1 Review y x The Distance Formula Midpoint Formula 1

Transcription

1 1 Review y x The Distance Formula Midpoint Formula 1

2 Note. The x and y intercepts are coordinates, not a single real number. Ex: (a, 0) and (0, b) where a, b R. Definition. An equation of the circle with center (h, k) and radius r is (x h) 2 + (y k) 2 = r 2 This is called the standard form for the equation of the circle. Example 1. Find the equation of the circle with given endpoints P ( 1, 3) and Q(7, 5) Example 2. Show that the equation represents a circle, find the center and radius of the circle. x 2 + y 2 + 6y + 2 = 0 2

3 Definition. Symmetry With respect to the x axis With respect to the y axis With respect to the origin Example 3. Test the equation for symmetry. (a) x = y 4 y 2 (b) x 2 + y 2 = 1 3

4 ( ) 1 Example 4. Find the equation of the line that passes through 2, 0 line 4x 8y = 1. and is perpendicular to the Definition. The standard form of the equation of a line is where A and B are not both zero and A, B R. Ax + By + C = 0, Note. Look at the proof of The Quadratic Formula on pg 118. Definition. The discriminant of the quadratic equation ax 2 + bx + c = 0 (a 0) is D = b 2 4ac. 1. If D > 0, then the equation has two distinct real solutions. 2. If D = 0, then the equation has exactly one real solution. 3. If D < 0, then the equation has no real solution. Example 5. A ball is thrown straight upward at an initial speed of 40 ft/s, its height is expressed by the given formula h = 16t t. When does the ball reach a height of 48 ft? 4

5 Definition. A complex number is an expression of the form a + ib where a, b R and i 2 = 1. It can also be denoted as z = a + ib and we define its complex conjugate to z = a ib. Example 6. Let z = 3 2i, find z. Definition. If r is negative, then the principal square root of r is r = i r. The two square roots of r are i r and i r. Example 7. Evaluate the radical expression and express the result in the form a + ib. ( 3 4)( 6 8) Example 8. Find all real solutions to the following equations. ( ) 2 ( ) x + 1 x + 1 (a) = 0 x x 5

6 6 (b) x x 1/2 6x 3/2 = 0

7 Recall: x = { x x 0 x x < 0 Note. When using the definition, you should always check your solutions. Example 9. Solve 5 2x + 6 = 14. Example 10. Solve 2 x > 3. 7

8 Definition. Direct Variation If the quantities x and y are related by an equation y = kx for some constant k 0, we say that y varies directly as x, or y is directly proportional to x, or simply y is proportional to x. The constant k is called the constant of proportionality. Definition. Inverse Variation If the quantities x and y are related by the equation y = k x for some constant k 0, we say that y is inversely proportional to x or y varies inversely as x. The constant k is called the constant of proportionality. Now, Combining Different Types of Variation. For example: Definition. If the quantities x, y, and z are related by the equation z = kxy then we say that z is proportional to the product of x and y. We can also express this relationship by that z varies jointly as x and y or that z is jointly proportional to x and y. If the quantities x, y and z are related by the equation z = k x y we say that z is proportional to x and inversely proportional to y or that z varies directly as x and inversely as y. Example 11. In the short growing season of the Canadian arctic territory of Nunavut, some gardeners find it possible to grow gigantic cabbages in the midnight sun. Assume that the final size of a cabbage is proportional to the amount of nutrients it receives and inversely proportional to the number of other cabbages surrounding it. A cabbage that received 20 oz of nutrients and has 12 other cabbages around it grew to 30 lb. What size would it grow to if it received 10 oz of nutrients and had only 5 cabbage neighbors? 8

9 2 Functions Graphs of the functions f and g are given. 1. Which is larger, f(6) or g(6)? 2. Find the values of x for which f(x) = g(x)? 3. Find the values of x for which f(x) g(x)? 4. Find the values of x for which f(x) > g(x)? 2.1 Functions Definition. A function f is a rule that assigns to each element x in a set A exactly one element, called f(x), in a set B. Example 12. Evaluate the function at the indicated values: h(t) = t + 1 ( 1 t ; h( 1), h ( ) 2 1 h. x ), h(t 1), 9

10 Definition. The net change in the value of a function f as the input changes from a to b (where a b) is given by f(b) f(a) Example 13. Find the net change in the value of the function between the given inputs. f(x) = 4 5x; from 3 to 5. Example 14. Find f(a + h) f(a), where h 0, given f(x) = 3x h 10

11 Example 15. Find the domain of the function g(x) = x 2x 2 + x 1. Example 16. The population P (in thousands) of San Jose, California, from 1980 to 2010 is shown in the table (midyear estimates are given). Draw a rough graph of P as a function of time t. t P

12 2.2 Graphs of Functions Definition. If f is a function with domain A, then the graph of f is the set of ordered pairs {(x, f(x)) x A} plotted in a coordinate plane. Note. A piecewise defined function is defined by different formulas on different parts of its domain. x if x 0 Example 17. Sketch a graph of the piecewise defined function f(x) = 9 x 2 if 0 < x 3 x 3 if x > 3. 12

13 Definition. The greatest integer function is defined by x = greatest integer less than or equal to x. Example 18. A taxi company charges $2.00 for the first mile (or part of a mile) and 20 cents for each succeeding tenth of a mile (or part). Express the cost C (in dollars) of a ride as a piecewise defined function of the distance x traveled (in miles) for 0 < x < 2, and sketch the graph of this function. Definition. A curve in the coordinate plane is the graph of a function if and only if no vertical line intersects the curve more than once. This is also known as the Vertical Line Test. Note. Look at pg 202 in your book to recognize the different kind of functions and their graphs. 13

14 2.3 Getting Information from the Graph of a Function Definition. The domain and range of a function y = f(x) can be obtained from a graph of f. The domain is the set of all x-values for which f is defined, and the range is all the corresponding y-values. Example 19. Find the domain and range of g. Definition. The solution(s) of the equation f(x) = g(x) are the values of x where the graphs of f and g intersect. The solution(s) of the inequality f(x) < g(x) are the values of x where the graph of g is higher than the graph of f. Graphs of the functions f and g are given. 1. Which is larger, f(6) or g(6)? 2. Find the values of x for which f(x) = g(x)? 3. Find the values of x for which f(x) g(x)? 4. Find the values of x for which f(x) > g(x)? 14

15 Definition. Increasing and Decreasing Functions f is increasing on an interval I if f(x 1 ) < f(x 2 ) whenever x 1 < x 2 in I. f is decreasing on an interval I if f(x 1 ) > f(x 2 ) whenever x 1 < x 2 in I. f is increasing f is decreasing Definition. Local Maxima and Minima of a Function The function value f(a) is a local maximum value of f if f(a) f(x) when x is near a. In this case we say that f has local maximum at x = a. The function value f(a) is a local minimum value of f if f(a) f(x) when x is near a. In this case we say that f has local minimum at x = a. 15

16 Example 20. The graph shows the population P in a small industrial city from 1950 to The variable x represents the number of years since (a) Determine the intervals on which the function P is increasing and on which it is decreasing. (b) What was the maximum population, and in what year was it attained? (c) Find the net change in the population P from 1970 to

17 2.4 Average Rate of Change of a Function Definition. Average Rate of Change The average rate of change of the function y = f(x) between x = a and x = b is average rate of change = change in y change in x = y x f(b) f(a) =. b a The average rate of change is the slope of the secant line between x = a and x = b on the graph f, that is, the line that passes through (a, f(a)) and (b, f(b)). Example 21. The graph shows the population P in a small industrial city from 1950 to The variable x represents the number of years since What was the average rate of change of P between x = 20 and x = 40? 17

18 Example 22. A woman is running around a circular track that is 200 m in circumference. An observer uses a stopwatch to record the runner s time at the end of each lap, obtaining the data in the following table. Calculate the woman s speed for each lap. Is she slowing down, speeding up, or neither? Time (s) Distance (m)

19 2.5 Linear Functions and Models Definition. A linear function is a function of the form f(x) = ax + b; a, b R. Definition. For the linear function f(x) = ax + b, the slope of the graph of f and the rate of change of f are both equal to a, the coefficient of x. Example 23. Meilin and Brianna are avid mountain bikers. On a spring day they cycle down straight roads with steep grades. The graphs give a representation of the elevation of the road on which each of them cycles. Find the grade of each road. 19

20 Example 24. The manager of a furniture factory finds that it costs $2200 to produce 100 chairs in one day and $4800 to produce 300 chairs in one day. 1. Assuming that the relationship between cost and the number of chairs produced is linear, find a linear function C that models the cost of producing x chairs in one day. 2. Draw a graph of C. What is the slope of this line? 3. At what rate does the factory s cost increase for every additional chair produced? 20

21 2.6 Transformations of Functions Example 25. Sketch the graph of f(x) = 3 2(x 1) 2. 21

22 Definition. Let f be a function. f is even if f( x) = f(x), for all x in the domain of f. f is odd if f( x) = f(x), for all x in the domain of f. Example 26. Determine whether the function f in even, odd, or neither. (a) f(x) = x (b) g(x) = 3x 3 + 2x (c) h(x) = x + 1 x 22

23 2.7 Combining Functions Definition. Let f and g be functions with domain A and B. Then the functions f + g, f g, fg, and f/g are defined as follows: (f + g)(x) = f(x) + g(x) Domain A B (f g)(x) = f(x) g(x) Domain A B (fg)(x) = f(x)g(x) Domain A B ( ) f (x) = g f(x) g(x) Domain {x A B g(x) 0} Example 27. Find f + g, f g, fg, and f/g and their domains given f(x) = 16 x 2 g(x) = x 2 1. and 23

24 Example 28. Use graphical addition to sketch the graph of f + g. Definition. Given two functions f and g, the composite function f g (also called the composition of f and g) is defined by (f g)(x) = f(g(x)). Example 29. Use f(x) = 2x 3 and g(x) = 4 x 2 to evaluate the expression. (a) (f g)( 2) (b) (g g)( 1) (c) (g f)(x) 24

25 (d) (f f)(x) (e) (g g)(x) Example 30. Find the function g f and g g and their domains. (a) f(x) = x 2 ; g(x) = x 3 25

26 (b) f(x) = 2 x ; g(x) = x x

27 2.8 One-to-One Functions and their Inverses Definition. A function with domain A is called a one-to-one function if no two elements of A have the same image, that is, f(x 1 ) f(x 2 ) whenever x 1 x 2. Definition. A function is one-to-one if and only if no horizontal line intersects its graph more than once. Example 31. Determine whether the function is one-to-one. (a) f(x) = 3x 2 (b) g(x) = 1 x 2 (c) h(x) = 1 x 2 ; 0 x 1 27

28 Definition. Let f be a one-to-one function with domain A and range B. Then its inverse function f 1 has domain B and range A and is defined by for any y B. f 1 (y) = x f(x) = y Definition. Inverse Function Property Let f be a one-to-one function with domain A and range B. The inverse function f 1 satisfies the following cancellation properties: (f 1 f)(x) = x for every x A (f f 1 )(x) = x for every x B Conversely, any function f 1 satisfying these equations is the inverse of f. Example 32. Use the Inverse Function Property to show that f and g are inverses of each other. f(x) = 2 5x; g(x) = 2 x 5 Example 33. Find the inverse function of f, if it exist, f(x) = 3x

29 Example 34. Find the inverse function of f, if it exist, f(x) = x 2 x + 2. Example 35. Use the graph of f to sketch the graph of f 1. 29

30 3 Polynomial and Rational Functions 3.1 Quadratic Functions and Models Definition. A polynomial function of degree n is a function of the form P (x) = a n x n + a n 1 x n a 1 x + a 0 ; a 0. A quadratic function is a polynomial function of degree 2. function of the form So a quadratic function is a f(x) = ax 2 + bx + c, a 0. Definition. A quadratic function f(x) = ax 2 + bx + c can be expressed in the standard form f(x) = a(x h) 2 + k. Where the vertex is given by (h, k). a > 0 a < 0 Example 36. Express f in standard form, find the vertex, sketch the graph, minimum (or maximum value), domain and range. f(x) = 3x 2 + 6x 2. 30

31 Example 37. Find the minimum value of f(x) = ax 2 + bx + c where a 0, (assume a > 0). Example 38. Find the maximum (or minimum) value of g(x) = 2x 2 + 8x

32 Example 39. A rain gutter is formed by bending up the sides of a 30-inch wide rectangular metal sheet as shown in the figure. 1. Find a function that models the cross-sectional area of the gutter in terms of x. 2. Find the value of x that maximizes the cross-sectional area of the gutter. 3. What is the maximum cross-sectional area for the gutter? 32

33 3.2 Polynomial Functions and their graphs Definition. A polynomial function of degree n is a function of the form where n is a nonnegative integer and a n 0. P (x) = a n x n + a n 1 x n a 1 x + a 0 The numbers a 0, a 1,, a 2,..., a n are called the coefficients of the polynomial. The number a 0 is the constant coefficient or constant term. The number a n, the coefficient of the highest power, is the leading coefficient, and the term a n x n is the leading term. Note. The graph of a polynomial function is continuous. breaks, holes, corners, or sharp points (cusp). Thus, the graph does not have any Definition. End Behavior of Polynomials The end behavior of the polynomial P (x) = a n x n + a n 1 x n a 1 x + a 0 is determined by the degree n and the sign of the leading coefficient a n. 33

34 Example 40. Determine the end behavior of each polynomian without graphing it. (a) P (x) = x 3 (x + 2)(x 3) 3 (b) Q(x) = (x + 3)(x 1)x(4 x) Definition. Real Zeros of Polynomials If P is a polynomial and c R, then the following are equivalent: 1. c is a zero of P. 2. x = c is a solution of the equation P (x) = x c is a factor of P (x). 4. c is an x intercept of the graph P. Definition. Intermediate Value Theorem for Polynomials If P is a polynomial function and P (a) and P (b) have opposite signs, then there exists at least one value c between a and b for which P (c) = 0. 34

35 Example 41. Sketch the graph of Q(x) = (x + 3)(x 2)x(4 x). Example 42. Sketch the graph of P (x) = x 3 + 3x 2 4x

36 Example 43. Sketch the graph of R(x) = (x + 1)x 2 (x + 2) 3 Definition. If c is a zero of P of multiplicity m, then the shape of the graph of P near c is as follows. m is odd, m > 1 m is even, m > 1 Note. If P (x) = a n x n + a n 1 x n a 1 x + a 0 is a polynomial of degree n, then the graph of P has at most n 1 local extrema. 36

37 3.3 Dividing Polynomials Definition. If P (x) and D(x) are polynomials, with D(x) 0, then there exist unique polynomials Q(x) and R(x), where R(x) is either 0 or of degree less than the degree of D(x), such that P (x) D(x) = Q(x) + R(x) D(x) or P (x) = D(x) Q(x) + R(x). The polynomials P (x) and D(x) are called the dividend and divisor, respectively, Q(x) is the quotient, and R(x) is the remainder. Example 44. Find the quotient and remainder x3 + 3x 2 + 4x + 3 3x

38 Example 45. Find quotient and remainder 2x5 7x x 2 6x + 8. Note. Synthetic Division is a quick method of dividing polynomials, it can be used when the divisor is of the form x c. Example 46. Find the quotient and remainder x5 + 3x 3 6. x 1 38

39 Definition. Remainder Theorem If the polynomial P (x) is divided by x c, then the remainder is the value of P (c). Definition. Factor Theorem c is a zero of P if and only if x c is a factor of P (x). Example 47. Show that the given value(s) of c are zeros of P (x), and find all other zeros of P (x). P (x) = x 3 5x 2 2x + 10 given c = 5. 39

40 Example 48. Find a polynomial of the specified degree that satisfies the given condition. Degree 4; zeros 1, 0, 2, 1 2 ; coefficient of x3 is 3. 40

41 3.4 Real Zeros of Polynomials Theorem Rational Zeros Theorem If the polynomial P (x) = a n x n + a n 1 x n a 1 x + a 0 has integer coefficients (where a n 0 and a 0 0), then every rational zero of P is of the form p where p, q Z and q p is a factor of the constant coefficient a 0. q is a factor of the leading coefficient a n. Example 49. Find all the real zeros of the polynomial and write the polynomial in factored form. P (x) = 2x 4 7x 3 + 3x 2 + 8x 4 41

42 Example 50. Find all the real zeros of the polynomial and write the polynomial in factored form. P (x) = x 5 4x 4 x x 2 + 2x 4 42

43 3.5 Complex Zeros and the Fundamental Theorem of Algebra Theorem The Fundamental Theorem of Algebra Every polynomial P (x) = a n x n + a n 1 x n a 1 x + a 0 with complex coefficients has at least one complex zero. Note. Any real number is also a complex number. Theorem Complete Factorization Theorem If P (x) is a polynomial of degree n 1, then there exist complex numbers a, c 1, c 2,, c n (a 0) such that P (x) = a(x c 1 )(x c 2 )... (x c n ) Example 51. Let P (x) = x 4 + 3x 2 4. Find all the zeros of P and find the complete factorization of P. 43

44 Theorem Zeros Theorem Every polynomial of degree n 1 has exactly n zeros, provided that a zero of multiplicity k is counted k times. Example 52. Find the complete factorization and all three zeros of the polynomial: P (x) = x 3 6x x 8. Example 53. Find a polynomial P (x) of degree 4, with zeros 2 and 0, where 2 is a zero of multiplicity 3. Also, P (1) = 4. 44

45 Theorem Conjugate Zeros Theorem If the polynomial P has real coefficients and if the complex number z is a zero of P, then its complex conjugate z is also a zero of P. Example 54. Find all the zeros of P (x) = x Example 55. Find a polynomial P (x) of degree 3 that has integer coefficients and zeros 1 2 and 2 i. 45

46 Definition. Linear and Quadratic Factors Theorem Every polynomial with real coefficients can be factored into a product of linear and irreducible quadratic factors with real coefficients. Note. A quadratic polynomial with no real zeros is called irreducible over the real numbers. Example 56. P (x) = x 4 + 8x 2 9. Factor P into linear and irreducible quadratic factors with real coefficients. 46

47 3.6 Rational Functions Recall: A rational function is a function of the form r(x) = P (x) Q(x), Q(x) 0 where P and Q are polynomials. We also assume that P (x) and Q(x) have no factor in common. Let s look at f(x) = 1 x. 47

48 Graphing Rational Functions Example 57. Graph r(x) = 2x2 4x + 5 x 2 2x + 1 and state the domain and range. 48

49 Definition. Finding Asymptotes of Rational Functions Let r be the rational function r(x) = a nx n + a n 1 x n a 1 x + a 0 b m x m + b m 1 x m b 1 x + b 0 1. The vertical asymptote of r are the lines x = a, where a is a zero of the denominator 2. a) If n < m, then r has horizontal asymptote y = 0. b) If n = m, then r has horizontal asymptote y = a n b m. c) If n > m, then r has no horizontal asymptote. Example 58. Graph r(x) = x2 x 6. State x-int, y-int, Horizontal and Vertical Asymptote (if x 2 + 3x any), Domain and Range. 49

50 Note. A graph can intersect the Horizontal Asymptote. What if they have a common factor? Example 59. Graph r(x) = x3 2x 2 3x. x 3 50

51 Example 60. Graph: r(x) = x2 + 2x x 1 51

52 3.7 Polynomial and Rational Inequalities Recall: Theorem Intermediate Value Theorem for Polynomials If P is a polynomial function and P (a) and P (b) have opposite signs, then there exists at least one value c between a and b for which P (c) = 0. Example 61. Solve the inequality: x 4 + 3x 3 3x 2 + 3x 4 > 0 52

53 Example 62. Solve the inequality: x 3 2x

54 4 Exponential and Logarithmic Functions 4.1 Exponential Functions Definition. The exponential function with base a is defined by f(x) = a x where x R, a > 0, and a 1. Example 63. Let f(x) = 4 x. Evaluate: 1. f(2) 2. f( 3/2) 3. f(π) 4. f( 2) Example 64. Now let s look at the graphs of f(x) = 4 x and g(x) = ( ) x

55 Definition. Graphs of Exponential Functions The exponential function f(x) = a x, a > 0, a 1 has domain R and range (0, ). The line y = 0 is a horizontal asymptote of f. f(x) = a x for a > 1 f(x) = a x for 0 < a < 1 Example 65. Using the graph of f(x) = 4 x, graph the following: g(x) = 4 x 1, h(x) = 4 x and k(x) = 4 x+1. 55

56 Example 66. Suppose you have an amount of money, lets call it P, and you want to invest it at an interest rate i per time period, then after one time period you will have: A = P + P i = P (1 + i) Definition. A(t) = P ( 1 + r n ) nt where, A(t) = amount after t years P = principal r = interest rate per year n = number of times interest is compounded per year t = number of years Annual = Semiannual = Quarterly = Monthly = Daily = 56

57 Example 67. If $500 is invested at an interest rate of 3.75% per year, compounded quarterly, find the value of the investment after the given number of years years 2. 3 years 3. 6 years Example 68. If $4000 is borrowed at a rate of 5.75% interest per year, compounded semiannual, find the amount due at the end of the given number of years years 2. 8 years 57

58 4.2 The Natural Exponential Function n ( ) n n Hence, e ,000 10, ,000 1,000,000 Definition. The natural exponential function is the exponential function with base e. f(x) = e x Example 69. (a) e 4 (b) 2e.035 Example 70. Let s look back at: A(t) = P ( 1 + r n ) nt. Now, let m = n r. 58

59 Definition. Continuously compounded interest is calculated by the formula A(t) = P e rt where, A(t) = amount after t years P = Principal r = interest rate per year t = number of years Example 71. If $2000 is invested at an interested rate of 3.5% per year, compounded continuously, find the value of the investment after 4 years. 59

60 4.3 Logarithmic Functions Recall: f(x) = a x, where a > 0 and a 1 Definition. Let a R + with a 1. The logarithmic function with base a, denoted log a, is defined by where x > 0. log a x = y a y = x Example 72. Graph of y = log 2 x. 60

61 Note. Logarithmic Properties log a 1 = 0 log a a = 1 log a a x = x a log a x = x Definition. The logarithm with base 10 is called the common logarithm and is denoted by: log x = log 10 x. Definition. The logarithm with base e is called the natural logarithm and is denoted by ln x = log e x Note. Logarithmic Properties ln 1 = 0 ln e = 1 ln e x = x e lnx = x Example 73. Find the domain of the function f(x) = ln(9 x 2 ). 61

62 4.4 Laws of Logarithms Definition. Let a R +, where a 1. Let A, B, C R, with A, B > 0. log a (AB) = log a A + log a B (4.1) ( ) A log a = B log a A log a B (4.2) log a (A C ) = C log a A (4.3) 62

63 Definition. Change of Base Formula log b x = log a x log a b Example 74. Use the Laws of Logarithms to expand the expression: ( ) x log 3 1 x 63

64 Example 75. Use the Laws of Logarithms to combine the expression: 4 log x 1 3 log(x2 + 1) + 2 log(x 1) 64

65 4.5 Exponential and Logarithmic Equations Example 76. Solve: 6 3x = 6 2x 1 Example 77. Solve: 1 + e 4x+1 = 20 65

66 Example 78. Solve: 2 x 10(2 x ) + 3 = 0 66

67 Note. You much check solutions when dealing with logarithms. Example 79. Solve: log 5 x + log 5 (x + 1) = log 5 20 Example 80. log 3 (x + 15) log 3 (x 1) = 2 67

68 Example 81. A woman invest $6500 in an account that pays 6% interest per year, compounded continuously. 1. What is the amount after 2 years? 2. How long will take for the amount to be $8000? Example 82. A sum of $1000 was invested for 4 years, and the interest was compounded semianually. If this sum amounted to $ in the given time, what was the interest rate? 68

69 4.6 The Natural Exponential Function Definition. Exponential Growth (Doubling Time) If the initial size of a population is n 0 and the doubling time is a, then the size of the population at time t is n(t) = n 0 2 t/a where a and t are measured in the same time units (minutes, hours, days, years, and so on). Example 83. A certain culture of the bacterium Rhodobacter sphaeroides initially has 25 bacteria and is observed to double every 5 hours. 1. Find and exponential model n(t) = n 0 2 t/a for the number of bacteria in the culture after t hours. 2. After how many hours will the bacteria count reach 1 million? 69

70 Definition. A population that experiences exponential growth increases according to the model n(t) = n 0 e rt where, n(t) = population at time t n 0 = initial size of the population r = relative rate of growth (expressed as a proportion of the population) t = time Example 84. The count in a culture of bacteria was 400 after 2 hours and 102,400 after 6 hours. Find a function that models the number of bacteria n(t) after t hours. Definition. Half-Life m(t) = m 0 2 t/h = m 0 ( 1 2 ) t/h 70

71 Definition. If m 0 is the inital mass of a radioactive substance with half-life h, then the mass remaining at time t is modeled by the function where r = ln 2 h is the relative decay rate. m(t) = m 0 e rt Example 85. A wooden artifact from an ancient tomb contains 25% of the carbon-14 that is present in living trees. How long ago was the artifact made? (The half-life of carbon-14 is 5730 years.) 71

72 5 Systems of Equation and Inequalities 5.1 Systems of Linear Equations in Two Variables 5.2 Systems of Linear Equations in Several Variables Note. There are three possible ways in solving the following system of linear equations in which it will lead to three possible solutions. Example 86. { 2x y = 6 x y = 3 Example 87. Solve the above system by using Gaussian Elimination: x y + 2z = 2 3x + y + 5z = 8 2x y 2z = 7 72

73 x 2y + 3z = 10 Example 88. Solve the above system by using Gaussian Elimination: 3y + z = 7 x + y z = 7 73

74 5.3 Partial Fractions Case 1. The Denominator is a product of distinct linear factors. Suppose that we can factor Q(x) as Q(x) = (a 1 x + b 1 )(a 2 x + b 2 ) (a n x + b n ) with no factor repeated. In this case the partial fraction decomposition of P (x) Q(x) P (x) Q(x) = A 1 a 1 x + b 1 + A 2 a 2 x + b A n a n x + b n takes the form Example 89. Find the partial fraction decomposition of the rational function: 7x 3 x 3 + 2x 2 3x 74

75 Case 2. The Denominator is a product of distinct linear factors, some of which are repeated. Suppose the complete factorization of Q(x) contains the linear factor ax + b repeated k times; that is, (ax + b) k is a factor of Q(x). Then, corresponding to each such factor, the partial fraction decomposition for P (x) Q(x) contains A1 ax + b + A 2 (ax + b) A k (ax + b) k Example 90. Find the partial fraction decomposition of the rational function: 3x 2 + 5x 13 (3x + 2)(x 2 4x + 4) 75

76 Case 3. The Denominator has irreducible quadratic factors, none of which is repeated. Suppose the complete factorization of Q(x) contains the quadratic factor ax 2 + bx + c (which can t be factored further). Then, corresponding to this, the partial fraction decomposition of P (x) will have the form Q(x) Ax + B ax 2 + bx + c Example 91. Find the partial fraction decomposition of the rational function: 4x2 + 4x + 3 x 3 + x 76

77 Case 4. The Denominator has a repeated irreducible quadratic factor. Suppose the complete factorization of Q(x) contains the quadratic factor (ax 2 + bx + c) k, where ax 2 +bx+c can t be factored further. Then, corresponding to this, the partial fraction decomposition of P (x) will have the form Q(x) A 1 x + B 1 ax 2 + bx + c + A 2x + B 2 (ax 2 + bx + c) + + A kx + B k 2 (ax 2 + bx + c) k Note. The techniques that we have done apply only to rational functions P (x) in which the degree Q(x) of P is less than the degree of Q. If this isn t the case, we must first use long division. Example 92. Find the partial fraction decomposition of the rational function: x 5 3x 4 + 3x 3 4x 2 + 4x + 12 (x 2) 2 (x 2 + 2) 77

78 5.4 Systems of Nonlinear Equations Use either the substitution method to find all solutions of the system of equations. { yx = 24 Example 93. 2x 2 y = 0 Example 94. { x 4 + y 3 = 17 3x 4 + 5y 3 = 53 78

79 { x 2 + 2y 2 = 2 Example 95. 2x 2 3y = 15 { 2x 2 + 4y = 13 Example 96. x 2 y 2 = 7/2 79

80 Example 97. A circular piece of sheet metal has a diameter of 20 in. The edges are to be cut off to form a rectangle of area 8 in 2. What are the dimensions of the rectangle? 80

81 5.5 Systems of Inequalities Recall: f(x) = x 4 Now let s look at: f(x) < x 4 f(x) > x 4 f(x) x 4 f(x) x 4 Example 98. Graph the inequality 3x y 7 x 2 + y 2 < 16 81

82 Example 99. Graph the solution of the system of inequalities. Find the coordinates of all vertices { x 2 + y 2 4 and determine whether the solution set is bounded. x y > 0 Example 100. x 2 y 0 x + y < 6 x y < 6 82

83 Example 101. A farmer has 300 acres of arable land on which she wants to plant cauliflower and cabbage. The farmer has $17, 500 available for planting and $12, 000 for fertilizer. Planting 1 acre of cauliflower costs $70, and planting 1 acre of cabbage costs $35. Fertilizer costs $25 for 1 acre of cauliflower and $55 for 1 acre of cabbage. 1. Find a system of inequalities that describes the number of acres of each crop that the farmer can plant with the available resources. Graph the feasible region. 2. Can the farmer plant 155 acres of cauliflower and 115 acres of cabbage? 3. Can the farmer plant 115 acres of cauliflower and 175 acres of cabbage? 83

84 6 Matrices and Determinants 6.1 Matrices and Systems of Linear Equations Definition. An m n is a rectangular array of number with m rows and n columns. a 11 a 12 a 1n a 21 a 22 a 2n A mn = a m1 a m2 a mn We say that the matrix has dimension m n. The number a ij are the entries of the matrix. The subscript on the entry a ij indicates that it is in the i-th row and the j-th row. Example 102. A 23 = ( 4 5 ) A 15 = ( ) 4 2 A 31 = Note. We can write a system of linear equations as a matrix, called the augmented matrix of the system, by writing only the coefficients and constants that appear in the equations. 2x 3y + 5z = 14 Example x y 2z = 17 x y + z = 3 Definition. A matrix is in row-echelon form if it satisfies the following conditions: The first nonzero number in each row (reading from left to right) is 1. leading entry. This is called the The leading entry in each row is to the right of the leading entry in the row immediately above it. All rows consisting entirely of zeros are at the bottom of the matrix. A matrix is in reduced row-echelon form if it is in row-echelon form and also satisfies the following condition: Every number above and below each leading entry is 0. 84

85 Not in row-echelon form Row-echelon form Reduced row-echelon form Example 104. Solve this using Gaussian Elimination combined with row-echelon form

86 Note. Another way of doing it is by using Gauss-Jordan elimination ( Using the reduced rowechelon form to solve a system is called Gauss-Jordan elimination.) Definition. The Solutions of a linear system in Row-Echelon Form Suppose the augmented matrix of a system of linear equations has been transformed by Gaussian elimination into row-echelon form. Then exactly one of the following is true: No Solution. If the row-echelon form contains a row that represents the equation 0 = c, c 0, then the system has no solution. Hence, it is inconsistent. One Solution. If each variable in the row-echelon form is a leading variable, then the system has exactly one solution. Infinitely Many Solutions. If the variables in the row-echelon form are not all leading variables and if the system is not inconsistent, then it has infinitely many solutions. Hence, the system is called dependent. 86

87 4x y + 36z = 24 Example 105. Solve the system of linear equations by using matrices: x 2y + 9z = 3 2x + y + 6z = 6 87

88 6.2 The Algebra of Matrices Definition. Equality of Matrices The matrices A = [a ij ] and B = [b ij ] are equal iff they have the same dimension m n, and corresponding entries are equal, that is, for i = 1, 2,..., m and j = 1, 2,..., n. a ij = b ij Let A = [a ij ] and B = [b ij ] be matrices of the same dimension m n, and let c R. A + B = [a ij + b ij ] A B = [a ij b ij ] ca = [ca ij ] Example 106. Let A = 5 0 B = C = ( 12 4 ) D = ( 1 2 ) Find the following operations if possible: 1. B + A 2. C + B 3. D C 4. 2A 88

89 Definition. Properties of Addition and Scalar Multiplication of Matrices Let A, B, and C be m n matrices and let c, d R. A + B = B + A (A + B) + C = A + (B + C) (c + d)a = ca + da c(da) = (cd)a c(a + B) = ca + cb Definition. Matrix Multiplication If A = [a ij ] is an m n matrix and B = [b ij ] and n k matrix, then their product is the m k matrix C = [c ij ] where c ij is the inner product of the i-th row of A and j-th column of B. We write the product as C = AB Example 107. Let ( ) F = G = Find F G. 89

90 Definition. Properties of Matrix Multiplication Let A, B, and C be matrices for which the following products are defined. Then, A(BC) = (AB)C A(B + C) = AB + AC (B + C)A = BA + CA Example 108. Given F and G from above, find GF and compare it to F G. Example 109. Write the system of equation as a matrix equation: 6x y + z = 12 2x + z = 7 y 2z = 4 90

91 6.3 Inverses of Matrices and Matrix Equations Definition. A square matrix is one that has the same number of rows as columns. a 11 a 12 a 1n a 21 a 22 a 2n A nn = a n1 a n2 a nn Definition. The main diagonal of a square matrix consists of the entries whose row and column numbers are the same. a 11.. a A nn = a nn Definition. Identity Matrix The identity matrix I n is the n n matrix for which each main diagonal entry is a 1 and for which all other entries are A nn = Definition. Inverse of Matrix Let A be a square n n matrix. If there exists an n n matrix A 1 with the property that AA 1 = A 1 A = I n then we say that A 1 is the inverse of A. 91

92 Example 110. Let ( ) 4 1 A = 7 2 B = ( 2 ) Find AB and BA. 92

93 Definition. Inverse of a 2 2 Matrix ( ) a b If A =, then c d If ad bc = 0, then A has no inverse. A 1 = 1 ad bc ( d b c a Note. ad bc is also known as the determinant of the 2 2 matrix. Example 111. Find A 1 given that A = ( ) ). 93

94 4 2 3 Example 112. Find B 1 given that B = Note. Read each row carefully while trying to find an inverse matrix since not all matrices will have an inverse. 94

95 Note. Being able to find the inverse of a matrix can help us solve a system of linear equations. Definition. Solving a Matrix Equation If A is a square n n matrix that has an inverse A 1 and if X is a variable matrix and B a known matrix, both with n rows, then the solution of the matrix equation is given by AX = B X = A 1 B. Example 113. Solve: { 3x + 4y = 10 7x + 9y = 20 95

96 6.4 Determinants and Cramer s Rule ( ) a b Definition. The determinant of the 2 2 matrix A = c d det(a) = A = a b c d = ad bc Example 114. ( 7 5 Given B = ). Find B. Definition. Let A be an n n matrix. The minor M ij of the element a ij is the determinant of the matrix obtained by deleting the i-th row and the j-th column of A. The cofactor A ij of the element a ij is A ij = ( 1) i+j M ij Example 115. Given A = Find M 23 and A

97 Definition. If A is an n n matrix, then the determinant of A is obtained by multiplying each element of the first row by its cofactor and then adding the results. a 11 a 12 a 1n a 21 a 22 a 2n det(a) = A =..... = a 11 A 11 + a 12 A a 1n A 1n.. a n1 a n2 a nn Example 116. Evaluate the determinant of the matrix A = Note. We could have expanded by a different row. Expand along the third row and see if you get the same thing. 97

98 Note. An important use of the determinant is to determine whether a square matrix has an inverse without actually calculating the inverse. Definition. The Invertibility Criterion If A is a square matrix, then A has an inverse iff det(a) Example 117. Given the following square matrix. Find A. A = Definition. It A is a square matrix and if the matrix B is obtained from A by adding a multiple of one row to another row or a multiple of one column to another, then det(a) = det(b). 98

99 Definition. Cramer s Rule The linear system { ax + by =r cx + dy =s has the solution provided that a c r s x = a c b d 0. b d b d y = a c a c r s b d Example 118. Use Cramer s Rule to solve the system. { 2x y = 9 x + 2y = 8 99

100 Definition. If a system of n linear equations in the n variables x 1, x 2,..., x n is equivalent to the matrix equation DX = B, and if D 0, then its solutions are x 1 = D x 1 D x 2 = D x 2 D... x n = D x n D x y + 2z = 0 Example 119. Use Cramer s Rule to solve the system. 3x + z = 11 x + 2y = 0 100

101 7 Conic Sections 7.1 Parabolas Definition. A parabola is the set of all points in the plane that are equidistant from a fixed point F (called the focus) and a fixed line l (called the directrix). 101

102 Parabola with Vertical Axis The graph of the equation x 2 = 4py is a parabola with the following properties. Vertex Focus Directrix V (0, 0) F (0, p) y = p the parabola opens upward if p > 0 or downward if p < 0. Note. The line segment that runs through the focus perpendicular to the axis, with endpoints on the parabola, is called the latus rectum, and its length is the focal diameter of the parabola. 102

103 Definition. Parabola with Vertical Axis The graph of the equation y 2 = 4px is a parabola with the following properties. Vertex Focus Directrix V (0, 0) F (p, 0) x = p the parabola opens to the right if p > 0 or to the left if p <

104 Example 120. Find the equation for the parabola that has its vertex at the origin and satisfies the given condition(s). (a) Focus: (0, 6) (b) Directrix: x = 1 8 Example 121. Find the focus, directrix, and focal diameter of the parabola. (a) x 2 = 8y (b) 5x + 3y 2 = 0 104

105 7.2 Ellipses Definition. An ellipse is the set of all points in the plane the sum of whose distances from two fixed two points F 1 and F 2 is a constant. These two fixed points are the foci (plural of focus) of the ellipse. 105

106 The graph of each of the following equations is an ellipse with center at the origin and having the given properties: Equation: x 2 a 2 + y2 b 2 = 1 x 2 b 2 + y2 a 2 = 1 a > b > 0 a > b > 0 Vertices: (±a, 0) (0, ±a) Major Axis: Horizontal, length 2a Vertical, length 2a Minor Axis: Vertical, length 2b Horizontal, length 2b Foci (±c, 0), c 2 = a 2 b 2 (0, ±c), c 2 = a 2 b 2 Example 122. Graph 3x 2 + 4y 2 = 12 and state the vertices and foci. Example 123. Given the Foci: (0, ±4) and Vertices: (0, ±5) find the equation of the ellipse. 106

107 Example 124. Length of minor axis: 10, foci on y-axis, ellipse passes through the point ( 5, 40). 107

108 7.3 Hyperbolas Definition. A hyperbola is the set of all points in the plane, the difference of whose distances from two fixed points F 1 and F 2 is a constant. These two fixed points are the foci of the hyperbola. The graph of each of the following equations is a hyperbola with center at the origin and having the given properties: Equation: x 2 a 2 y2 b 2 = 1 y 2 a 2 x2 b 2 = 1 a > 0, b > 0 a > 0, b > 0 Vertices: (±a, 0) (0, ±a) Transverse Axis: Horizontal, length 2a Vertical, length 2a Asymptotes: y = ± b a x y = ±a b x Foci: (±c, 0), c 2 = a 2 + b 2 (0, ±c), c 2 = a 2 + b 2 Example 125. Graph 9x 2 16y 2 = 144 and state the foci and vertices. 108

109 Example 126. Find an equation for the hyperbola that satisfies the given conditions: 1. Vertices: (0, ±6), asymptotes: y = ± 1 3 x 2. Foci: (±5, 0), vertices: (±3, 0) 109

110 7.4 Shifted Conics Shifting Graphs of Equations If h and k are positive real numbers, then replacing x by x h or by x + h and replacing y by y k or y + k has the following effect(s) on the graph of any equation in x and y: Replacement x replaced by x h x replaced by x + h y replaced by y + k y replaced by y k How the graph is shifted Right h units Left h units Downwards k units Upwards k units Shifted Ellipse x 2 a + y2 (x h)2 = b2 a 2 (y k)2 b 2 = 1 110

111 Example 127. Graph (x 1)2 (y + 2)2 + = 1 and state the new foci and vertices Example 128. Find the equation for the ellipse with the given properties: vertices ( 1, 4) and ( 1, 6) and foci ( 1, 3) and ( 1, 5). 111

112 Shifted Parabolas p > 0: x 2 = 4py (x h) 2 = 4p(y k) p < 0: x 2 = 4py (x h) 2 = 4p(y k) p > 0: y 2 = 4px (y k) 2 = 4p(x h) p < 0: y 2 = 4px (y k) 2 = 4p(x h) 112

113 Shifted Hyperbolas x 2 a y2 (x h)2 = 1 2 b2 a 2 (y k)2 b 2 = 1 y 2 a x2 (y k)2 = 1 2 b2 a 2 (x h)2 b 2 = 1 Example 129. A shifted conic has the equation x 2 5y 2 2x + 20y = 44. Find the center, vertices, foci, and asymptotes of the hyperbola and sketch the graph. 113

114 Note. The graph of the equation Ax 2 + Cy 2 + Dx + Ey + F = 0 where A and C are not both 0, is a conic or degenerate conic. In the nondegenerate cases the graph is a parabola if A or C is 0. an ellipse if A and C have the same sign (or a circle if A = C). a hyperbola if A and C have opposite signs. Example 130. Determine whether the graph of the equation is an ellipse, a parabola, a hyperbola, or a degenerate conic. (a) 2x 2 + y 2 = 2y + 1 (b) x 2 + 4y x 40y = 0 114

115 8 Sequences and Series 8.1 Sequences and Summation Notation Definition. A sequence is a function a whose domain is the set of natural numbers. The terms of the sequence are the functions values a(1), a(2), a(3),..., a(n),... We usually write a n instead of the function notation a(n). So the terms of the sequence are written as a 1, a 2, a 3,..., a n,... The number a 1 is called the first term, a 2 is called the second terms, and in general, a n is called the nth term. Example 131. Suppose we have the function a(n) = 3n. Thus, a(1), a(2), a(3),... Example 132. Find the first five terms and the 100 th term of the sequence defined by 1. t n = n n r n = ( 1) n (n 2 1) 115

116 Example 133. Find the nth term of a sequence whose first several terms are given. 1. 3, 0.3, 0.03, 0.003, , 1 2, 3, 1 4, 5, 1 6,... The Fibonaci Sequence 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89,

117 Definition. For the sequence a 1, a 2, a 3, a 4,..., a n,... the partial sums are S 1 = a 1 S 2 = a 1 + a 2 S 3 = a 1 + a 2 + a 3 S 4 = a 1 + a 2 + a 3 + a 4. S n = a 1 + a 2 + a a n S 1 is called the first partial sum, S 2 is the second partial sum, and so on. S n is called the n th partial sum. The sequence S 1, S 2, S 3,..., S n,... is called the sequence of partial sums. Example 134. Find the first four partial sums and the nth partial sums of the sequence given by a n = 2 3 n 117

118 n Note. S n can be written as a k = a 1 + a 2 + a a n. k=1 Example 135. Find the sum of each: 1. 4 k 3 2. k=1 3 k=1 k 1 k k=1 2 Let a 1, a 2, a 3,... and b 1, b 2, b 3,... be sequences. Then for every possible integer n and any real number c the following properties hold. n (a k + b k ) = k=1 n (a k b k ) = k=1 n a k + k=1 n a k k=1 ( n n ) c a k = c a k k=1 k=1 n k=1 n k=1 b k b k Example 136. Write the sum using sigma notation. 1 2 ln ln ln ln ln

119 8.2 Arithmetic Sequences Definition. An arithmetic sequence is a sequence of the form a, a + d, a + 2d, a + 3d, a + 4d... The number a is the first term, and d is the common difference of the sequence. The nth term of an arithmetic sequence is given by a n = a + (n 1)d (a) -31, -19, -7, 5,... (b) 7, 8, 10, 13,... Example 137. The twelfth term is 118, and the eighth terms is 146. Find the first term, 1001-th term and the n-term. 119

120 Let s find the sum of the first n terms of the arithmetic sequence whose terms are a k = a+(k+1)d. S n = n [a + (k 1)d] k=1 Definition. For the arithmetic sequence given by a n = a + (n 1)d, the nth partial sum S n = a + (a + d) + (a + 2d) + (a + 3d) + (a + 4d) + [a + (n 1)d] is given by either of the following formulas. 1. S n = n 2 [2a + (n 1)d] 2. S n = n ( a + an 2 ) 120

121 Example 138. A partial sum of an arithmetic sequence is given. Find the sum, ( ) Example 139. An architect designs a theater with 15 seats in the first row, 18 in the second, 21 in the third, and so on. If the theater is to have a seating capacity of 870, how many rows must the architect use in his design? 121

122 8.3 Geometric Sequences Definition. A geometric sequence is a sequence of the form a, ar, ar 2, ar 3, ar 4, ar 5, The number a is the first term, and r is the common ratio of the sequence. The nth term of a geometric sequence is given by a n = ar n 1 (a) 27, -9, 3, -1,... (b) 3, 48, 93, 138,... Example 140. The fourth term is 12 and the seventh term is 32. Find the second term and the 9 nth term. 122

123 Definition. For the geometric sequence defined by a n = ar n 1, the nth partial sum is given by S n = a + ar + ar 2 + ar 3 + ar ar n 1 r 1 S n = a 1 rn 1 r Example 141. Find the sum

124 An expression of the form is called an infinite series. a k = a 1 + a 2 + a 3 + a 4 + a 5 + k=1 S n = n Note. In general, if S n gets close to a finite number S as n gets large, we say that the infinite series coverages (or is convergent). If it does not, then it diverges (or is divergent). 124

125 An infinite geometric series is a series of the form a + ar + ar 2 + ar ar n 1 + The nth partial sum of such a series is given by the formula Suppose that r < 1 as n. Then, S n = a 1 rn 1 r, r 1 S n If r < 1, then the infinite geometric series a 1 r converges and has the sum If r 1, the series diverges. = a + ar + ar 2 + ar 3 + k=1 S = a 1 r Example 142. Determine whether the infinite geometric series is convergent or divergent. If it is convergent, find its sum. (a) (b)

126 8.4 Mathematical Induction 1 = = = = (2n 1) = n 2 We will be proving the above condition using Mathematical Inductions. Principle of Mathematics Induction (PMI) Let n N and P (n) be a statement depending on n. Suppose that the following two condition are satisfied. P (1) is true. k, P (k) is true P (k + 1) is true. Then P (n) is true n. 126

127 Some helpful formulas n 1 = n i=1 n k = i=1 n(n + 1) 2 n k 2 = i=1 i=1 n(n + 1)(2n + 1) 6 n k 3 = n2 (n + 1) 2 4 Example 143. Use PMI to provo the following: n 2 = n(n + 1)(2n + 1) 6 127

128 Example 144. Prove, using PMI, that 8 n 3 n is divisible by 5 n N. 128

129 8.5 The Binomial Theorem Let s look at: (a + b) 1 = (a + b) 2 = (a + b) 3 = (a + b) 4 =. After a closer look at the expansions above, you might see some patterns Pascal s Triangle and its relationship to (a + b) n 129

130 Factorial: n! = (n 1) n *Note: 0! = 1 Definition. The Binomial Coefficient Let n, r Z + with r n. The binomial coefficient is denoted by ( n r) and is defined by Find ( ( 7 5) and 7 ) 2 ( n r) = n! r!(n r)! ( ) ( ) n n = r n r Now, lets rewrite Pascal s Triangle using this idea: 130

131 For any r, k N + with r k, ( k ( r 1) + k ) ( r = k+1 ) r. Now we can state the Binomial Theorem ( n (a + b) n = 0 = n k=0 ) a n + ( ) n a n k b k. k ( ) n a n 1 b + 1 ( ) ( ) n n a n 2 b ab n n 1 ( ) n b n n Example 145. Use the Binomial Theorem to expand (x 2y) 4. General Term of the Binomial Expansion The term that contains a r in the expansion of (a + b) n is ( n ) r a r b n r Example 146. Find the term containing x 4 in the expansion of (x + 2y)