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FUNCTNS TRIG & LNR STM MATH 150
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Section 44 Math 150 Lecture Notes Exponential and Logarithmic Equations To Solve an Exponential Equation 1 Isolate the exponential expression on one side of the equation 2 Change from exponential to logarithmic form 3 Use the Laws of Logarithms to solve for the variable To Solve a Logarithmic Equation 1 Use the Laws of Logarithms to combine condense the logarithms into one term 2 Change from logarithmic to exponential form 3 Solve the resulting equation for the variable Example 1 Find the solution of each exponential equation correct to four decimal places 2334 e3 16 10 2 16 Example 2 Solve the equation xzex xex 7 ex 0 Section 44 Example 3 Solve each logarithmic equation for x ln2 x l logx 7 4 3 Example 4 Solve each logarithmic equation for x 2 logxlog2 log3x74 log5xlog5 x l log5 20 Section 45 Math 150 Lecture Notes Applications of Exponentials and Logarithms A population that experiences exponential growth increases according to the model nt mequot where nt population at time I no initial size of the population r relative rate of growth expressed as a proportion of the population I time If m0 is the initial mass of a radioactive substance with half life h then the mass remaining at time t is modeled by the function mt moe39quot where r 11172 According to Newton s Law of Cooling if D0 is the initial temperature difference between an obj ect and its surrounding and if it surroundings have temperature Ts then the temperature of the obj ect at time tis modeled by the function Tt T s Doe39k where k is a positive constant that depends on the type of obj ect Chemists measure the acidity of a solution on the pH Scale pH logH where H is the concentration of hydrogen ions measured in moles per liter M According to the Richter Scale the magnitude M of an earthquake is M log where I is the intensity of the earthquake measured by the amplitude of a seismograph reading taken 100 km from the epicenter of the earthquake and S is the intensity of an earthquake whose amplitude is 1 micron 10394 cm The intensity level B measured in decibels dB is B lOlogIL 0 Section 45 Example 1 A culture contains 1500 bacteria initially and doubles every 30 minutes a Find a function that models the number of bacteria nt after tminutes b Find the number of bacteria after 2 hours c After how many minutes will the culture contain 4000 bacteria Example 2 After 3 days a sample of radon222 has decayed to 58 of its original amount a What is the halflife of radon222 b How long will it take the sample to decay to 20 of its original amount Section 45 Example 3 A kettle full of water is brought to a boil in a room with temperature 20 C After 15 minutes the temperature of the water has decreased from 100 C to 75 C Find the temperature after another 10 minutes Illustrate by graphing the temperature function Example 4 The Northridge California earthquake of 1994 had a magnitude of 68 on the Richter scale A year later a 72magnitude earthquake struck Kobe Japan How many times more intense was the Kobe earthquake than the Northridge earthquake Section 43 Math 150 Lecture Notes Laws of Logarithms Laws of Logarithms Let a be a positive number with a 7i 1 Let A B and C be any real numbers with1 gt 0 andB gt 0 1 log AB logaA logaB 2 log logaA flogaB 3 10gaAC C logaA Change of Base Formula loga x 10 x gb loga b Example 1 Evaluate the expression 10g3 100 710g3 18 710g3 50 2 Example 2 Use the Laws of Logarithms to expand the expression log a J b HE 3x2 Example 3 Use the Laws of Logarithms to expand the expression 111W x Section 43 Example 4 Use the Laws of Logarithms to combine condense the expression logs x2 71710g5x71 Example 5 Use the Laws of Logarithms to combine condense the expression logax 1 logx 4 logx4 x2 1 Example 6 Use the Change of Base Formula and a calculator to evaluate the logarithm correct to four decimal places log 92 Section 51 Math 150 Lecture Notes The Unit Circle The unit circle is the circle of radius 1 centered at the origin in the xyplane Its equation is x2 y2 1 If a distance I gt 0 is marked along the unit circle starting at 1 0 and moving in a counterclockwise direction the arrival point is called the terminal point The reference number associated with a real number I is the shortest distance along the unit circle between the terminal point determined by t and the xaxis 52 Example 1 Show that the point 3 is on the unit circle Example 2 Gwen that P 1s on the un1t c1rcle nd the m1ss1ng coord1nate of P 3 1n quadrant 11 3 Example 3 Find P given that the x coordinate of P is positive and the ycoordinate of P is am Examgle 4 Fmd rand me lemma pmm delenmned by to increases m mtremmls ufn fur my mm m me gure Examgle s Fmd me 12mm pmm PO47 un me umt male dela mme by Examgle a Fmd thereferencenumba39 fun and fun 797 Section 32 Math 150 Lecture Notes Division of Polynomials If Px and Dx are polynomials with Dx 0 then there exist unique polynomials Qx and Rx where Rx is either 0 or of degree less then the degree of Dx such that POC DOC Q06 ROG The polynomials Px and Dx are called the dividend and divisor respectively Qx is the quotient and Rx is the remainder The long division process for polynomials mirrors the process for long division of integers Synthetic division is a quick method of dividing polynomials writing only the essential parts that can be used when the divisor is of the form x 7 c Remainder Theorem If the polynomial Px is divided by x 7 c then the remainder is the value Pc Factor Theorem 0 is a zero ofP iifx 7 c is a factor ofPx Example 1 Divide Px by Dx Px 7 x5 x4 7 2x3 x 1 Dx 7 x2 x 7 1 Example 2 Divide Px by Dx Px 7 6x4 10x3 5x2 x 1 Dx 7 3x 2 Section 31 Math 150 Lecture Notes Polynomial Functions A polynomial function of degree n is a function of the form Pxax an1x 391a1xa0 Where 7 is a nonnegative integer and an at 0 The numbers a0 a1 a2 an are called the coef cients of the polynomial The number a0 is the constant coef cient or constant term The number an the coef cient of the highest power is the leading coef cient and the term aux is the leading term Graphs of polynomial functions are smooth curves with no breaks or corners The end behavior of a polynomial is a description of what happens as x becomes large in the positive or negative direction Notation x gt 00 means x becomes large in the positive direction x gt 00 means x becomes large in the negative direction For any polynomial the end behavior is determined by the term that contains the highest power of x Section 31 If P is a polynomial function then c is called a zero of P if Pc 0 In other words the zeros of P are the solutions or roots of the polynomial equation Px 0 If P is a polynomial and c is a real number then the following are equivalent 0 is a zero ofP 2 x c is a solution of the equation Px 0 3 x 7 c is a factor ofPx 4 x c is an x intercept of the graph of P Intermediate Value Theorem for Polynomials If P is a polynomial function and Pa and Pb have opposite signs then there exists at least one value 0 between a and b for which Pc 0 Graphing Polynomials Functions 1 Find all the real zeros or xintercepts Break it down into linear factors by factoring methods and or quadratic formula 2 Mthe xintercepts and determine the shape near the intercepts according to the multiplicity of the factor 3 Determine the end behavior of the polynomial 4 Make a table of values including test points to determine where the graph is above or below the xaxis and 15intercept 5 Plot the test points and yintercept and sketch a smooth curve passing through the points and having the required end behavior IfPx a xquot a xquot391 alx a0 is a polynomials of degree n then the graph ofP has at most 71 7 1 local extrema Example 1 Sketch the graph of the function by transforming the graph of the parent function Indicate all x and yintercepts on the graph x 12x7 25 16 Section 31 Example 2 Example 3 Example 4 Example 5 Sketch the graph of the polynomial function Px x 7 12 x 23 Sketch the graph of the polynomial function Px 1A 2x4 3x3 7 16x 7 242 x52xzx Determine the end behavior of P Graph the polynomial and determine how many local maxima and minima it has y 712995 375x4 7 7x3 715x2 18x Section 36 Math 150 Lecture Notes Rational Functions PUG ROG A rational function is a function of the form rx where P and Q are polynomials The domain of a rational function consists of all real number x except those for which the denominator is zero The line x a is a vertical asymptote of the function f if fx approaches i 00 as x approaches a from the right or left The line y b is a horizontal asymptote of the function f if fx approaches b as x approaches i 00 Asymptotes of Rational Functions anxquot a x39 1 a1x a0 bmx39quot bmilxm391 b1x b0 1 The vertical asymptotes of r are the lines x a where a is a zero of the denominator Let r be the rational function rx 2 a If n lt m then V has horizontal asymptote y 0 b If n m then V has horizontal asymptote y 4 m c If n gt m then V has no horizontal asymptote To Sketch a Graph of a Rational Function Factor the numerator and denominator Note that a factor common to numerator and denominator indicates where there is a hole in the graph Find x and 15intercepts and graph them Find any vertical asymptotes and graph them with dotted lines Find the horizontaloblique asymptote and graph it with a dotted line Make a table of values including test points to determine behavior near asymptotes and plot additional points as needed to determine the rest of the graph Elk59 Se ctio 36 2 Examg le 1 Sketchthe g raphz x EXam p iI Qf Sketchthe graph r c 4 Section 36 M 23 392 Example3 Sketchthe graph rx x x x Section 18 Math 150 Lecture Notes The Cartesian Coordinate System The Coordinate Plane The Cartesian Coordinate System is formed by two perpendicular number lines that intersect at 0 on each line The horizontal axis is the x axis the vertical axis is the y axis The point of intersection is the origin The two axes divide the plan into four quadrants Any point P in the coordinate plane can be described by a unique ordered pair of numbers a b for which a is the x coordinate or abscissa and b is the y coordinate or ordinate Distance and Midpoint The distance between two pointsAx1y1 and Bxz yz is dA B x2 x12 y2 yl2 The midpoint of the line segment from Ax1 yl to Bxz yz is Graphs of Equations in Two Variables An equation in two variables expresses a relationship between two quantities A point x y satis es the equation if it makes the equation true when the values for x and y are substituted into the equation The graph of an equation in x and y is the set of all points x y in the coordinate plane that satisfy the equation Intercepts The x intercept is the point where the graph of an equation intersects the xaxis The y intercept is the point where the graph of an equation intersects the yaxis Section 18 Circles A circle is the set of all points Px y in a plane that are the same distance r from a given point Ch k which is the center of the circle The distance r is the radius of the circle The standard form for the equation of the circle with center h k and radius r is xihzyik2 r2 If the center of the circle is the origin 0 0 then the equation is x2 y2 r2 Smmetgy When a graph is symmetric with respect to the x axis it is a mirror image of itself across the x aXis when y is replaced with 7y in the equation the result after simpli cation is the same equation For every point x y on the graph the point x y is on the graph When a graph is symmetric with respect to the y axis it is a mirror image of itself across the y aXis when x is replaced by 7x in the equation the result after simpli cation is the same equation For every point x y on the graph the point x y is on the graph When a graph is symmetric with respect to the origin it gives the same graph when it is rotated 180 about the origin or when it is re ected across the xaXis then the yaXis or the y aXis then the xaXis When x is replaced by 7x w y is replaced by 7y the resulting equation is the same after simpli cation For every point x y on the graph the point x y is on the graph Section 65 Math 150 Lecture Notes The Law of Cosines The Law of Cosines In any triangle ABC we have a2 b2 c2 72bc cosA b2a2 czi2ac cosB c2azbzi2ab cosC Since an angle and its supplement have the same sine knowing the sine of an angle does not uniquely specify the angle However every angle between 0 and 180 has a unique cosine Therefore when it is possible to use the Law of Cosines to solve a triangle ambiguity is not present A hearing is a navigation direction that is an acute angle measure from due north or due south Heron s Formula The area oftriangle ABC is given by AA 1ss as bs c where s is the semiperimeter s 12 a b c ofthe triangle Example 1 Solve triangle ABC given that a 52 b 66 and A C 55 Example 2 Solve triangle ABC given that a 20 b 25 and c 35 Section 65 Example 3 Find the area ofthe triangle given that a 12 b 25 and c 14 Example 4 Two tugboats that are 130 ft apart pull a barge Ifthe length of one cable is 220 ft and the length ofthe other is 250 ft nd the angle formed by the two cables Semnll Math 150 Lecture Notes Average Rate of Change fls increasing on an interval 1 Mn gm Whenever x1lt X2 in 1 fls decreasing on an interval 1if x1gt x2wheneverx1 lt mi 1 Examgle 1 Deteimme the intervals on which the function is increasing and decreasing USGS USGS 00110500 Navasota Rv nr Easterly TX A A A A A A A A A A A A A A a a L N e e u N A q n a n m E g u m E 5m any 10 any 2n any 22 any 24 any 25 any 23 any an Provisional nate Subject te Revision 1 Nedian daily statistic lt47 unars X neasured discharge Discharge Soume haghwmerdmausy yv Section 23 The average rate of change of the function y x between x a and x b is Average rate of change m M change 1n x b a The average rate of change is the slope of the secant line between x a and x b 0n the graph of f that is the line that passes through afa and bfb Example 2 Determine the average rate of change of the given function between the given values of the variable 2 x7 xl xl6 J Example 3 Determine the average rate of change of the given function between the given values of the variable Section 25 Math 150 Lecture Notes Maxima and Minima A maximum of minimum value of a function is the largest or smallest value of the function an interval A quadratic function is a function f of the form x axz bx c where a b and c are real numbers and a 7i 0 The graph of any quadratic function is a parabola A quadratic function x axz bx c can be expressed in the standard form x ax 7 h2 k by completing the square The graph of f is a parabola with veltex h k the parabola opens upward if a gt 0 or downward if a lt 0 Let f be a quadratic function with standard form x ax 7 h2 k The maximum or minimum value offoccurs at x h If a gt 0 then the minimum value of f is h k If a lt 0 then the maximum value of f is h k Example 1 Express the given quadratic function in standard form Find the vertex and x and yintercepts Sketch the graph x 3x2 6x 2 Example 2 Find the vertex of the general quadratic function x ax2 bx c Section 25 2 b The max1mum or m1n1mum value of a quadrat1c functlon x ax bx 0 occurs at x 7 a Example 3 Find the local maximum and minimum values of the function and the value of x at which each occurs State the answer correct to two decimal places fx xxx x2 Example 4 Find the local maximum and minimum values of the function and the value of x at which each occurs State the answer correct to two decimal places x g x2xl Secumm mmugnna Math 150 Lecture Notes Introduction to Conic Sections Parabola cuttan plane parallel to side ofco e Ctrcle and Elllpse Hyperbclas ht enwiki ediaor wikiConic section ht mathworldwolframcomConicSectiunhtml Seam m4 mg m The general equation for a com semen 15 Az 23 nyDxEyF 0 Resultm Eqnznnn Ax2Ay2 DxEyF 0 ACD0 By2EyF0 O r 35E Ax2DxF0 Ax28y2nyDxEyF0 bothposmve andAB Aanchave sz8y2nyDxEyF0 dAfferent sxgns L m dArecmx A 721112 rm 5 a constant mm mathworld Wolfram comEllipse html Sechnn NH thmngh ma A L A L points F1 and F2 is a constant These two xed points are the foci ofthe hyperbola hmenw1kigediamywikiHmerbola Example 1 Find the vertices and graph the conic a 12x o 2 2 7 Example 2 Find the vertices and graph the conic 9 V 3913K 16y 43 0 Section 101 through 103 Example 3 Find the vertices and graph the conic 9x2 4y2 7 18x 16y7 11 0 Practice Problems httpwww 39 39 39 39 nrg quqonq 19 on ncnYV leAlgebra conics circle le httpwww 39 39 39 org quqonq 19 on ncnyV leAlgebra conics ellime le httpWWW 39 39 39 39 m9 le om 16 on aQnYV leAlgebra conics hvnerbolaxml Section 12 Math 150 Lecture Notes Exponents and Radicals Integer Exponents Ifa is any real number and n is a positive integer then the nth power ofa is aquot a a a 7 factors of a The number a is the base and n is the exponent i a If a 7i 0 is any real number and n is a positive integer then a0 l and a39quot Laws of Exponents 1 am an amn 2 an 3 amquot amquot 4 A Q w V M Q w Scienti c Notation A positive number x is said to be written in scienti c notation if it is expressed as follows x a X 10quot Where Is a lt10 71 is an integer Radicals fb iff 1326 and 1320 If n is any positive integer then the principal nth root of a is de ned as Mb means bquota Section 12 Properties of nth Roots 1 W ZKE a quota 2 n7 3 MawZ 4 Waifnisodd 5 W aiifniseven Rational Exponents For any rat10na1 exponent 7 1n lowest terms where m and n are 1ntegers and n gt 0 n Rationalizing the Denominator A useful procedure that involves rewriting an expression without a radical in the denominator by multiplying both numerator and denominator by an appropriate expression is called rationalizing the denominator It is sometimes useful instead to rationalize the numerator Section 16 Math 150 Lecture Notes Applications Involving Equations Application Solving Process Identify the variables Express all unknown quantities in terms of the variables Set up an equation or model that expresses a relationship Solve the equation and check the answer 59 Example 1 A pot contains 6 gallons of brine salt water at a concentration of 120 ounces per gallon How much of the water should be boiled off to increase the concentration to 200 ounces per gallon Example 2 John Jerry and Sue have been hired to paint sheds Working together they can paint a shed in 40 of the time it takes Jerry alone It takes John 4 hours to paint a shed alone and it takes Sue one hour longer than it takes Jerry How long does it take Jerry to paint a shed alone Example 3 Keisha drove from San Bueno to Junction a distance of 250 miles She increased her speed by 10 miles per hour for the trip from Junction to Tuno a distance of 360 miles If the total trip took 11 hours what was her speed from San Bueno to Junction Section 21 Math 150 Lecture Notes In oduc ontoFunc ons The term function is used to describe a dependence of one quantity on another A function f is a rule that assigns to each element x in setA exactly one element called x in a set B The setA is called the domain of the function SetB is the range of the function The symbol that represents a number in the domain of a function is called an independent variable The symbol that represents a number in the range is called a dependent variable Representations of functional relationship Function Machine x x input output f When the domain of a function is not explicitly stated then by convention the domain of the function is the set of all real numbers for which the related expression is de ned Functions may be represented or described in four ways verbally algebraically graphically numerically e333 Section 21 Example 1 Evaluate the piecewise de ned function at the indicated values 3x if x lt 0 x xl z39fOSxSZ x 22 x gt 2 39s Example 2 Use the function to evaluate the indicated expression and simplify rock xl a2 fah fahfa h where h 0 4 Example 3 Find the domain of the function f x 2x 6 x x Example 4 Find the domain of the function gx v x2 2x 8 Section63 Math 150 Lecture Notes Trigonometric Functions of Angles Let 9 be an angle in standard position and let Px y be a point on the terminal side If r x2 y2 is the distance from the origin to the point Px y then sin6Z cos6E tan6Zx 0 r r x csc6y 0 sec6 x 0 cot6Zy 0 y x r Quadrantal angles are angles that are coterminal with the coordinate axes Let 9 be an angle in standard position The reference angle 6 associated with 9 is the acute angle formed by the terminal side of 9 and the xaXis To evaluate a trig function for an angle 1 Find the reference angle 6 for the angle 9 Determine the sign of 9 by guring the quadrant in which 9 lies E The absolute value of the trig function of 9 is the same as 6 so you must attach the sign from step 2 Reciprocal Identities csc6 sec6 cot67 s1n6 cos6 tan6 sin 6 cos 6 cot 6 cos 6 sm 6 tan 6 Even Odd Identities Sine cosecant tangent and cotangent are odd functions cosine and secant and even functions sin 6 sin 6 cos 6 cos 6 tan 6 tan 6 csc 6 csc 6 sec 6 sec 6 cot 6 cot 6 Pythagorean Identities sin2 6 cos2 61 tan2 6 1 sec2 6 cot2 6 1 csc2 6 The area of a triangle with sides of lengths a and b and with included anglee is A 12 ab sin 9 Section 63 Example 1 Find the reference angle for the following 60 7700 237239 567239 Example 2 Find the exact value of the following trig functions 627239 tan 135 sin 660 cot sec 3 Example 3 Find the quadrant in which 9 lies given that cot 9 lt 0 and csc 9 lt 0 Example 4 Find the values of the trig functions of 9 given that csc 9 5 and cos 9 lt 0 Example 5 The time in seconds that it takes for a sled to slide down a hillside inclined at an d angle 9 is t where d is the length of the slope in feet Find the time it takes to slide s1n down a 2500f00t slope inclined at 45 Section 11 Math 150 Lecture Notes Real Numbers The natural or counting numbers are l 2 3 4 The whole numbers consist of the natural numbers and 0 01 2 3 4 The integers consist of the natural numbers together with the negatives and 0 4 3 2 l 01 2 3 4 The rational numbers are numbers that can be written as an integer divided by an integer or a ratio of integers Examples 12 019 427 31 The irrational numbers are numbers that cannot be written as an integer divided by an integer Examples J3 11 3 5 6 Properties of Real Numbers Commutative Property for Addition a b b a for Multiplication ab ba Associative Property for Addition a b c a b c for Multiplication abc abc Distributive Property abcabac or bcaabac Additive Identity a00aa Subtraction is the inverse operation for addition undoes addition and is the same as adding the negative of the number to be subtracted a 7 b a b Properties of Negatives la a 7a a 61b 6143 ab ab ab 7a b a 7 b 7a 7 b b 7 a 9959 seem H Multiplicative Identity 1 1 1 a a mm multiplying by me reciprocal a b 7 Properties of Fractions 1 c 1 7 b 2 b 3 5 c 4 5 b E be 51f5imenadbc b d TheRealLine n L we ordered Geomeuically in lt bd1en 1 lies to the le OH on the number line 72 cs 18 In 3 2 31 n s I 2 2 5 1 3 Sets and Intervals A set is a collection ofobject called elements ome set Nomequot a e Smeans a is an element ofset s 2 means is not an element ofquot betweenO and 5 nintersection uunion gsubset Cpropersubset qnotasubset Interval Notation abxaltxltb abxa x b eoin nity Section 11 Absolute Value and Distance The absolute value of a number is the distance from the number to 0 on the number line H a z39faZO a a z39falt0 Properties of Absolute Value 1 MEG 2 aa 3 abaHb EM b W If a and b are real numbers then the distance between the points a and b on the real line is da b b 7 61 Is a b b a Why or why not Section 27 Math 150 Lecture Notes Composition of Functions Let f and g be functions with domains A and B Then the functions f g f 7 g fg and g are de ned as follows f gx x gx Domain A n B 2 gx x gx Domain A n B ngx xgx Domain A n B rikx fx Domain x e A MB l gx 0 Kg J gx Given two function f and g the composite function f o g also called the composition of f and g is de ned by f 0 gx f gx The domain of f o g is the set of all x in the domain of g such that gx is in the domain off fog 11200 Example 1 Findf gf7 g fg and g and their domains fx M 4 gx x2 Section 27 Example 2 For x 3x 7 5 and gx l 7 x2 evaluate the following gg3 g 0 f 3 f 0 g3 g 0 f x Example 3 Find the functions fo g g o f f o f g o g and fo g o h and their domains 1 x xi 5 gee xE hoe x3 2 7 intheform fogoh 35Z Example 4 Express the function F x Section 15 Math 150 Lecture Notes Solving Equations An equation is a statement that two mathematical expressions are equal Solving an equation is the process of nding the solutions or roots the values for the variables that make the equation true Equivalent equations are equations with exactly the same solutions Properties of Equality 1 AB ACBC 2 ABCgtCACB C 0 Linear Eg uations A linear equation in one variable is an equation equivalent to one of the form ax b 0 where a and b are real numbers and x is the variable 5 Quadratic Eg uations A quadratic equation is an equation of the form ax2 bx c 0 where a b and c are real numbers with a 7i 0 Zero Product Property 1430 ifandonlyif 140 orB0 The solutions ofthe equation x2 c are x xE and x Completing the Square is a term for a process that puts the quadratic equation in the form above so that the left side of the equation is a perfect square The Quadratic Formula The roots of the quadratic equation axz bx c 0 where a 7i 0 are bib2 4ac 2a The discriminant of the general quadratic ax2 bx c 0 a 7i 0 is D b2 7 4ac 1 If D gt 0 then the equation has two distinct real solutions 2 If D 0 then the equation has exactly one real solution 3 If D lt 0 then the equation has no real solution Section 15 Other Types of Equations When we use a process that does not produce equivalent equations to solve an equation such as one involving a radical where we square both sides of the equation we may have one or more extraneous solutions which are solutions of the resulting equation but not of the original equation In these cases checking the answers in the original equation is not solely for the purpose of nding mistakes in our work but a necessary part of the process to eliminate extraneous solutions An equation of quadratic type is one of the form aW2 bW c 0 where Wis an algebraic expression 4 2 35 Example 1 7 7 x l xl xZ l nn 1 Example 2 Solve for n S 2 Example 3 2x2 6x3 0 Example 4 x 5x5 Example 5 3W 40 Section 28 Math 150 Lecture Notes Inverses and 11 Functions The inverse of a function is a rule that acts on the output of the function and produces the corresponding input It undoes or reverses what the function did Not all inverses of functions are themselves functions A function with domainA is called a one to one function if xl i xz whenever x1 at x Horizontal Line Test A function is onetoone iff no horizontal line intersects its graph more than once If f is a onetoone function with domainA and range B then its inverse function f 391 has domain B and range A and is de ned byf1y x c x y Inverse Function Property Let f be a onetoone function with domainA and range B The inverse function f 391 satis es the following properties f391fx x for every x inA f391x x for every x in B Conversely any function f 391 satisfying these equations is the inverse off The graph off1 can be obtained by re ecting the graph of f across the line y x Example 1 Determine whether the function is onetoone fx x4 5 0 S x S 2 Example 2 Assume f is a onetoone function If f 44 2 f1ndf2 Example 3 If x x 4x with x Z 2 f1ndf3915 Section 28 Example 4 Find the inverse function of x 2x l Example 5 Draw the graph of f and use it to determine whether the function is onetoone x x39lxl Example 6 The given function is not onetoone Restrict its domain so that the resulting function is onetoone Find the inverse of the function with the restricted domain Fx27x2 Section 62 Math 150 Lecture Notes Trigonometry of Right Triangles The Trigonometric Ratios sim9M mazm tamQZoppine hYp Otenuse hypotenuse ad ac em cscghyLeIiuse sec w cot 9ach nt Opposlte ad acent Opposne Special Right Triangles 30 60 90 Triangle 60 2 600 1 1 1 2 30 300 I I 43 2 45 45 90 Triangle 450 450 1 45 1 7 45 450 1 Q 2 To solve a triangle means to determine all the side lengths and all the angle measures When an observer is looking at an object the line from the eye to the object is called the line of sight The angle of elevation is the angle from the horizontal up toward an object The angle of depression is the angle down from the horizontal The angle of inclination refers to an angle up an inclined plane Section 62 Example 1 Find the exact values of the six trig ratios of the angle 9 in the triangle A 40 Example 2 Find the value of x D 24 Example 3 Sketch a triangle with acute angle 9 where sin 9 and nd the other ve trig ratios of 9 Example 4 Solve the right triangle Section 64 Math 150 Lecture Notes The Law of Sines Recall that the following properties are triangle congruences ie there is exactly one possible triangle that can be drawn with the given information SSS SAS ASA AAS An AAA congruence does not produce triangle congruence although the triangles are similar A SSA congruence ASS does not produce a triangle congruence since there are sometimes two triangles that meet the given criteria To solve atriangle given ASA AAS or SSA use the Law of Sines To solve atriangle given SAS or SSS use the Law of Cosines often followed by the Law of Sines Law of Sines sinA sinB sinC b c In triangle ABC When two sides and an angle not between them SSA is known there may or may not be a unique triangle This situation is often called the ambiguous case I If A is acute and a lt b a If bsinA lt a lt b then 2 A s b If bsinA a then 1 A c If bsinA gt a then 0 A s II If A A is acute and a gt b then 1 A 111 If A A is obtuse a Ifagtbthen1A b IfaSbthen0A Example 1 Sketch the triangle and then solve it using Law of Sines 4A23 4B100 650 Section 64 Example 2 Sketch the triangle and then solve it using Law of Sines 4B3l 4C50 b45 Example 3 Sketch the triangle and then solve it using Law of Sines 6150 b60 4A35 Section 84 Math 150 Lecture Notes Introduction to Vectors Quantities that are determined only by magnitude ie length mass temperature area are called scalars A vector is a line segment with magnitude and an assigned direction An arrow is used to specify the direction Vector AB has initial pointA and terminal point B The magnitude or length of the vector is the length of the segment AB and is denoted by Two vectors are equal if they have equal magnitude and the same direction Vector A C is the sum of vectors EB and B C when it is the displacement u A B followed by the displacement v BC Multiplication of a Vector by a Scalar If a is a real number and v is a vector then av is a vector of magnitude lal lvl and has the same direction as v ifa gt 0 or the opposite direction as v if a lt 0 The difference of two vectors u and v is de ned by u 7 v u v In the coordinate plane a vector v can be represented as an ordered pair of real numbers v a b where a is the horizontal component of v and b is the vertical component of v Component Form of a Vector If a vector v is represented in the plan with initial pointP x1 y1 and terminal point Q x2 yz then v x2 7x1 yz y1gt Two vectors are equal iii their corresponding components are equal The magnitude or length ofa vector v a b is lvl x a2 b2 SecmnRe Algebraic Opemu39uns an Vecturs Ifu lta1 b1gt and v lta2 b2gt then u v a ltat Wr Cu ltca1cb1gt 6 e9 Pmpenies nf Vecturs Vector A ddrtror Muluphcauon by a Scalar u vu cuv cucv uvwuvw cducudu u 0 u mu com dcu lu u u V 0 Length of a Vector Cut 6 M 60 Vecmrs in Terms an andj The vectorv lta bgt can be expressed m terms ofi andj byv lta bgtai bj Hnriznnlzl and Vertical Cnmpunents Hf z Vectnr Letv be avectorwrth magmude M and drrectror 9 Then v a bgtai bj where a v c056 and b sme H M We can express v as v M cos ei M sm 9 Examglel sketch u Zv usmg vectors u and v m the gure Secnan 274 c Examplel V ector 39 39itialpoint 5 Example 3 Findu 72v and 3 4v for vectors u lt2 5 andv lt31gt Example 4 Find w m w v m and u v for vectors u 3i 2j andv 2ij Example 39 39 39 direction and write the vector in terms of the vectors i andj u60 e1zo Section 84 Example 6 Find the magnitude and direction in degrees of the vector u 5 12 Example 7 A river ows due south at 4 miles per hour An alligator heads due east swimming at 3 miles per hour relative to the water Find the true velocity of the alligator as a vector httn39 hvnerte 1l39mnk h U r 39 quot quotaddition 10 39 n1 t 4 ndf h n39 hV Pm thonk h Ic 39 quot v t additi0nW0r39 39 quot quot ndf httpthe P mnk ph I y r 39 quot v t additi0nw0rlltsheetw0rdspdf Section 75 Math 150 Lecture Notes Trigonometric Equations To solve a trigonometric equation 1 Use rules of algebra to isolate the trig function on one side of the equal sign 2 Use knowledge of the values of the trig functions to solve for the variable Example 1 Find all solutions of each equation sinxl0 litanzx0 cos3xsin3x tansx9tanx 4sinxcosx2sinx72cosxl Example 2 Find all solutions of each equation in the interval 0 211 tanx3cotx 25in2xcosxl 3csc2x4 Section 75 Example 3 a Find all solutions of the equation b Use a calculator to solve the equation in the interval 0 211 correct to four decimal places 3tanx15 3sin2xl 2sin2xc0sx Example 4 Use an addition or subtraction formula to simplify the equation Then nd all solutions in the interval 0 211 cos x cos 2x sinx sin 2x 12 Example 5 Use a double or halfangle formula to solve the equation in the interval 0 211 x tani s1n x 2 Section 54 Math 150 Lecture Notes More Trigonometric Graphs The tangent and cotangent functions have period 11 tan x1ttanx cot x1tcotx The cosecant and secant functions have period 211 csc x 211 csc x sec x 211 sec x The functions y a tan kx and y a cot kx k gt 0 have period The functions y a csc kx and y a sec kx k gt 0 have period 27 Example 1 Find the period and graph the function fx 3 sec x Example 2 Find the period and sketch the graph of gx 3 csc x Section 54 Example 3 Find the period and graph the function hx tan x E 4 Section 73 Math 150 Lecture Notes DoubleAngle HalfAngle and ProductSum Formulas Double Angle Formulas sin2x2sinxcosx cos 2x cos2 x 7 sin2 x Half Angle Formulas 1 Sin 7 i cos x 2 172 sinzx 2 cos2x71 2tanx x 1 cosx sinx tan2x 2 tan7 7 1 tan x 2 s1nx Lowering Powers 2 1 cos2x s1n x 2 Product to Sum Formulas 1cosx Determination of or 7 sign depends on quadrant of i 2 1cos2x 2 1 cos2x cos x tan x 2 1cos2x sin u cos v 12 sinu v sinu 7 v cos u sin v 12 sinu v 7 sinu 7 v cos u cos v 12 cosu v cosu 7 v sin u sin v 12 cosu 7 Sum to Product Formulas x s1nxs1ny2s1n y cos 2 x cosxcosy2cos y cos 2 x v 7 cosu v y x x s1nxs1ny2cos y sin y 2 x 2 x x cosxcosy2 s1n y y sm 2 Section 73 4 Example 1 Find sm 2x cos 2x and tan 2x g1ven that tan x 3 and x 1s 1n quadrant IV Example 2 Use the formulas for lowering powers to rewrite the expression in terms of the rst power of cosine cos4 x sin2 x Example 3 Use an appropriate halfangle formula to nd the exact value of the expression tan 7 12 Example 4 Simplify the expression by using a doubleangle formula or a halfangle formula cos2 50c sin2 50c Section 71 Math 150 Lecture Notes Trigonometric Identities Fundamental Trigonometric Identities Reciprocal Identities CSCX sin x cos x tan x Even Odd Identities Sine cosecant tangent and cotangent are odd functions cosine and secant and even functions sin x sin x cos x cos x tan x tan x csc x csc x sec x sec x cot x cot x Pythagorean Identities 2 2 2 2 2 2 s1n xcos xl tan xlsecx cot xlcscx Cofunction Identities 7239 7239 7239 s1n i x cosx tan i x cotx sec i x cscx 2 2 2 7239 7239 7239 cos i x s1nx cot i x tanx csc i x secx 2 2 2 To Prove a Trigonometric Identity Start with one side and try to transform it in to the other side using identities It is usually easier to start with the more complicated side Use algebra and trig identities to change the side you started with Rewrite fractions with equivalent fractions using a common denominator Use factoring and trig identities to simplify expressions When all else fails try rewrite all functions in terms of sine and cosine Note You may work backwards from the other side but you cannot leave the proof in that form Use that information only to help you gure out how to think about the proof N E Section 71 Example 1 Simplify the trig expression seer an x W Simplify the trig expression tan x cos x csc x l sin x 2 Example 3 Prove the 1dent1ty 7 sec x tan x l s1n x tan xsinx tanx sinx Example 4 Prove the 1dent1ty tanxs1nx tanxsmx Section 71 tan x cot x Example 5 Prove the 1dent1ty sm xcos x tan x cot x SecuanAJ Math 150 Lecture Notes Exponential Functions Em L 39 an Icnl X where a gt 0 and a 1 The exponential function x d d gt 0 d 1 has domain oo no and range 0 no The line y 0 the xaxis is a horizontal asymptote off The graph of fhas one ofthe following hapes 39 39 39 lcwithbasec Itiso en referredto as the exponential function Compound interest is calculated by the formula At P1 L where n At amount a er 2 years P principal r interest rate peryear number of tim es interest is compounded per year 2 number ofyears Continuously compounded interest is calculated by the formula 42 P2quot where At amount a er 2 years P 39nterest rate peryear 2 number ofyears Example 1 Use a calculator to evaluate the function x 2 1 at the values 07 r a ln 23 Round your answer to three decimal places semm Examgle 2 End the exponential functionx m whose graph 5 gwen Examgle 3 wmh othe followmg s thefuncnon graphed below7 x 5x x 75 x 5x 3 x 5quot x 5 Section 41 Example 4 Graph the function using transformations of the parent function y e 3 4 State the domain range and asymptote Example 5 Assume that a population of rabbits behaves according to the logistic growth model 300 quot0 005 0050 55 n where no is the initial rabbit population 0 a If the initial population is 50 rabbits what will the population be after 12 years b Draw graphs of the function nt for no 50 500 2000 8000 and 12000 in the Viewing rectangle 0 15 by 0 12000 c From the graphs in part b observe that regardless of the initial population the rabbit population seems to approach a certain number as time goes on What is that number This is the number of rabbits that the island can support Sectioner Math 150 Lecture Notes Graphs of Functions lffis afunction with domain A then the graph ofIis the set ofordered pairs Xv x X EA In other words the graph of fis the set of all points a y such that y a A linear luncu39on is afunctionfofthe form x mx b which represents aline with slope m A 39t t w L 39 39 s thefunction 39 39 andits r r graph is the horizontal line y h one my to graph afunction is to make atable ofvalues plot the points and join them with a smooth curve I I uu c u u u u yVa1ues Th ofthe function To graph apiecewise lunchon sketch the graph ofeach piece with adotted line Then use a solid tar r mumuni n in on u u 39 ie than or equal to a an Vertical Line Test A curve in the coordinate plane is the graph ola luncu39on iffno vertical line intersects the curve more than once Example 1 Determine whether the curve is the gmph ofafunction ofx Ifit is state the domain and ranve ofthe function A 74 Example 2 Determine whether the equation de nes y as afunction ofx 5 y 6 Example 3 Determine whether the equation de nes y as afunction ofa 2a M 0 Section 72 Math 150 Lecture Notes Addition and Subtraction Formulas Addition and Subtraction Formulas sin A B sinA cosB cosA sinB sin A 73 sinA cosB icosA sinB cos A B cosA cosB isinA sinB cos A 73 cosA cosB sinA sinB tan A 3 tanAtanB 1 tanAtanB Hum173 tanA tanB 1tanAtanB Sums 0f Sines and Cosines Ifa and b are real numbers then a sinA b cosA a2 b2 sin x and satis es a b xa2 b2 cos and sin v a2 b2 Example 1 Use an addition or subtraction formula to nd the exact value of each expression cos 75 tan 195 cos ml sin Example 2 Use an addition or subtraction formula to write the expression as a trigonometric function of one number Find the exact value sin 10 cos 80 cos 10 sin 80 cos 137 cos sin 137 sin 15 5 15 5 Section 72 mph 3 Prove the identity tan x E L9H 4 tan x 1 Example 4 Express the function in terms of sine only Graph the function Fx cosxsinx
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