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Math and Humanity MATH 116
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Section 83 Properties of Triangles Practice HW from Mathematical Excursions Textbook not to hand in p 502 119 odd 4355 odd In this section we look at the basic concepts of similar triangles We then examine the relationship between the sides of a right triangle Similar Triangles Similar triangles have exactly the same shape but not necessarily the same size The following two triangles are similar AA The following de nition gives a mathematical description of similarity De nition Two triangles are similar if 1 All pairs of corresponding angles are equal 2 All pairs of corresponding sides are proportional that is for the diagrams above Ag DE DF EF Note If wig ampQ then ad be We explore the similarity of triangles in the following examples Example 1 For the similar triangles F C 12 ft A B D E nd EF Solution Note The concept of similarity of two triangles extends to all sides of two triangles including their heights Example 2 For the similar triangles F E C B 10 cm 20 cm A D Find the height of triangle ABC Solution Formulas for the Area and Perimeter of a Triangle Given the triangle ABC C A B l Perimeter Sum of the Lengths of the 3 sides AB BC AC 2 Area baseheight bh Example 3 The triangles ABC and DEF are similar Find the perimeter of triangle ABC and the area of triangle DEF 12 in A B D E 4in 10in Solution To nd the perimeter of triangle ABC we need side BC We can nd side BC by using the proportion Q a DE EF Since AB 4 DE 10 and EF 12 we have 4BC 10 12 Cross multiplying the equations and solving for BC gives 10 BC 4 12 10 BC 48 BC E 48 in 10 Thus we have Perimeter of Triangle ABC AB BC AC 3 in 4m 48in 118 in To nd the area of triangle DEF we need its height h This is found using the proportion height of triangle ABC AB height of triangle DEF DE Setting height oftriangle ABC 5 AB 4 DE 10 and setting h height oftriangle ABC we have next page 2 i h 10 Cross multiplying gives 5 10 4 h 4h 50 h a 125 in 4 110125 625in2 Area of Triangle DEF gbh E Example 4 Suppose a 6 ft tall man who is standing in front of a tree casts a 14 ft shadow Suppose the man is standing 20 ft from the tree Find the height of the tree Solution The Pythagorean Theorem In a right triangle triangle whose largest angle is 90 degrees the sum of the squares of the two legs is equal to the square of the hypotenuse The gives the Pythagorean Theorem Pythagorean Theorem Formula For the right triangle c a b we have a2 b2 02 Example 5 Find the unknown side of the triangle Round to the nearest tenth 6m 8 11 Solution Section 61 Rectangular Coordinates Practice HW from Mathematical Excursions Textbook not to hand in p 107 1 735 odd 4151 odd In this section we review the concept of the coordinate plane We then review the basics of functions Rectangular Coordinate System Consider the x y coordinate plane Quadrant II Quadrant I xlt0ygt0 xgt0ygt0 x Quadrant III Quadrant IV xlt0ylt0 xgt0ylt0 Points on the x y plane are represented using ordered pairs x y Example 1 Graph the points 2 l 0 4 3 l 4 3 3 1 Solution Graph of an Equation The graph of an equation is a geometrical drawing of all the ordered pairs that an equation satis es The easiest way to get a rough sketch of any equation is to plot points Example 2 Make a rough sketch of the equation y 2x 1 Solution A Example 3 Make a rough sketch of the equation y x2 2 Solution Functions A function is a rule that assigns to each number x in one set called the domain of the function a unigue one Q only one number y in another set called the range Notation A function is represented by y f x Note To evaluate a function y f x for a given value x a replace every occurrence of x in the formula for f x with the value a Example 4 Evaluate the function f x 1 3x at x 2 Solution Example 5 Evaluate the function gt 2t3 4t2 l at t 2 Solution Section 84 Volume and Surface Area Practice HW from Mathematical Excursions Textbook not to hand in p 518 137 odd In the section we look at some important surfaces in 3 dimensions Two important measurements that we will consider are the concepts of volume and surface area Volume You will speci cally look at how to nd the volume of various three dimension geometric objects such as rectangular solids and cylinders The volume of an object is the amount space occupied by the object On the next page is a list of some of the key formulas used to nd volume We will use these formulas in the following examples Example 1 Find the volume of rectangular solid that is 3 feet by 2 feet by 4 feet 4 feet 2E6 3 feet Solution Geometric Shape Sketch Volume Formula Rectangular Solid V lwh Right Circular Cylinder V 7 r2 h V s3 Cube Regular Square Pyramid V lbzh 3 Sphere Vlzrr2h Right Circular Cone Example 2 Find the volume of a cylinder with a radius of 3 in and height of 4 inches Solution The problem is modeled by the following diagram Substitute the values of the radius r 3 in and the height h 4 in into the volume formula V rzh V 7r3in24in V 7r9in24in V36m39n3 Thus the volume i V 3677 1131in3 Example 3 Suppose you have a cylinder shaped hot water heater that has a height of 5 feet and suppose the diameter of the bottom and top is radius of 2 feet How much water can the hot water heater hold Solution Example 4 The height of a regular square pyramid is 8 m and the length of a side of the base is 9 In What is the volume of the pyramid Solution Surface Area Every three dimensional object has both volume and surface area The volume as stated earlier measured the amount of space occupied by a three dimension object The surface area of a three dimension object measures amount of surface of the object The surface area of object such as a cube rectangular solid pyramid or cylinder is found by find the area of each face and nding the sum of the face On the next page is a list of some of the key formulas used to find surface We will use these formulas in the following examples Example 5 Find the surface area of a sphere with a diameter of 15 cm Solution Example 6 Find the surface area ofa cube whose sides measure 3 in Solution Geometric Shape Sketch Surface Area Formula Rectangular Solid S 21w 21h 2wh Right Circular Cylinder S 27172 2 r h V 632 Cube Regular Square Pyramid Sb22sl Sphere V 4 2 V 7139 r2 7139 rl Right Circular Cone Example 7 Suppose you want to mail a rectangular solid shaped package that measures 20 cm by 15 cm by 12 cm How postal wrap do you need to completely cover the package If the postal rate is 10 cents per cubic centimeter find the cost to mail the package Solution Section 6263 Rectangular Coordinates Practice HW from Mathematical Excursions Textbook not to hand in p 350 1 735 odd 4351 odd p 360 121 odd Linear Functions A linear function is a function of the form y f x mxb where m slope and b represents the ycoordinate of the y intercept 0 b Note The graph of a linear function is a straight line To graph a line we need at least two points A quick way to sketch a graph of a linear function is to nd its intercepts De nition For a linear function y f x mx b we de ne the intercepts as follows yz39ntercept 7 The point where the graph of the function crosses the yaXis To nd the y intercept set x 0 and solve for y This will give the point 0 b xintercept 7 The point where the graph of the function crosses the xaXis To nd the x N intercept sety 0 and solve for x This will give the point E 0 a Example 1 Find the x and y intercepts of the graph of the equation f x 2x 4 Example 2 Find the x and y intercepts of the graph of the equation 4x 3y 8 Solution Example 3 An approximate linear model that gives the remaining distance in miles a plane must travel from Los Angeles to Paris is given by st 6000 500t where st is the remaining distance thours after the ight begins Find and discuss the meaning in the context of the application of the intercepts on the vertical and horizontal axes Solution Facts about Lines 1 The slope intercept eguation of a line is given by yfxmxb where m slope and b represents the ycoordinate of the yintercept 0 b 2 Ifm gt 0 then the line goes up from left to right If m lt 0 then the line goes down from left to right 3 Suppose we are given the points two points x1y1 and x2 y2 on the following line my We de ne the slope through these points as follows changeiny ri y2 y1 change in x run x2 x1 4 If m 0 then y fx 0x b or fx b This gives a constant function A y 4 A line with a unde ned slope is a vertical line ofthe form x a A ky 5 To graph a line plot at least two points and draw a line connecting them If you have at least one point you can use the slope to nd additional points Example 4 Graph the line that passes through the point 2 l and has slope of3 Solution Writing the Equation of a Line To write the equation of any line we need the slope and at least w point Using y mx b we substitute the value of value of m and the coordinates of the point in for x and y and solve for b Example 5 Find the equation of the line that passes through the point 2 l and has slope of 3 Solution Example 6 Find the equation of the line through the points 1 2 and 5 8 Sketch the graph Solution Modeling Problems Using Linear Functions Many problems can be modeled using linear functions We illustrate this in the upcoming examples It is important to note that the equation y mx b is read as y is a function of x Points on the graph of this function are given by using ordered pairs x y Example 7 A car rental agency charges 100 plus 120 per mile to rent a car Determine a linear function that will find the amount the agency will charge when a customer drives x miles Use this equation to find out how much the agency charges when a customer rents a car and drives 1000 miles Solution 10 Fact For the equation y fx mx b the slope m means that when x increases by one unit y increases if m gt 0 or decreases if m lt 0 by m units Example 8 During a brisk walk a person burns about 38 calories per minute If a person has burned 191 calories in 50 minutes determine a linear function that models the number of calories burned after t minutes Use it to determine the number of calories burnt after 1 hour 60 minutes Solution Here we want a linear function that represents the number of calories burned as a function of the number of minutes I If we let C the number of calories burned and t the number of minutes then the linear function that models this problem is Cmtb To complete the equation of this linear function we need the slope and a point that satisfies this function Since a person burns 38 calories per minute when we increase the time by 1 minute the calories burnt increases by 38 Thus the slope is m 38 Hence substituting this value gives C38tb To find b we need a point that satisfies this equation Points on this equation are of the form t C Ifa person has burned 191 calories in 50 minutes this says I 50 when C 191 thus giving the point 50 191 Substituting these values gives 191 3850b 01 1913850b l9ll90b 191 190 b bl Thus the linear function describing the number of calories burnt is C38tl To find the number calories burnt after 60 minutes we simply substitute t 60 into this equation Number of Calories Burnt after 60 minutes C 3860 l 229 calories I 60 Example 9 As a weather balloon rises in altitude from sea level the temperature decreases at a fairly constant rate If the temperature is 590 F at sea level and 5550 F at 1000 ft nd a linear function that relates the altitude to the temperature What is the temperature at an altitude of 24000 ft Section 31 Logic Statements and Quanti ers Practice HW from Mathematical Excursions Textbook not to hand in p 123 19 odd 2349 odd In this section we introduce the basic ideas of compound statements and symbolic logic We start with introducing the concept of a statement Statements A statement is a sentence that is either true g false It cannot be both true and false Note Questions opinions and commands do not represent statements Example 1 Determine which of the following sentences are statements a Neil Sigmon teaches math b Do your homework c Neil Sigmon has taught math for 100 years d Does Neil Sigmon teach history Math is fun D Compound Statements Compound statements are statements made up of one or more simple statements In this section we introduce the ve most common compound statements and the symbols used to represent them Types of Compound Statements 1 N Negation 7 the denial of a statement frequently formed by inserting the word not Notation The negation symbol is given by N Example Let P It is raining NP Conjunction 7 consists of two statements joined by and Notation Use the A symbol for and Example Let P He is tall Q He is strong P A Q Disjunction 7 consists of two statements joined by or Notation Use the v symbol for or Example Let P Terri goes to school Q Terri works PvQ Conditional 7 uses the If P then Q phrasing P is called the hypothesis and Q is Called the J or 39 39 of the Jquot 39 Notation The conditional If P then Q is represented by P Q Example Let P Players lift weights Q Players will get stronger P gtQ 5 Biconditional 7 uses the P if and only if Q phrasing Notation The biconditional P if and only if Q is represented by P Q Example Let P We eat at Burger King Q Wendy s is closed Plt Q Example 2 Write each statement in symbolic form representing each component of the sentence with the letter indicated in parentheses a Neil teaches math m and Neil talks slow s b If you got to class c then you will do well w c I will major in history h or in political science p d It is not true that math is hard h Example 3 Let c you go to class h you do your homework and p you pass this class Write each symbolic statement in words a c v h bh p c c h Quanti ers Quanti ers are used are used to denote the existence of something or denote that every element of a set satis es some condition We will introduce two types of quanti ers in this section Two Types of g 2uanti ers l Existential Quanti ers 7 used as pre xes to assert the existence of something Fact The words some there exists and at least one are phrases representing essential quanti ers Examples Some people are good in math There exists some people who are not cut out for college N Universal Quanti ers 7 used as pre xes to assert that every element of a set satis es a condition or to deny the total existence of something Fact The words all and every are phrases representing universal quanti ers to assert that every element of a set satis es a condition The words none or no are phrases representing universal quanti ers to deny the existence of something Examples All baseball players are well liked No student likes homework N0te Universal Quanti ers can be expressed as a conditional If then statement For example the statement All baseball players are well liked can be rewritten as If you are a baseball player then you are well liked Negation 0f Quanti ers The negation of statement changes its truth value For example the negation of the sentence I am a nice person is I am not a not person Also the negation ofthe statement I do not like sports is I do like sports However it is easy to misinterpret the negation of quanti ed statements For example the negation of the statement All people are nice is not All people are not nice or No people are nice This is too narrow an interpretation The negation of All people are nice should be Some people are not nice Section 43 Different Base Systems Practice HW from Mathematical Excursions Textbook not to hand in p 205 2127 odd 17 119 odd Recall that the number system that we use the HinduArabic system is a base 10 system since all numbers can be written as a sum of the powers of 10 In this section we learn about other modern place value systems In particular we discuss how to convert between bases used in modern times and bases that computers use Binary Numbers Binary Numbers are base 2 numbers are made up only of 0 s and 1 s Computers use these numbers to represent data internally Examples of binary numbers are 0 which represents the number 0 100 which represents the number 4 1001 which represents the number 9 and 1011000 which represents the number 88 We now give a formal de nition of a binary number De nition A binary number bkbkil b2b1b0 where bi 0 or1 represents the base 10 decimal number given by bkzk 112k 1l222 1312113020 We illustrate this de nition in the following examples Example 1 Find the base 10 decimal representation of the binary number1002 Solution Example 2 Find the base 10 decimal representation of the binary number10110002 Solution Example 3 Find the base 10 decimal representation ofthe binary number 11100010112 Solution Note To convert a decimal base 10 number to binary we compute the powers of 2 starting with 20 that are less than the given number Then write the number as a sum of these powers of 2 from largest to smallest writing a coefficient of l in front the power of 2 that occurs in the sum and a 0 in front of the power of 2 that does not occur Reading off the coefficients from left to right gives the binary representation We illustrate this technique in the following examples Example 4 Convert 77 to binary Solution Example 5 Convert 320 to binary Solution Example 6 Convert 5413 to binary Solution We start by computing the powers of 2 that are less than 5413 This gives 20 1 24 16 28 256 212 4096 21 2 25 32 29 512 213 8192gt4096 STOP 22 4 26 64 210 1024 23 8 27 128 211 2048 Then we can write 5413 as 541340961317 40961024 293 4096102425637 40961024256325 409610242563241 21221028252220 12120211 1210029 128 027026 125024023122 021120 Thus reading off the coef cients we see that the binary representation base 2 representation of 5413 is 10101001001012 Conversion to Numbers Involving Other Bases We can extend the above concepts to arbitrary base conversions We extend the concept of binary numbers to arbitrary bases De nition A number of base a given by bkbk71b2b1b0 where 0 S b lt a represents the base 10 decimal number given by bkak bk1ak1 b2a2 b1a1 boao Fact The number of digits a number of base a can have is always one less than a For example For a base a 2 binary number the allowed digits are 0 and 1 For a base a 3 number the allowed digits are 0 l and 2 For a base a 10 number the allowed digits are 0 l 2 3 4 5 6 7 8 and 9 We illustrate this de nition in the following examples Example 7 Convert the numeral 201d ee to base 10 Solution Example 8 Convert the numeral 5476eight to base 10 Solution Notes 1 Numbers with bases larger than 10 need additional digits The additional digits are added starting with the letter A and adding subsequent letters when need The letters in base 10 represent increasing values For example A 10 B 11 C 12 D 13 etc For example the following represent the digits and their numerical base 10 values for the base 12 and base 16 IBaselZDigithl123456789 lBaselOValuel 0 1 2 3 4 5 6 7 8 9 10 11 IBasel6Digit0123456789A B C ID El lBaselOValuelOll2345678910I1112131415 2 The bases two binary eight octal and sixteen hexadecimal are number systems that are important in computer science Example 9 Convert the numeral EA3Sixteen to base 10 Solution Converting From Base Ten Numbers Back to an Arbitrary Base Converting from base ten numbers back to numbers of an arbitrary base is a slightly more difficult process The basic behind goes back to the process of long division you learned in high school We review this concept in the following example 40 Example 10 Cons1der 40 3 Determine the quot1ent and remainder and wr1te the result as an equation Solution To determine the quotient and remainder for division of larger numbers we can use a calculator for assistance The process involves getting the quotient by performing the division on the calculator and truncating or chopping all the digits to the right of the decimal point We illustrate the process in the following example Example 11 Calculate the quotient and remainder of the division 1024 37 using a calculator Solution Note To convert a decimal base 10 number to and arbitrary base a we compute the powers of a starting with do that are less than the given number Then take the highest power of a that divides into the given number The quotient of this division will represent the coef cient of the power of this power of a that occurs in the base a representation The remainder is then taken the power of a is decreased by l and the process is repeated until a0 l is reached We illustrate this technique in the following examples Example 12 Convert 687 to base ve Solution Section 21 Basic Properties of Sets Practice HW from Mathematical Excursions Textbook not to hand in p 61 183 odd Basic Concepts of Sets De nition A set is a collection or groups of objects Examples of Sets S spring summer fall winter V a e i o u 0 13 579 Aabcdexyz Fact The members of a set are called elements Notation 6 means an object is an element of set g means an object is not an element of set Example 1 Determine if the numbers 1 l3 and 14 are elements of the set 0 13 579 Solution Example 2 Determine if the letters b m and Y are elements of the setA ab c d e X yz Solution Note The empty set is the set with Q elements Notation for empty set or G Basic Number Sets Natural Numbers or Counting Numbers N 1 2 3 4 5 6 Whole Numbers W 0 l 2 3 4 5 6 Integers I 5 4 3 2 l 01 23456 Rational Numbers Q the set of all terminating numbers or repeating decimals that is the set of numbers of the form E where p and q are integers q and q 0 Irrational Numbers J the set of all numbers that are not a terminating number or repeating decimal not rational Real Numbers R the set of all rational numbers or irrational numbers Ways of Describing Sets 1 Roster Form When the elements of the set are written out inside a pair of braces The sets S spring summer fall winter V a e i o u 0 1 3 5 7 9 A a b c d e x y z are examples of sets in roster form Example 3 In roster form write the set of people on the US dollar bills for the set of bills less than or equal to 100 Solution Example 4 In roster form write the set of whole numbers less than 8 Solution Example 5 Write the set of integers x that satisfy x 5 3 in roster form Solution Example 6 Write the set of natural numbers x that satisfy x 5 3 in roster form Solution 2 Verbal Description of Sets describe set in words Example 7 Write a description of the set April June September November Solution Example 8 Write a description ofthe set x l x e N and x lt 5 Solution Here this set is read as the set of x such that x is a natural number counting number that is less than 5 We can easily write the elements of this set which in roster form is 1 2 3 4 3 SetBuilder Notation Uses a short hand notation to describe sets usually large ones Notation is given by the form x l x Read as the set ofx such thatx is Example 9 Use setbuilder notation to write the set spring summer fall winter Solution Example 10 Use setbuilder notation to write the set 10 20 30 90100 Solution One solution could be x i x is an integer multiple of 10 greater than or equal to 10 and less than or equal to 100 Well De ned Set A set is well de ned is the elements of the sets are clearly de ned that is if it is obVious what elements belong it If a set is well de ned then there should not be any confusion of what the elements are in the set Examples of well de ned sets V a e i o u 0 13 579 Examples of sets that are not well de ned The set nice people K 310100 21 0 Cardinality of a Set The cardinality of a nite setA denoted by nA is the number of elements in that set Example 11 Determine cardinality ofthe seta e i o u Solution Example 12 Determine cardinality of the setB of countries that border the USA Solution Equal and Equivalent Sets SetA is equal to set B denoted by A B if and only ifA and B have the same elements regardless of order SetA is equivalent to set B denoted by A N B if and only if A and B have exactly the same number of elements regardless of order Note The order that the elements of a set are listed does not matter If the elements are the same the sets are equal Also each element of a set is listed just once The elements of a set in general are not repeated Example 13 Determine whether the sets a e i o u and u o i a e are equivalent equal or neither Solution Example 14 Determine whether the sets 1 2 3 and 4 5 6 are equivalent equal or neither Solution Since the sets have the same number of elements in this case 3 they are equivalent sets However the elements are not the same Thus they are not equal Section 64 Quadratic Modeling Practice HW from Mathematical Excursions Textbook not to hand in p 372 117 2545 odd Quadratic Functions A quadratic mction is a function of the form yfxax2bxc a 0 Note The graph ofa quadratic function is a parabola The following are example graphs ofthe functions y x2 2x 3 and y x2 5 W yZX QrTxr mezxM 5 Definition The vertex ofa parabola is the highest or lowest point on its graph Note For the parabolay x2 2x 3 the Vertex is given by the point 1 4 For the parabolay x2 5 the Vertex is giVen by 0 5 Formula for the Vertex The formula for the Vertex ofa parabola y fx ax2 bx c a at 0 is giVen by b b Ky Zgtf EJ Example 1 Algebraically nd the vertex of the equation y x2 2x 3 Solution Example 2 Algebraically nd the vertex of the equation y x2 5 Solution Finding the x intercepts of a Parabola The x intercepts of a quadratic function y f x ax2 bx c a at 0 are the points where the graph crosses the x aXis To nd the xintercepts of a quadratic function we set y fx 0 and solve for x Example 3 Algebraically nd the x intercepts of the equation y x2 2x 3 Solution Example 4 Algebraically nd the nd the x intercepts of the equation y x2 5 Solution Maximum and Minimum Values of a Quadratic Function Given a quadratic function y f x m2 bx c a at 0 the maximum value is the largest y value that the graph of the function obtains The minimum value is the smallest y value that the graph of the function obtains Using this idea we can algebraically obtain a quick sketch algebraically of a quadratic function by performing the following steps A Quick Way to Sketch a Quadratic Function and Determine the Maximum and Minimum Values To sketch the graph of a quadratic function y f x m2 bx c a a6 0 we use the following steps 1 Find the x intercepts by setting y f x 0 and solving for x b b 2 Find the vertex x y i f i 2a 2a 3 Determine the shape of the graph If agt0 then the graph is cupped upward and the y coordinate of the vertex b y f is a minimum The graph has no maximum at If alt 0 then the graph is cupped downward and the y coordinate of the vertex b y f is a maximum The graph has no minimum a Example 5 For the quadratic function f x 3x2 6x nd the xintercepts and vertex Sketch the graph and determine the minimum or maximum value of the function Solution Example 6 For the quadratic function f x x2 4x 3 nd the xintercepts and vertex Sketch the graph and determine the minimum or maximum value of the function Solution Applications of Quadratic Equations Using the vertex of a quadratic equation we can nd the maximum or minimum of quantities that are modeled by quadratic equations We illustrate this in the following examples Example 7 Suppose that the path traveled by a golf ball can be modeled with the quadratic equation y 002x2 058x where x is the distance in yards from the point it was hit and y is the height of the golf ball in yards Assuming the ground is level and rounding your answers to 1 digit to the right of the decimal find the maximum height the golf ball reaches How far from where it was hit does the ball reach the ground Solution Example 8 A mining company has determined that the cost c in dollars per ton of mining a mineral is given by Ct 02t2 2t 12 where t is the number of tons of mineral that are mined Find the number of tons of the mineral that should be mined to minimize the cost What is the minimum cost Solution Section 32 Truth Tables Equivalent Statements and Tautologies Section 33 The Conditional and Biconditional Practice HW from Mathematical Excursions Textbook not to hand in p 124 5157 odd p 135 13 15 17 29 33 37 39 41 43 45 p 144 15 17 2535 odd 4751 odd In this section we introduce basic truth tables of compound statements Truth Tables A truth table gives the truth value of a compound statement for all possible truth values of the simple statements that make it up We will consider the truth tables for negation the conjunction the disjunction and the conditional 1 Truth Table for Negation Given by not P and represented by N P the truth value is simply reversed Truth Table For Negation 2 Conjunction pA q p and q Truth Table for Conjunction A Note A conjunction p A q can only be true if both simple statements p and q are true If either p and q or both are false the nal statement is false Example I am a Radford student and I am in Math 116 can only be true if you are both a Radford student and are indeed taking Math 116 3 4 Disjunction p v q p or q Truth Table for Disjunction V Note A disjunction p v q can only be false if both simple statements p and q are false If either p and q or both are true the nal statement is true Example I will go my Math 116 class or stay in bed is only false ifyou do not go to y your Math 116 and do not stay in bed Conditional p q If p then q Truth Table for Conditional Note A conditional p q can only be false if the hypothesis p is true and the conclusion q is false Example Suppose p you attend class and q you will pass and we form the conditional p q that says If you attend class then you will pass This conditional can only be false if you attended the class but still did not pass the class Example 1 Determine whether each statement is true or false a 3 S 9 b The United States borders Canada and Australia c If pigs can y then orioles are birds Construction Truth Tables of Compound Statements Involves knowing the values of the four basic truth tables negation conjunction disjunction and conditional These are summarized at the top of the next page Basic Truth Tables Truth Table For Negation Truth Table for Disjunction Truth Table for Conditional Example 2 Construct a truth table for the compound statement pA N q Solution Example 3 Construct a truth table for the compound statement N pv N q Solution Example 4 Construct a truth table for the compound statement p N q Solution Example 5 Construct a truth table for the compound statement q A N N p v q Solution Example 6 Construct a truth table for the compound statement p p q v q Solution Example 7 Construct a truth table for the compound statement p v N q N N p A q Solution The following is the truth table with intermediate steps included Note that it is best to construct what is to the left and right of the middle conditional arrow rst NP N9 v Equivalent Statements Two statements are equivalent if they both have the same truth values for all possible truth values of their component statements that is if the nal results of their truth tables the same Notation If statement p is equivalent to statement q we write p E q Example 8 Show that N p A q is equivalent to N p v N q Solution Example 9 Show that p q is equivalent to N p v q Solution Using Equivalent Statements t0 Rewrite Sentences We can use equivalent statements to rewrite sentences that communicates the same statement logically Common Eguivalent Statements 1 De Morgan s Laws for Statements For statements p and q pvqE NPANq pAqE vaNq Example 10 Use one of De Morgan s Laws to rewrite the statement It is not the case that I did my homework and came to class in an equivalent form Solution Example 11 Use one of De Morgan s Laws to rewrite the statement I did not fail this class and I did not unk out of school Solution Symbolically if we let p I did fail this class and q I did unk out of school this sentence can be represented as NPANq Since De Morgan s Laws says N pA N q E N p v q we can rewrite the sentence as It is not the case that I failed this class or unked out of school 2 Equivalent Disjunctive Form of the Conditional For statements p and q P q E N P V q Example 12 Write the conditional statement If Virginia Tech wins then Radford students will rejoice in its equivalent disjunctive form Solution 3 Negative of the Conditional For statements p and q N p gt 1 E M N 617 Example 13 Write the negation of the conditional statement If Virginia Tech wins then Radford students will rejoice Solution If we let p Virginia Tech wins and q Radford students will rejoice then we have the conditional statement p q Since the negation of the conditional statement symbolically is N p q E pA N q the negation of the this conditional can be written as Virginia Tech wins or Radford students will not rejoice Section 65 Exponential Functions Practice HW from Mathematical Excursions Textbook not to hand in p 386 19 odd 1529 odd In this section we talk about the basics of exponential functions Before de ning what an exponential function we review two basic exponent laws that can be useful Laws of Exponents 1b01 1 2 b quot b7 b a6 0 Exponential Functions The exponential function with base b is denoted by fx bquot where b gt 0 b at l and x is any real number Example 1 Given fx 5X1 f1nd f0 f3 and f 2 Solution 2x Example 2 GivenHx nd H3 H 2 and Hl Solution Graphs of Exponential Functions Example 3 Graph fx 2x and gx 3quot Solution Emmlz u Graph x 2x and gx 3x Snllnjnn 1n gaml annan Willxalx Emu 5 Gmph m 2 3 Snlmjnn The Number 2 The number 6 is an irrational number approximated by e m 27182818 The function f x ex is called the exponential function of base 6 Example 6 Determine the number 6 82 and 6 3 on a calculator Solution Example 7 Graph fx ex and gx 67quot on the same graph Applications of Exponential Functions We now look at some basic application problems involving exponential functions Compound Interest Compound interest is where the interest is always calculated on the current amount in an account The amount in the account after a certain time can be calculated using the following formula Compound Interest Formula r nt AP1i 7 P principal the original amount of money invested at time t 0 where 0 r annual yearly interest rate in decimal n the number of times per year interest is compounded n 1 annually 71 2 semiannually n 4 quarterly n 12 monthly 7 365 daily I the number of years the money grows A the future amount the amount of money the investment grows to If interest is compound continuously interest is incrementally always being added to the acco The following formula is the compound interest formula for continuous compounding squot 1111 Compound Interest Formula A Pequot where P principal the original amount of money invested at time t 0 0 r annual yearly interest rate in decimal t the number of years the money grows A the future amount the amount of money the investment grows to Example 8 Suppose 5000 was invested at 7 annual interest How much money would be in the account after 10 years if interest is compounded a semiannually Solution b monthly Solution c continuously Solution Example 9 The radioactive isotope iodine131 is used to monitor thyroid activity The number of grams N of iodine131 in the body thours after injection is given by 1 11937 NI l5 7 2 Find the number of grams of the isotope 24 hours after an injection Round to the nearest tenthousandth Solution For this problem we want to nd the number of grams N after t 24 hours We substitute t 24 in to the given formula to obtain Number of Grams after 1 241937 t 24 hours N2415E 1 50 501239029427 1509177016186 1 1376552428 Rounding to the nearest tenthousandth four places to the right of the decimal we see that there are approximately 13766 grams of iodine131 24 hours after injection Supplemental Section 1 Perspec ve Practice HW not to handm pp 14717 at the end of these notes Perspective Have you everlooked at a pathtthg ofapretty seehe of the country state and the picture Orperhaps you wondered how the amst was able to create aLhree objects m manner that creates three dAmEnsxonal effector depth m the atath For and athree dJmEnSlonal etfeet is not obtamed Figure 1 Example whae vu39y lime paspec ve is used There are several ways to create aperspecuve m an whteh wtu he1p obtath a three h dw orpamnng As tttums out there are several types of perspective Here we wtu study four types of perspective Types of Perspective 1 Overlapping Shapes 2 Diminishing Sizes 3 Atmosphere Perspective 4 One Point Perspective As we study each type of perspective we will look both art work and photographs that exhibit each type of perspective Overlapping shapes In overlapping shapes depth perception is created by using overlapping shapes If one shape is placed in front of another it givens the perception that the scene is not at Here are a few pictures that exhibit perspective using overlapping shapes Figure 2 Example of overlapping shapes Diminishing sizes In diminishing sizes depth is created by systematically making objects smaller Objects that are closer to the eye are naturally larger than objects that are further from the eye In some drawings or paintings artist use variances in sizes to create depth perception Diminishing sizes also are prevalent in photographs In the following examples you can see how diminishing sizes occur In the Golden Gate Bridge photograph notice that the rst tower of the Golden Gate Bridge is much larger that the second tower A depth perception is created naturally in the photo due to the different sizes of the objects in the photograph If we look closer at the photo we can also see that the suspension cables of the bridge get smaller as the bridge get further away Figure 3 Example of diminishing sizes The next example is drawing called the matching necklace The artist uses different size beads or gems to create depth perception Figure 4 Example of diminishing sizes Atmospheric Perspective In atmospheric perspective depth is created by making objects that are farther away less clear by diminishing both color and shading If you have ever looked at objects such as mountains at a distance you Will notice that they seem less vivid than objects that are closer This effect can be captured in a painting or drawing by using different shading on objects closer to the naked eye than object farther from the eye Again this perspective can also be clearly seen in a photograph as well In next photograph you can see how atmospheric perspective appears to the naked eye in real life Notice that closer objects appear to be much more vivid Where as objects further away the mountains and clouds are less vivi Figure 5 Example of atmospheric perspective R my Amen Blersmdl ISM Figure 6 Example of atmospheric perspective OneePoim Perspective 1r onerpomt perspectlve amsts ear represent threerdlmenslonal spaee on atwo dlmenslonal earwas by usmg the slmple geometry ofconverglng llnes When onepoml perspeeave ls used objects seem to converge on a slngle xed polnt Objects are drawn m onepoml perspeeave usmg the followmg steps 1 Turn your paper horizontal landscape orientation Turn your paper horizontal 2 Draw a horizontal llne somewhere on the page Thls ls your eye level Draw a harlzon ma 3 Draw a dotln the mlddle ofyourhonzon llne Thls ls yourvanlshlng polnt Make vanishing ppm are orthogonal Draw a square or rectangle E Draw arthogonnls from shape corners to vanishing point These 6 Draw a horizontal 39 Draw a horizontal line To and our form 7 Draw a Vertical line down from the horizontal line to compl ete the side Draw a vertical line to make the farm39s side 9 Erase the remammg onhogonals Erase the arthogannls Now you have a 3D form in Emapain pgrspec vel 10 Add detads and expenmem Draw nmrhzr forml Add wmdows and dour M V e u n where the y vamshmg pomts are In this photograph we can see that the cable car tracks meet at a vanishing point in the photograph Notice that the vanishing point is located at a point near the end of the street or below the tower of the San Francisco Bay Bridge In this next photograph you can see how the railroad tracks meet a vanishing point to create a one point perspective Using Proportions to Calculate Distances in a Drawing Suppose we asked to make a drawing in onepoint perspective with a few trees in it and we were asked to draw the trees using diminishing sizes to create a perspective in the drawing How could we use math to gure out how to space the trees apart from one another The answer to the problem is to use similar triangles and proportions to nd the correct distances Before we work example like this let s review a few things about similar triangles Similar Triangles and Proportions In the diagram below is a picture of two triangles that are similar Since these triangles are similar the sides of two similar triangles are proportional Given that the sides of the h d tr1angle are proportlonal we can set up the follow1ng ratlo between the s1des hi 672 1 We consider the problem of trying to nd one of these sides in the following example Example 1 Given that the triangles below are proportional nd hl Solution Set up a proportion and solve for the missing variable which is h1 h2 d2 hl d1 5 8 Substitute in given values hl 14 8111 145 Cross multiply 8111 70 Simplify 70 h1 g 875 ft Solve We can use proportions to help draw objects correctly in onepoint perspective We illustrate in the following examples 11 Example 2 For the picture below ifa 5 in b 3 in and e 12 in nd d and 0 Round your answer to 2 decimal places Solution Example 3 Use the picture below ifa 10 in d 18 in and e 30 in nd b and 0 Round your answer to 2 decimal places Solution 13 Example 4 In the diagram below the distance between the telephone poles is c the height of the two telephone pole is a and b respectively the distance from rst telephone pole at to the vanishing point is e and the distance from second telephone pole to the vanishing point is 61 Using the diagram to nd distance d and c given that a 4 in b 3 in and e 12 in Solution Using the given values of a 3 in b 3 in and e 12 in we rst use similar triangles and the concept of proportionality to nd side 61 AMAQ w 5ampmiamp 312 4d 4d 36 E 4 4 Since cde ce d12in 9in3in Exercises 1 Name the four types ofpaspectwe dxscussed m ths semen z Gwe a bnefdescnpuon ofeach othe four types ofperspectwe 3 Section 66 Logarithmic Functions Practice HW from Stewart Textbook not to hand in p 400 141 odd 4961 Logarithmic Functions De nition For b gt 0 b at l the logarithmic function of base b is denoted by ylogbx means by x Fact The logarithm is the exponent b must be raised to in order to get x Example 1 Write the logarithmic equation log 4 16 2 in exponential form Solution l Example 2 Wr1te the logar1thm1c equat1on log3 a 4 1n exponent1al form Solution Example 3 Write the exponential equation 53 125 in logarithmic form Solution 1 Example 4 Write the exponential equatlon 2 4 E 1n logarithmlc form Solution Example 5 Evaluate log2 32 Solution Example 6 Evaluate log 5 Solution Example 7 Solve the equation log7 x 2 for x Solution Converting the equation to exponential form we have log7 x 2 Special Types of Logarithms Logarithm Base 10 Given by ylogxlog10 x Note logxlog10 x y means 10y x Example 8 Evaluate 10g1000 Solution Example 9 Solve logx 25 for x Solution Natural Logarithm Function Given by y lnx 10ge x Note loge x y means ey x Example 10 Solve lnx 25 for x Solution First note that the equation says that by the de nition of the natural logarithm lnx loge x 25 Ifwe convert loge x 25 x e25 z 121825 to exponential form we obtain Graph of Logarithmic Function The logarithmic function of base b is given by fx 10gb x Fact Graphically y logb x is the inverse of the exponential function y bx This means if the ordered pair x y is on the graph of f x bx then the ordered pair y x is on the graph of fx logb x Example 11 Graph fx log2 x Solution quotquot 39 hanem 39 39 39 ofbaseb fxlogbx Calculating The Natural Logarithm and Log Base 10 on a Calculator Use the log key for the log base 10 and the ln key for the natural logarithm Example 12 Approximate ln7 ln04 log14 log2 ln0 logZand ln3 on a calculator Solution Applications of Logarithmic Functions Logaritth are useful in applications that have large numerical values Logarithms can scale the values in a more useful form to understand We look at some of these applications next The pH of a Solution Chemistry uses logarithm to determine the pH of liquid The pH of a liquid measures the acidity or alkalinity of a liquid A liquid with a pH of l is a very strong acid and a liquid with a pH of 14 is a very strong base Speci cally the acidity ofa substance is a function of its hydrogenion concentration The pH of substance can be determined by the taking the logarithm of its hydrogen concentration H The following formula calculates the pH of a substance Formula for the pH of a Solution The pH of a solution with a hydroniumion concentration of H moles per liter is given by pH logH Example 13 Find the pH of a sample of orange juice that has a hydrogenion concentration of 34x 10 4 mole per liter Solution The Richter Scale for an Earthquake The Richter scale uses a logarithmic scale to measure the magnitude of an earthquake Let I be the intensity of an earthquake Often the intensity I is given in terms of the constant I 0 where I 0 is the intensity of the smallest earthquake called a zerolevel earthquake that can be measured on a seismograph near the earthquake s epicenter The following formula is used to compute the Richter scale magnitude of an earthquake The Richter Scale Magnitude of an Earthquake An earthquake with an intensity of I has a Richter scale magnitude of M logi 10 where I 0 is the measure of the intensity of a zerolevel earthquake Example 14 The earthquake on November 17 2003 in the Aleutian Islands of Moska had an intensity of I 63095734 I 0 Find the Richter scale magnitude of the earthquake Round to the nearest tenth Solution Example 15 Suppose an earthquake measured 81 on the Richter scale Find the intensity of the earthquake in terms of I 0 Round to the nearest whole number Solution World Oil Supply Example 16 The time it will take the world s oil supply to be depleted can be modeled by the following formula where r is the estimated oil reserves in billions of barrels Tr1429ln00041r1 a Use the model to nd out how much time it will take to use 500 billions barrels Solution Here we substitute r 500 into the function Time it takes to use T 500 14291 00041500 1 14291 305 16 r 500 billionbarrels n n 3 years Seamn23Sahmnumns Practice HW from Mathematical Excursions Textbook not to hand in p 71 119 odd 2945 odd In this section we look at the most basic of set operations union and intersection In addition we look at how Venn diagrams can be used to examine sets Intersection and Union of Two Sets The intersection of two sets A and B denoted as A n B is the set of elements that are in bothA w B that is the set of elements A and B have in common The union oftwo sets A and B denoted as A U B is the set of elements that are in A in B Q in bothA and B To nd just take the elements of A and put them together with the elements of B without repetition of elements that is write the common elements of both sets only once Example 1 Find A n B and A U B for the following pairs of sets a A Radford UNCA NC STATE B Virginia Tech Radford UVA bAmath Bfun cA jL BQ Example 2 Let U 0 1 2 3 4 5 6 7 8 9 A 1 3 4 5 7 B 2 3 4 5 6 and C 0 2 4 6 8 9 nd each ofthe following a A39 d A n B e A39nB f A UB39 g AuBnC h AuB39nC39 Note Parts d and f from Example 2 illustrate a more general result known as De Morgan 3 Laws De Morgan s Laws For sets A and B AuB39A39nB AnB39A39uB39 Venn Diagrams Venn Diagrams give a geometric way involving circles to represent sets Note It is probably easiest to number your regions and compute the set results using the ordinary de nition of set and union Then shade in the nal result Example 3 Draw a Venn diagram to ShOW the following sets a AnB gt U b AUB gt U c A39UB A B IV d AnB39 Solution A B IV Note that the shaded region comes from the following observations A Regions I II These are the regions inside circle A B39 Regions I IV These are regions not in circle B A n B39 Region I Intersection of circles A and B39 7 regions they have in common e A n B gt U f AUBnC gt 4 B C VIII Section 45 Primes Numbers Practice HW from Mathematical Excursions Textbook not to hand in p 228 1120 2940 The purpose of this section is to study prime numbers and how prime numbers are the building blocks of all numbers Prime Numbers Recall that a prime number p is a number whose only divisors are 1 and itself 1 and p A number that is not prime is said to be composite The following set represents the set of primes that are less than 100 2 3 5 711131719 23 29 31 37 41 43 47 53 59 61 67 71 73 79 83 89 97 A larger list ofprimes can be found in the Barr text on pp 370372 Facts About Primes 1 There are an in nite number of primes 2 Every natural number can be factored into a product of primes Fundamental Theorem of Arithmetic Determining the Primality of Larger Positive Integers Because of its use in cryptology and other applications mathematical techniques for determining whether large numbers are prime have been targets of intense research We study some elementary factors for determining the primality of numbers Fact 2 is the only even prime Any even number larger than 2 is not prime since 2 is a divisor Example 1 Is 10000024 prime Solution How do we determine if large positive integers are prime The next example illustrates an elementary method for doing this Example 2 Is 127 prime Solution Example 2 provides the justi cation for the following primality test for prime numbers Sguare Root Test for Determining Prime Numbers Let n gt 1 be anatural number Ifno prime number 2 3 5 7 ll 13 less than J is a diVisor of n than n is prime Example 3 Determine if 839 is prime Solution Example 4 Determine if 1073 is prime Solution Example 5 Determine if 1709 is prime Solution As the numbers tested get larger the square root test for primality does have limitations The next example illustrates this fact Example 6 Determine if 958090550049 is prime Solution Being able to determine whether large numbers are prime has important applications in topics such as cryptography the science of secret message writhing To deal with larger numbers much more sophisticated tests for primality testing have been developed and are an on going topic of research The largest prime number discovered up to 2006 was the 232582657 number l which is a 9808358 digit prime number Factorization of Composite Numbers A number that is not prime is said to be composite Ifa number is composite it can be factored into prime factors other than 1 and itself This is guaranteed by the following fact Theorem The Fundamental Theorem ofArithmetic Every natural number larger than 1 is a product of primes This factorization can be done in only one way if is disregarded For example to factor 30 we can compute 30 65 235325523 same prime factorization dixregarding order An elementary way to obtain prime factorizations with small prime factors involves the use of a calculator and a factor tree The following examples illustrate this technique Example 7 Factor 380 into a product of prime factors Solution Note Prime factorizations of numbers using exponential notation with the factors arranged in order of increasing magnitude is called the canonical form factorization of the number Example 8 Factor 3267 into a product of prime factors in canonical form Solution Section 22 Complements Subsets and Venn Diagrams Practice HW from Mathematical Excursions Textbook not to hand in p 71 149 odd In the section we continue with our discussion of basic set concepts The Universal Set and Compliment of a Set The universal set is the set Uof all possible elements being 391 J for a particular problem Example 1 Find an appropriate universal set for the elements Virginia Texas Virginia and Ohio Solution De nition The compliment of setA is the set of all elements in the universal set U that are not in the set A Notation for Compliment Given a set A the compliment is given byA39 Example 23 Find the compliment of the following sets given that the universal set is U 0 1 23456 7 89 3 0 1 4 5 6 b xlxlt6 and er c or Subsets If eve element ofa setA is also an element ofa set B we sayA is a subset ofthe set B Notation The notation A g B denotes that the setA is a subset of the set B The notation A B denotes that the setA is not a subset of the set B Facts 1 Every set is a subset of itself that is given a set A then A g A 2 The empty set is a subset of every set that is for any setA E lt A Example 3 For the following sets insert g or Z in the blank space between the sets to make a true statement 21 121 45 12141618 b red white yellow the colors in the American ag c baseball football all sports d the set of rational numbers the set of integers Proper Subsets A setA is aproper subset ofa set B denoted by A c B if every element ofA is an element ofB and A at B For example the set baseball football is a proper subset of the set baseball football basketball and we write baseball football C baseball football basketball However the set USA Mexico Canada is not a subset ofthe set Mexico Canada USA since the sets are equal However we can write USA Mexico Canada g Mexico Canada USA Example 4 Given the sets U a e i o u W a i o u X a u and Y 6 determine whether the following is true or false aXgW Listing the Subsets of a Set Fact A subset containing 71 elements contains 2quot subsets Example 5 List the number and all of the subsets of the set dog cat Solution
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