 19 pts) Submitted: Apr 19 at 12:43pm Question 1 Question Grade: 1.0 Weighted Grade: (1/1.0) If q and f are inverse functions and q(−2) = 8, what is f (9) ? Your Answer: cannot be determined Correct Answer: cannot be determined Comment: If q and f are inverse functions and q(a) = b, then f (b) = a. However, since 9 is not the given domain value for q, the answer cannot be determined. Question 2 Question Grade: 1.0 Weighted Grade: (1/1.0) Choose any false statements regarding the graph. Select all that apply. Choice Selected Points The graph is a function. Yes +1 The graph is a function that has an inverse function. Yes +1 The graph is a one-to-one function. Yes +1 The inverse of the graph is not a function. No The graph passes the horizontal line test. Yes +1 The graph passes the vertical line test. Yes +1 Number of available correct choices: 5 Comment: A vertical line can be drawn through the graph intersecting the graph in more than one place, so the graph fails the vertical line test. Therefore, the graph does not represent a function. A horizontal line can be drawn through the graph intersecting the graph in more than one place. So, the graph fails the horizontal line test. Therefore, the graph is not a one-to-one function...
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...Algebra 2 Quarter 4 Review Name: ________________________ Class: ____________ Date: _______________ Section 1: Logarithms and Exponential Relations Definitions to Know: * Natural Logarithm * Common Logarithm * Mathematical * Exponential Growth * Exponential Decay Question 1) Change the following from exponential form to logarithmic form (1 mark each): a) b) Question 2) Change the following from logarithmic form to exponential form (1 mark each): a) b) Question 3) Solve for WITHOUT using a calculator. Show all of your work. (Hint: Use the definition of a logarithm.) (2 marks each) a) b) c) d) Question 4) Apply the Change of Base Formula to rewrite the logarithms with the common logarithm. (1 mark each) a) b) Question 5) Solve for the variable. Show all of your work and all of your steps. (Hint: Use the properties of logarithms.) (4 marks each) a) b) c) d) Question 6) Solve for the variable. Show all of your work and all of your steps. Show the answer to 4 decimal places. (Hint: Use the common logarithm.) (4 marks each) a) b) c) Question 7) Solve for . Show all of your work and all of your steps. Show the answer to 4 decimal places. (Hint: Use the natural logarithm and the definition of a logarithm.) (4 marks each) a) b) c) Question 8) Ms. Mary bought a condo for $145 000. Assuming that the value of the condo will appreciate at most 5% a year, how much will the condo be worth in 5 years...
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...For the benefit of the students, specially the aspiring ones, the question of JEE(advanced), 2013 are also given in this booklet. Keeping the interest of students studying in class XI, the questions based on topics from class XI have been marked with ‘*’, which can be attempted as a test. For this test the time allocated in Physics, Chemistry & Mathematics and Physics are 22 minutes, 21 minutes and 25 minutes respectively. FIITJEE SOLUTIONS TO JEE(ADVANCED)-2013 CODE PAPER 2 3 Time: 3 Hours Maximum Marks: 180ase read the instructions carefully. You are allotted 5 minutes specifically for this purpose. INSTRUCTIONS A. General: 1. This booklet is your Question Paper. Do not break the seals of this booklet before being instructed to do so by the invigilators. 2. Blank papers, clipboards, log tables, slide rules, calculators, cameras, cellular phones, pagers and electronic gadgets are NOT allowed inside the examination hall. 3. Write your name and roll number in the space provided on the back cover of this booklet. 4. Answers to the questions and personal details are to be filled on a two-part carbon-less paper, which is provided separately. These parts should only be separated at the end of the examination when instructed by the invigilator. The upper sheet is a machine-gradable Objective Response Sheet (ORS) which will be retained by the invigilator. You will be allowed to take away the bottom sheet at the end of the examination. 5. Using a black ball point pen darken...
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...UNIVERSITI TUNKU ABDUL RAHMAN ACADEMIC YEAR 2012/2013 APRIL EXAMINATION FHMM1014 MATHEMATICS I THURSDAY, 25 APRIL 2013 TIME: 5.00 PM – 7.00 PM (2 HOURS) FOUNDATION IN SCIENCE SOLUTIONS This is not an official document of UTAR. The University is not responsible for any errors found in the solutions. Solutions to FHMM1014 Mathematics I (April 2013) 2 FHMM1014 MATHEMATICS I Q1. (a) (i) a + bi = +i 1 (a − 2) + bi ⇒ a + bi = a − 2) + bi )(1 + i ) (( a + bi = (a − 2 − b) + (a − 2 + b)i a a = − 2 − b So b = a−2+b ⇒ ⇒ − b=2 a= 2 z= 2 − 2i (ii) (iii) z= 4+4 = 2 2 ⇒ −π θ= 4 −2 = tan θ =−1 2 = arg( z ) θ= −π 4 (b) (i) 3log x x log 3 = 3log x =81 ⇒ ⇒ 2 × 3log x 162 = 3log x =34 log x = 4 ⇒ x = 104 (ii) = log10 x + log(1 + 2 x ) log 5x + log 6 log10 x (1 + 2 x )= log 5 x × 6 10 x (1 + 2 x ) = x × 6 5 Let ⇒ 2 x (1 + 2 x ) = 6 y = 2 x then y 2 + y − 6 = 0 ( y − 2)( y + 3) = 0 y − 2 = 0 ⇒ y = 2 ⇒ 2x = 2 ⇒ x = 1 y + 3 = ⇒ y = 3 ⇒ 2 x = 3 impossible − − 0 Solutions to FHMM1014 Mathematics I (April 2013) 3 FHMM1014 MATHEMATICS I Q1. (Continued) (c) P ( x)= A( x − 2)( x − (2 − i ))( x − (2 + i )) P( x) = A( x − 2)( x 2 − 4 x + 5) P( x) = A( x3 − 6 x 2 + 13 x − 10) P(0) = −5 P( x) = ⇒ 1 A= 2 1 3 ( x − 6 x 2 + 13 x − 10) 2 (d) A(B ( A − ( B C ) )′ = C )′ ( )′ = A′ ( B C ) = ( A′ B ) ( A′ C ) = ( A B′ )′ ( A C ′ )′ =B )′ ( A − C )′ (A− Q2. (a) x3...
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...QUESTION PAPER CODE 65/1/1 EXPECTED ANSWERS/VALUE POINTS SECTION - A Q. No. 1-10. 1. x = 25 2. x = 6. 2x3/2 + 2 10. 1 5 Marks 3. 10 π 12 4. x = 2 5. x = + 6 2π 3 x +c 7. 8. 5 9. ˆ ˆ { r – (aˆi + bˆj + ck ) }⋅ (ˆi + ˆj + k ) = 0 or r⋅ ˆ+ˆ+k =a+b+c i j ˆ ( ) 1×10 =10 m SECTION - B 11. ∀ (a, b) ∈ A × A a + b = b + a ∴ (a, b) R (a, b) ∴ R is reflexive For (a, b), (c, d) ∈ A × A If (a, b) R (c, d) i.e. a + d = b + c ⇒ c + b = d + a then (c, d) R (a, b) ∴ R is symmetric For (a, b), (c, d), (e, f) ∈ A × A If (a, b) R (c, d) & (c, d) R (e, f) i.e. a + d = b + c & c + f = d + e Adding, a + d + c + f = b + c + d + e then (a, b) R (e, f) ∴ R is transitive ∴ R is reflexive, symmetric and transitive 1m 1m ⇒ a+f = b+e 1m hence R is an equivalance relation [(2, 5)] = {(1, 4), (2, 5), (3, 6), (4, 7), (5, 8), (6, 9)} ½m ½m 2 12. 1 + sin x + 1 – sin x cot–1 1 + sin x – 1 – sin x 2 2 x x x x cos + sin + cos − sin 2 2 2 2 2 2 x x x x cos + sin − cos − sin 2 2 2 2 = cot–1 2½ m x 2 cos 2 x x −1 = cot −1 = cot cot = 2 2 2 sin x 2 OR 5 2 1 1 LHS = 2 tan −1 + tan −1 + sec −1 7 5 8 1½ m 1 1 + −1 5 8 + tan −1 1 = 2 tan 7 1– 1 40 2⋅1 1 3 + tan −1 1 = tan −1 2 7 7 1– 1 3 1½+½ m = 2 tan −1 1 + tan −1 3 1m ...
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...1. What is the primary place to store log files on a local Linux system and what are recommended procedures for that location? Almost all logfiles are located under /var/log directory It is very important that the information that comes from syslog not be compromised. Making the files in /var/log readable and writable by only a limited number of users is a good start. 2. Why remote logging to a central server is considered a best practice? To identify a baseline system state with the use of the logs & to keep the information from prying eyes. 3. What is the syntax and file you would edit with the necessary entries to send syslogs from your Linux system to a logging server at 172.130.1.254? su –c ’ vi/etc/rsyslog.conf ‘, then remove the # from in front of $ModLoadimudp and $UDPServerRun514 if it hasn’t already been done. Then add a line below remote host with the following syntax *.*@@172.130.1.254:541 4. Why is the “Tripwire” application considered a file integrity checker? File Integrity Monitoring is available as a standalone solution or as part of Tripwire’s Security Configuration Management suite, where you have continual assurance of the integrity of security configurations and complete visibility and control of all change for your continuous monitoring, change audit and compliance demands. 5. Could rkhunter be considered a file integrity checker? Why or why not? Rootkit Hunter is considered a file integrity checker because it...
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...equation for x. e3x+5 = 36 | | x = -1/3 | | | x2 = 6 | | | x = -3 | | | x = 1/3 | | | x = 3 | 5 points Question 6 Write the logarithmic equation in exponential form. log8 64 = 2 | | 648 = 2 | | | 82 = 16 | | | 82 = 88 | | | 82 = 64 | | | 864 = 2 | 5 points Question 7 Write the logarithmic equation in exponential form. log7 343 = 3 | | 7343 = 2 | | | 73 = 77 | | | 73 = 343 | | | 73 = 14 | | | 3437 = 2 | 5 points Question 8 Write the exponential equation in logarithmic form. 43 = 64 | | log64 4 = 3 | | | log4 64 = 3 | | | log4 64 = -3 | | | log4 3 = 64 | | | log4 64 = 1/3 | 5 points Question 9 Use the properties of logarithms to simplify the expression. log20 209 | | 0 | | | -1/9 | | | 1/9 | | | -9 | | | 9 | 5 points Question 10 Use the One-to-One property to solve the equation for x. log2(x+4) = log2 20 | | 19 | | | 17 | | | 18 |...
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