Math 1400 - Mano Table of Contents - Review - page 1 of 2 Review Topics for MATH 1400 Elements of Calculus Table of Contents MATH 1400 Elements of Calculus is one of the Marquette Core Courses for Mathematical Reasoning. Pre-Requisites: Three ears of high school Mathematics. This course has students from various backgrounds. All students have taken three ears of high school mathematics, one of which is Algebra. Some have taken four ears of mathematics and the last course ma have been Pre-Calculus or Calculus. The course is taught so that students with various backgrounds should be able to successfull complete the course, if the put in the effort to do so. This means that ou must attend all classes and do the assigned problems in a timel manner (i.e. keep up with the class; do not fall behind). Some of ou have not taken a math course in several ears. For man of ou, our Algebra skills are weak or simpl a memor of things past. While I will review Algebra as we work through the development of Calculus, ou will be best served with a review of basic Algebra skills before embarking on learning and understanding the concepts of Calculus. What follows is seven Review topics. The first si should be review for all, the last review on logarithms ma be brand new material for man. The first si topics are encountered in the first chapter that we cover, Chapter 10 of College Mathematics for Business, Economics, Life Sciences and Social Sciences, Barnett, Ziegler, Bleen, 12 th Edition, Pearson Prentice Hall. Because we will be using the material in the first 4 Reviews when cover Section 10.1, I strongl suggest that ou review these topics before the semester begins. If ou cannot, I cannot stress enough the importance of understanding these topics before the first test of the semester. What follows is a table of contents for the review eercises when the will be needed. The first chapter that we will cover is Chapter 10, followed b Chapters 11, 12, 13 and parts of 14. Review 1 The Real Numbers this should be completed before the first class. The concepts in this are fundamental to understanding Calculus. We begin with section 10.1, which will take us three das to cover. Review 2 Functions this should be completed before the first class. The concepts in this are fundamental to understanding Calculus. Review 3 Multiplication of Polnomials this needs to be understood before completing Review 4. The topics in this review will be used during the entire semester. This should be completed before the 2nd da of class. Review 4 Factoring Polnomials This should be completed before the 3rd da of class, (at the ver latest). Review 5 Eponents This should be completed before we cover section 10.2 Review 6 Linear Equations This should be completed before we get to section 10.4. The Derivative (the basis for the rest of the semester.) After Section 10.4, we will be on a faster pace tring to cover approimatel one section per da. Review 7 Logarithms This can wait until immediatel after the first test. However, this is new material for some of ou. You ma want to go through it twice to be sure that ou are comfortable with it. I am more than willing to help ou with this outside of class. Answers to Review Questions
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Math 1400 - Mano Review 1 Review 1*- The Real Numbers To be completed prior to the first class *Review sheets contain material that should be a review for most students and are meant to be completed in less than 30 minutes in preparation for class. Feel free to discuss and complete these with our classmates. However, simpl coping someone's work defeats the purpose and is a waste of our time. Sets of numbers - see Tetbook Appendi A-1 for further eplanation You should be familiar with the following sets of numbers. The letter names given to them are their traditional names in all of mathematics. Integers: Z Rational Numbers: Q Integers include the counting numbers (1, 2, 3, ), their opposites and zero Rational numbers are numbers that can be epressed as the quotient of two integers and in decimal form are either terminating or repeating decimals. Irrational Numbers: I Irrational numbers are numbers that in decimal form are non-terminating and non-repeating. Real Numbers: The Real Numbers is the union of the Rational and Irrational numbers. Q1: The Venn diagram shows the relationships between these four sets of numbers. Complete the chart b checking each set to which the number belongs. Then place each number on the Real number line below. Q Z The Universe is I Indicate to which sets, each number belongs. number Z Q I 0
Review 1 - Page 2 of 2 Using Interval Notation see Tebook pages 6-7* for further eplanation We can indicate an interval of Real numbers graphicall, with a compound inequalit or using interval notation. The interval (-2, 1] does not include the endpoint -2 and does include the endpoint 1. graphicall inequalit interval -3-2 -1 0 1 2 3 On a number line, eclusion is shown with an empt (or open) circle and inclusion is shown with a solid circle. An inequalit,uses < or > to show eclusion and or to show inclusion. Interval notation uses a paren, ( or ), to show eclusion and a square bracket, [ or ], to show inclusion. Q2: In the interval, the largest number is and the smallest number is. Q3: Complete the chart to describe each interval graphicall, as an inequalit and using interval notation. graphicall inequalit interval This is also the wa we indicate a point in the coordinate plane. e.g. P is the point (3,5). The correct interpretation comes from the contet of what ou are doing. -3-2 -1 0 1 2 3 - and + (negative infinit and infinit) are never included in an interval since the are not numbers. The represent the idea that the interval continues without bound, or infinitel. The set of Real Numbers is (-, +) How man Real Numbers are there? Q4: A set of numbers is dense if, for an two numbers and in the set, where, there is another number in the set such that. (i.e between an two numbers in the set there is another number belonging to the set). A. Illustrate that "the set of Real Numbers is dense " b finding a real number between each and. -0.6-0.5 2 2.01-3.9686-3.9682 4.0039354 4.0039365 B. Given that the set of Real Numbers is dense, how man numbers are there between an two numbers? C. Is the set of integers dense? Eplain.
Math 1400- Mano Review 2 Review 2 Functions A complete review of functions can be found in our tetbook in Chapter 2. Before we state a formal definition of a function let's discuss, informall the concept of a function. Think of a function as a black bo that takes as input a number, does something to the input number and outputs a new number. For eample, below we show a function that outputs 5, when the input number is 2. 2 5 Continuing with the same function, here are some additional inputs into the function bo and outputs from the function bo. 3 10 4 17 5 26 It would be wrong to assume that onl positive integers can be input into the function. This function bo also accepts negative numbers, fractions and, in fact, accepts an real number. Here are some more eamples. -2 5-5 26 3 1.25 In general, we often think of the input number as name, such as. and the output number as, and we give the function a The statement sas is a function of The statement tells us that for the function when. Were ou able to determine what function is inside the black bo?
Review 2 - Page 2 of 7 Lets's look inside the bo. Definition: A function is a correspondence between two sets of numbers, the domain and the range, such that for each element in the domain, there eists one and onl one element in the range. Comments: The input numbers, belong to the domain and the output numbers, belong to the range. The definition sas that for ever value in the domain, there is one and onl one, value in the range. Therefore, the net two eamples are not functions. 2 This eample shows This is not possible if is a function since must have onl one value. This is wh we define the square root function, to mean the positive square root, i.e. -2 use the function. If we want the negative square root, we, i.e. does not eist A function must be defined at ever element in its domain. Therefore, if is not defined (ND), is not in the domain of the function. So, what are the domain and range for the function To determine the domain, we ask ourselves " Are there an real numbers that we cannot substitute for " Since an real number can be squared and added to 1, the answer is no, since an number can be substituted for. Therefore, the domain of the function is all real numbers, which can be represented as or, using interval notation,. We call the independent variable, since we can let equal an value in the domain. To determine the range, we ask ourselves "What values can this function output?" Since is alwas positive or zero, is alwas greater than or equal to 1. Therefore, using interval notation, the range is. We call the dependent variable, since the value of depends upon the value of
Review 2 - Page 3 of 7 Graphing a Function The epression, "a picture is worth a thousand words", is especiall true in mathematics. If ou look at the inside back cover of our tetbook, ou will see the graphs of some elementar functions. These are the tpes of functions we will be working with this semester in MATH 1400. You should be able to visualize these functions b looking at the function, b graphing a few points on the function, or b using our graphing calculator. Let's graph the function we have been talking about,, b plotting some points on the coordinate plane. The independent variable, in this eample, is alwas plotted on the horizontal ais and the dependent variable, in this eample, is alwas plotted on the vertical ais. First we list the coordinates of some points that satisf the function. Then we plot those points, and then connect the dots with a curve. List points Plot points Sketch curve -3-2 -1 0 1 2 3 is an eample of a second degree polnomial, or a quadratic function. Graphs of quadratic functions are parabolas. This parabola has a verte at the point (0,1) and is congruent to the graph of the function, which has its verte at (0,0). Another wa to sa this is that is the graph of shifted verticall up one unit. A Polnomial has the form, where each is an real number (including 0) and each is a positive integer or zero. The term with the highest power is called the leading or dominant term and its value of is the degree of the polnomial. A constant is a polnomial with degree 0. Eamples of polnomial funtions: Eamples of functions that are not polnomials:
1. For each of the functions, find Eercises A. B. C. Review 2 - Page 4 of 7 2. Let A. Find the value of when B. What is C. What values of give a value of 11? D. What values of give a value of 1? E. The domain is and the range is 3. For each function (continued on the net page) Complete the list of points (some ma be undefined, indicating the value is not in the domain) Plot the points and sketch the curve. List the domain and range using interval notation. -3-2 The domain is -1 0 1 The range is 2 3
Review 2 - Page 5 of 7-2 -1 The domain is 0 1 2 The range is -4-1 The domain is 0 1 4 The range is 9 Recall: The absolute value of a number is its positive distance from zero on the real number line. We know
Review 2 - Page 6 of 7-3 -2 The domain is -1 0 1 The range is 2 3 The functions in the net two eamples have a numerator and denominator. Remember that an fraction with 0 in the denominator is not defined. e.g. are undefined -3-2 The domain is -1 0 1 The range is 2 3
Review 2 - Page 7 of 7-4 -3 The domain is -2-1 -1/2 The range is -1/4 0 1/4 1/2 1 2 3 4
Math 1400 - Mano Filler page if ou are printing in duple mode.
Math 1400- Mano Review 3 Review 3 Multiplication of Polnomials Basic properties of Addition and Multiplication on the Set of Real Numbers. are Real numbers Properties of Addition Properties of Multiplication Commutative: e.g. Associative: e.g Additive Identit is 0: e.g. Additive Inverse of is : e.g. Distributive Propert of Multiplication over Addition: e.g. Commutative: e.g. Associative: e.g. Multiplicative Identit is 1: e.g. Multiplicative Identit of is if and and Multiplication of a Monomial (single term) and a Binomial (two terms) Multipl rewrite subtraction as addition of the opposite appl Distributive propert appl Commutative(re-order) and Associate(re-group) properties multipl rewrite as subtraction Multiplication of Two Binomials Multipl appl Distributive propert appl the Distributive propert two times multipl add like terms Some of ou ma have learned this using the FOIL method : First-Outer-Inner-Last. All of the above steps can be done in one step: First + (Outer + Inner) + Last =. As long as ou do it correctl, an method is fine. Squaring a Binomial Multipl F + (O+I) + L
Q1: Multipl the following: Review 3- Page 2 of 2
Math 1400 - Mano Review 4 - page 1 of 2 Review 4 - Eponents Definitions: (1) If is a positive integer then, Eamples: - times (2) If (3) If and are integers, (4) If is positive, then Note: is the reciprocal of and is the reciprocal of Q1: Rewrite using positive eponents onl. Omit an root smbols. Q2: Evaluate without a calculator. Check our answer with our calculator. Q3: Rewrite so that the epressions have no visible denominator and no root smbols.
Math 1400 - Mano Review 4 - page 2 of 2 Properties of Eponents Eamples Q4: Rewrite the epressions. Multipl factors when possible. All eponents should be positive and all root smbols should be removed.
Math 1400 - Mano Review 5: Factoring Polnomials - page 1 of 2 Review 5 Factoring Polnomials Tpe 1:, where and are integers We want to factor into two factors of the form where and can be either positive or negative. If we complete the multiplication, we get (and ou should verif this) Therefore, and The first clue in factoring this tpe of epression is to determine if is positive or negative. If is positive, then either both and are positive or both and are negative. o The sign of and is determined b the sign of. If is negative, and are negative. If is positive, and are positive. And, is the sum of and (i.e. ) If is negative, then and have opposite signs. o The sign of is the same as the sign of. Eample 1: Factor into In this eample and. We need two factors of whose sum is -5. Since is positive we know that and are either both negative or both positive. Since is negative, we know that and are both negative. Again, we need two factors of whose sum is -5. The possible factors are shown in the chart to the left: 6 1 2 3-6 -1-2 -3 Since the last entr in the table satisfies o o We conclude Eample 2: Factor into (Notice the similarit to Eample 1) In this eample and. Since is negative we know that and have opposite signs. Since is negative, the factor with the larger magnitude is negative. We need two factors of whose sum is -5. The possible factors are shown in the chart to the left: 6-1 2-3 -6 1-2 3 Since the third entr in the table satisfies o o We conclude
Math 71 - Mano Review 5 - page 2 of 2 Tpe 2: : the difference of two squares The difference of two squares alwas factors into the product of two conjugates Definition: conjugates are two binomials such that one is the sum of two terms and the other is the difference of the two terms. Eample 3: Factor Eample 4: Factor Tpe 3: an polnomial epression Look for common factors in all terms of the polnomial, which include common constant factors and variable factors. Eample : Factor o The three terms have constant factors of 6, 4 and 10, which have a common factor of 2 o The three terms have variable factors of, which have a common factor of o Therefore, we can pull a factor or from each of the three terms in the epression.
Math 1400 - Mano Review 6- page 1 of 2 Review 6 Linear Equations Read pages 13 20 in the tetbook to review. I will just do some eamples here, which are sufficient if ou remember the basics. If ou need more practice complete pgs 23-24: 1-4, 5 27 odd, 37, 41, 43, 45 1. Forms of a linear equation: A. form B. form, where is the slope and is the C. point - slope form, where is the slope and is a point on the line. 2. Calculating slope: Given two points on the line Eample 1: What is the equation of the line pictured to the right? Pick an two points on the line. I chose and The The equation of the line, Therefore, is Also, ou can calculate the slope directl from the graph. From the top point (-1,3) move verticall down 4 units and then horizontall to the right 2 units to the point (1, -1). i.e. Q1: Through observation, which line has positive slope? negative slope? no slope? 0 slope? Then, find the equation of each line above. A. B. C. D.
Math 1400 - Mano Review 6- page 2 of 2 Q2: Write the equation of a line with slope 3 and -intercept -5. Q3: Write the equation of the line that passes through the points (-2,5) and (-3, 2) Eample 2: The third form of a linear equation: form of a line If we know one point on a line and the slope, then all other points on the line must satisf What is the equation of the line that passes through the point (-6, 2) and has slope? All points on the line must satisf go directl to this statement if ou memorize the point-slope form Q4: Find the equations of the lines with given slope through the given point. This can be done if ou start with the slope-intercept form of the line or the point-slope form of the line. Tr these both was.
Math 1400 - Mano Review 7: page 1 of 4 Introduction / Review 7 Logarithms Compete before Sections 11.1 & 11.2 Some of ou have previousl studied logarithms in high school and some of ou have not. There is a good review of logarithms in the tetbook, Section 2.3 pages 105 118. I am including a brief introduction/review here that gives ou eactl what ou need to know to successfull complete the rest of this course. What is a Logarithm? A logarithm is an eponent. There are two related functions, the eponential and logarithmic functions. If we write the general eponential function as, then the related logarithmic function is For both of these function, is called the base of the function with the restrictions that and Eample 1: (read as ) Two special cases are and (read as "natural log" where is the natural number) if no is shown, the base is 10 ln is a logarithm with base Q1: Write the related eponential statement for each logarithmic function in the first column and write the related logarithmic statement for each eponential function in the second column. Q2: Calculate the following without a calculator. Note about For all possible values of following are true: the Q3: Your TI83 or TI84 has two kes LOG and LN in the left-hand column. Find the following rounded to 4 significant digits to the right of the decimal point.
Math 1400 - Mano Review 7: page 2 of 4 Inverse Functions The function is the inverse function of if whenever is a point on the graph of the point is a point on the graph of Eample 2: and are inverse functions. Since, the point is on the graph of And, since, the point is on the graph of The graph of is the reflection of the graph of about the line Q4: Complete this chart. eponential function logarithmic function -3-2 -1 0 1 2 3 Q5. Plot the points ou found in the charts above. The first point in each chart is alread plotted. The gre line is, the line about which the reflection occurs. The red line connecting the points and is perpendicular to and bisected b the line (i.e. the pts are reflections of each other), Connect the points that belong to each graph. Q6. The domain and range of are domain range and the domain and range of are domain range
Math 1400 - Mano Review 7: page 3 of 4 Because eponential functions and their related logarithmic functions are inverses of each other, what one function does to a number, the other function undoes it. This is a crude wa of saing Eample 3: Eample 4: When do we use these properties? A. When ou are solving for a value that is an eponent, take the log or ln(either will work) of both sides. Solve for : take the logarithm of both sides: i.e. B. When ou are solving for a value that is in the argument of a logarithmic function, raise both sides as the eponent of a power function with the same base as the log function. Solve for : raise both sides to a power of 10 (since 10 is the base of log) Q6. Solve for the unknown This can also be solved b simpl writing the related eponential statement
Math 1400 - Mano Review 7: page 4 of 4 So far we have discussed the Properties of Logarithms listed in the first column (#1 - #4). We will look at the other three properties (#5 - #7), that come from related properties of eponents. Each propert is written using the general base and rewritten using the natural logarithm, which we will usuall be using in Chapter 11. If OR Properties of Logarithms OR OR OR OR OR OR Eample 5: Since the, we know that Q7: Given the following logarithms (rounded to 4 decimal places) appl Logarithm Properties #5 - #7 to find the answers to A D. Do not use a calculator Eample 6: Proof of #5. Let and Then the related eponential statements are Then substitution propert of eponents appl substitute definitions of Q8. Using Eample 6 as our guide, prove propert #6.