MATH 142 Business Mathematics II. Summer, 2016, WEEK 1. JoungDong Kim

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1 MATH 142 Business Mathematics II Summer, 2016, WEEK 1 JoungDong Kim Week 1: A8, 1.1, 1.2, 1.3, 1.5, 3.1, 3.2, 3.3 Chapter 1. Functions Section 1.1 and A.8 Definition. A function is a rule that assigns to each element x in the domain exactly one element in the range. The domain is the set of all values of the independent variable (typically x) that produce real values for the dependent variable (typically y). The range is the set of all possible y values as x varies throughout the domain. If the correspondence is a function we use the notation f(x) read f of x. Ordered pair of the form (x,y) can be written as (x,f(x)). 1

2 Ex1) Find the domain of the following functions. 1. f(x) = 2x 4 2. g(x) = x 1 x 2 5x+6 3. h(x) = 5x 6 3 3x 5 In Sum, The domain of most algebraic functions is the set of all real numbers EXCEPT: 1. Any real numbers that cause the denominator of a function to be Any real numbers that cause us to take the square root (or any radical with an even index) of a negative number. and more... 2

3 Definition. Functions with domains divided into two or more parts with a different rule applied to each part are called piecewise-defined functions. The absolute value function, x, is such an example. x if x 0 x = x if x < 0 Graph of a functions The graph ofafunctionf consists of all points (x,y)such that x is in thedomain of f andy = f(x). Ex2) Graph the function f(x) = x. 3

4 Catalog of Basic Functions 1. Constant function f(x) = c, c : real constant 2. Identity function f(x) = x 3. Square function f(x) = x 2 4. Cubing function f(x) = x 3 5. Square root function f(x) = x 6. Reciprocal function f(x) = 1 x 4

5 Ex3) Graph the following piecewise defined function: x 2 2 if x < 0 f(x) = 2x+3 ifx 0 Vertical line test A graph in the xy-plane represents a function of x, if and only if, every vertical line intersects the graph in at most one place. 5

6 Finding domain and range from graphs To find the domain of f, vertical line scanning the x-axis. To find the range of f, horizontal line scanning the y-axis. 6

7 Ex4) Ataxi-cabcompany inacertaintown charges all customers a base feeof $5per ride. They then charge an additional 50 cents per mile for the first 10 miles traveled and $1/miles. Write a piecewise function, C(x), for the cost of a cab ride if x represents the number of miles traveled. 7

8 Increasing, Decreasing, Concavity, and Continuity A function y = f(x) is said to be increasing (denoted by ր) on the interval I if the graph of the function rises while moving left to right or, equivalently, if f(x 1 ) < f(x 2 ) when x 1 < x 2 on I A function y = f(x) is said to be decreasing (denoted by ց) on the interval I if the graph of the function falls while moving left to right or, equivalently, if f(x 1 ) > f(x 2 ) when x 1 < x 2 on I If the graph of a function bends upward ( ), we say that the function is concave up. If the graph of a function bends downward ( ), we say that the function is concave down. A straight line is neither concave up nor concave down. A function is said to be continuous on an interval if the graph of the function does not have any breaks, gaps, or holes in that interval. 8

9 Ex5) Given the graph of f(x) below, determine the interval(s) on which f(x) is y x 1. increasing 2. decreasing 3. cancave up 4. cancave down 5. continuous 9

10 Ex6) Let f(x) = 1 f(x+h) f(x), find and simplify. x 2 h 10

11 Special Functions Polynomial function A polynomial function of degree n is a function of the form f(x) = a n x n +a n 1 x n 1 + +a 1 x+a 0 (a n 0) wherea n,a n 1,,a 1,a 0, areconstantsandnisanonnegative integer. The domain is all real numbers. The coefficient a n is called the leading coefficient. A polynomial function of degree 1 (n = 1), f(x) = a 1 x+a 0, (a 1 0) is a linear function. A polynomial function of degree 2 (n = 2), f(x) = a 2 x 2 + a 1 x + a 0, (a 2 0) is a quadratic function. Note. Domain of polynomial is all real numbers. Polynomials are continuous for all real numbers. Graph is smooth. Power function A power function is a function of the form f(x) = kx r where k and r are any real numbers. r = n, where n is a positive integer. r = 1, where n is a positive integer. n r = 1 11

12 Rational function A rational function R(x) is the quotient of two polynomial functions and thus is of the form R(x) = f(x) where f(x) and g(x) are polynomial functions. the domain of R(x) is all real numbers for g(x) which g(x) 0. 12

13 Graphing Techniques Transformations Vertical and Horizontal Shifts: Suppose c > 0. To obtain the graph of y = f(x)+c, shift the graph of y = f(x) a distance c units upward. y = f(x) c, shift the graph of y = f(x) a distance c units downward. y = f(x c), shift the graph of y = f(x) a distance c units to the right. y = f(x+c), shift the graph of y = f(x) a distance c units to the left. Expansion, Contraction and Reflecting: Suppose c > 1. To obtain the graph of y = cf(x), stretch the graph of y = f(x) vertically by a factor of c. y = 1 f(x), shrink the graph of y = f(x) vertically by a factor of c. c y = f(x), reflect the graph of y = f(x) x-axis. 13

14 Ex7) Graph y = x, y = x+2, and y = x 1 on the same graph. Ex8) Graph y = x 3, y = (x 2) 3, and y = (x+1) 3 on the same graph. Ex9) Graph y = x, y = 2 x, and y = 2 x on the same graph. 14

15 Ex10) If g(x) = 2f(x+1) 3, describe how f(x) was transformed to get g(x). 15

16 Section 1.2 Mathematical Models A mathematical model is a mathematical description(often by means of a function or an equation) of a real-world phenomenon. The purpose of the model is to understand the phenomenon and perhaps to make predictions about future behavior. Mathematical Models of Cost, Revenue, and Profits In the linear cost model we assume that the cost m of manufacturing one unit is the same no matter how many units are produced. Thus, the variable cost is the number of units produced times the cost of each unit: variable cost = (cost per unit) (number of units produced) = mx where x is the number of units. If b is the fixed cost and C(x) is the cost, then C(x) = cost = (variable cost) +(fixed cost) = mx+b In the linear revenue model we assume that the price p of a unit sold by a firm is the same no matter how many units are sold. Thus, the revenue is the price per unit times the number of units sold. If we denote the revenue by R(x), R(x) = revenue = (price per unit) (number sold) = px Thus the Profit P is P(x) = profit = (revenue) (cost) = R(x) C(x) 16

17 Ex11) A manufacturer of garbage disposals, has a monthly fixed cost of $10,000 and a production cost of $20 for each garbage disposal manufactured. The unit sell for $60 each. 1. What is the cost function? 2. What is the revenue function? 3. What is the profit function? Break-Even Quantity: The value of x at which the profit is zero. (i.e., the point of intersection of the revenue and cost functions) Ex12) Find the break-even point for the garbage disposals in Ex11. 17

18 Mathematical Models of Supply and Demand If the price is assumed to be linear, then the graphs of demand equation and supply equation are straight lines. Market Equilibrium: The point at which the consumer and supplier agree upon. (i.e., the point of intersection of the supply and demand curves) 18

19 Ex13) For a particular commodity, it is found that 9 units will be supplied at a unit price of $10.50 whereas 3 units will be supplied at a unit price of $5.50. For this commodity, it is found that the consummers are willing to consume 8 units at a unit price of $7 but are only willing to consume 1 unit at a price of $ Assuming that both supply and demand functions are linear, find the a) supply equation b) demand equation c) market equilibrium 19

20 Practice1) For the quadratic function y = 2x 2 8x+6, what is the coordinate of the vertex point? 20

21 Ex14) It is found that the consumers of a particular toaster will demand 64 toaster ovens when the unit price is $35 whereas they will demand 448 toaster ovens when the unit price is $5. Assuming that the demand function is linear, and the selling price is determined by the demand function, a) Find the demand equation. b) Find the revenue function. c) Find the number of items sold that will give the maximum revenue. What is the maximum revenue? d) If the company has a fixed cost of $1,000 and a variable cost of $15 per toaster, find the company s linear cost function. e) What is the company s maximum profit? f) How many toasters should be sold for the company to break even? 21

22 Section 1.3 Exponential Models The exponential functions are the functions of the form f(x) = a x, where the base a is a positive constant with a 1. Properties of the Graphs of f(x) = a x 1. Domain is the set of all real numbers. 2. Range is the set of all positive real numbers. 3. All graphs pass through the point (0, 1). 4. The graph is continuous (no holes or jumps). 5. The x axis is a horizontal asymptote (but only in one direction). 6. If a > 1, the graph is increasing (exponential growth). 7. If 0 < a < 1, the graph is decreasing (exponential decay). f(x) = a x for a > 1 y f(x) = a x for 0 < a < 1 y x x Ex15) Graph f(x) = 2 x, g(x) = 3 x, h(x) = ( ) x 1, and k(x) = 2 ( ) x 1 on the same graph. 3 22

23 Law of exponents If a and b are positive numbers and x and y are any real numbers, then 1. a x a y = a x+y 2. a x a y = ax y 3. a x = 1 a x 4. a 0 = 1 for a 0 5. (ab) x = a x b x 6. ( a b) x = a x b x 7. (a x ) y = a xy 8. a x = a y if and only if x = y 9. For x 0, a x = b x if and only if a = b The Natural Exponential Function Any positive number can be used as the base for an exponential function, but some bases are used more frequently then others. For example, 2, 10 and e. The number e is defined as the value that ( 1+ 1 n) n approaches as n becomes large, written as ( e = lim 1+ 1 n n n) The Natural Exponential Function is the exponential function f(x) = e x with base e. It is often referred to as the exponential function. Since 2 < e < 3, the graph of the natural exponential function lies between the graphs of y = 2 x and y = 3 x. 23

24 Ex16) Simplify ex+4 e 4 x Ex17) Solve each equation for x a) 9 x 1 = 3 1+x b) x 2 e x 5xe x = 0 24

25 Growth and Decay Applications Function of the form y = Ce kt where C is the initial amount, k is the relative growth(or decay) rate, and t is time. If k is positive, it is exponential growth model, if k is negative, it is exponential decay model. Ex18) The population of a particular city grows continuously at a relative growth rate of 5.4%. If 30,000 people currently live in the city, what will be the population in eight years? Ex19) The population of an undesirable city is modeled by p(t) = ce 0.09t where t represents the number of years since 1950, p(t) represents the population in the t th year, and c is a constant representing the population in a) If the city had a population of 20,000 in 1990, what is the value of c? b) Use the model to predict the population in

26 Compound Interest If a principal P (present value) is invested at an annual rate r (expressed as a decimal) compounded m times a year, then the amount A (future value) in the account at the end of t years is given by A = P(1+ r m )mt Ex20) If $5,000 is invested in an account paying 2.5% compounded monthly, how much will be in the account at the end of 10 years? Continuously Compounded Interest A = Pe rt where P is principal, r is the annual interest rate compounded continuoulsy (as a decimal), t is time, and A is the accumulated amount at the end of t years. Ex21) What amount will an account have after five years if $1,000 is invested at an annual rate of 3.25% compounded continuously? 26

27 Effective Yield If $1000 is invested at an annual rate of 9% compounded monthly, then at the end of a year there is ( F = $ ) 12 = $ in the account. This is the same amount that is obtainable if the same principal of $1000 is invested for one year at an annual rate of 9.381%. We call the rate 9.381% the effective annual yield. The 9% annual rate is often refered to as the nominal rate. If we let r eff be the effective annual yield for 1 year, then r eff must satisfy ( P 1+ m) r m = P(1+reff ) and thus r eff = ( 1+ r m) m 1 r eff = e r 1 is the effective annual yield compounded continuously. Ex22) You have been doing some research and have found that you can either invest your money at an annual interest rate of 3.55% compounded monthly or 3.50% compounded continuously. Which one would you choose? 27

28 Calculator Functions TVM Solver: We can use the TVM Solver on our calculator to solve problems involving compound interest. To access the Finance Menu, you need to press APPS > 1:Finance (Please note that if you have a plain TI-83, you need to press 2nd x 1 to access the Finance Menu). Below we define the inputs on the TVM Solver: N = the total number of compounding periods I% = interest rate (as a percentage) P V = present value (principal amount). Entered as a negative number if invested, a positive number if borrowed. PMT = payment amount (0 for this class) F V = future value (accummulated amount) P/Y = C/Y = the number of compounding periods per year. Move the cursor to the value you are solving for and hit ALPHA and the ENTER. Effective Yield: We use the C:Eff(option on the Finance Menu to compute the effective rate of interest. The inputs are as follows: EF F(annual interest rate as a percentage, the number of compounding periods per year) 28

29 Section 1.5 Logarithms Every exponential function f(x) = a x, with a > 0 and a 1, is a one-to-one function and therefore has an inverse function. The inverse function f 1 is called the Logarithmic function with base a and is denoted log a. Definition of the Logarithmic function Let a be a positive number with a 1. The Logarithmic function with base a, denoted by log a, is defined by y = log a x a y = x 29

30 Common Logarithm The logarithm with base 10 is called the Common logarithm and is denoted by omitting the base logx = log 10 x Natural Logarithm The logarithm with base e is called the Natural logarithm and is denoted by ln lnx = log e x The natural logarithm function y = lnx is the inverse function of the exponential function y = e x which is lnx = y e y = x Ex23) Solve for x, y, or b without a calculator a) log 3 x = 2 b) log b e 4 = 4 c) log 49 ( 1 7 ) = y 30

31 Properties of Logarithm 1. log a 1 = 0 a 0 = 1 2. log a a = 1 a 1 = a 3. log a a x = x 4. a log a x = x 5. log a b m = mlog a b 6. log a nb = 1 n log ab 7. log a xy = log a x+log a y ( ) x 8. log a = log y a x log a y 9. log a b = log cb log c a and log ab = 1 log b a 10. log a M = log a N M = N Ex24) Find the domain of each funtion a) f(x) = 2+log 5 x b) g(x) = log(x 3) 31

32 Ex25) Solve the following for x: a) log b x = 2 3 log b27+2log b 2 log b 3 b) logx+log(x 3) = 1 Ex26) Find the domain of f(x) = 7 2x 9 log 2 (4x 1). 32

33 Ex27) Evaluate log Ex28) Solve x = 2 for x rounded to four decimal places. Ex29) Given log b 3 = and log b 4 = , find the following a) log b 48 b) log b ( 16 b 2 ) 33

34 Ex30) How long will it take for the amount in an account to double if the money is compounded continuously at an interest rate of 2.5% per year? 34

35 Chapter 3 Limits and the Derivative Section 3.1 Limits Definitions We write lim f(x) = L x a and say the limit of f(x), as x approaches a, equals L if we can make the values of f(x) arbitrarily close to L (as close to L as we like) by taking x to be sufficiently close to a but not equal to a. This says that the value of f(x) get closer and closer to the number L as x gets closer and closer to the number a (from either side of a), but x a. Left-Hand Limit: We write lim x a f(x) = L if f(x) is close to L whenever x is close to, but to the left of a. Right-Hand Limit: We write lim x a +f(x) = L if f(x) is close to L whenever x is close to, but to the right of a. Existence of a Limit If the function f(x) is defined near x = a but not necessarily at x = a, then if and only if both the limits exist and are equal to the same number L. lim f(x) = L x a lim x a f(x) and lim x a +f(x) 35

36 Ex31) Use the graph below to find the following limits: a) lim x 3 f(x) = b) lim x 3 +f(x) = c) lim f(x) = x 3 d) lim x 1 f(x) = e) lim x 1 +f(x) = f) lim f(x) = x 1 g) lim x 4 f(x) = h) lim x 4 +f(x) = i) lim f(x) = x 4 j) lim x 2 f(x) = k) lim x 2 +f(x) = l) lim f(x) = x 36

37 x 1 Ex32) Use a tabel of values to estimate lim. Confirm your result graphically. x 1 x

38 Limit Laws Suppose that c is a constant and the limits exist. Then limf(x) and lim g(x) x a x a lim[f(x)+g(x)] = limf(x)+ lim g(x) x a x a x a lim[f(x) g(x)] = limf(x) lim g(x) x a x a x a lim[cf(x)] = clim f(x) x a x a lim[f(x) g(x)] = limf(x) lim g(x) x a x a x a 5. f(x) lim x a g(x) = lim x af(x) lim x a g(x) if lim x a g(x) lim x a [f(x)]n = [limf(x)] n x a lim x a lim c = c x a lim x = a x a lim x a xn = a n lim x a n x = n a n f(x) = n f(a) where n is a positive integer. (If n is even, we assume that lim x a f(x) > 0). 38

39 Ex33) Determine the follwing limits: a) lim x 2 10 b) lim x 3 x 2 +3x 2 c) lim x 3 3x+2 x 4 39

40 Limits of Polynomial and Rational Functions 1. lim x a f(x) = f(a), f any polynomial function 2. lim x a r(x) = r(a), r any rational function with nonzero denominator at x = a Ex34) Find each limit. a) lim x 3 x 3 4x 2 +2 b) lim x 2 x 2 1 x+2 x 2 +3 if x 3 c) If f(x) = x 4 if x > 3, find i) lim x 0 f(x) ii) lim x 3 f(x) iii) lim x 3 +f(x) iv) lim x 3 f(x) x 2 4 d) lim x 2 x+2 40

41 How to find limits algebraically lim x a f(x) Try plugging a into the function: 1. If you get a real number, that is your answer (unless you are dealing with a piecewise function) 2. If you get 0 (indeterminate form), algebraically manipulate (usually factor), cancel, and plug in 0 a again. 3. If you get none zero number 0 Ex35) Evaluate each limit: a) lim x 6 x+3 then the limit does not exist. b) lim x 2 2x 2 x 6 x 2 c) lim x 5 x 5 x 5 d) lim x 4 16 x 2 e) lim x x 2 41

42 Ex36) Evaluate lim x 4 x x+4 Ex37) If f(x) = 3x 2 f(x+h) f(x) +4x 7 find lim. h 0 h 42

43 Ex38) Evaluate and examine the behavior of f(x) = 7 x x 2 at x = 0. Definition. The vertical line x = a is a vertical asymptote for the graph of y = f(x) if lim x a f(x) = ± or lim +f(x) = ± x a 43

44 Locating Vertical Asymptotes of Rational Functions If f(x) = n(x) is a rational function, d(c) = 0 and n(c) 0, then the line x = c is a vertical d(x) asymptote of the graph of f. Ex39) Find vertical asymptote(s) of f(x) = x 3 x 2 4x+3. 44

45 Ex40) Find each limit. x 2 a) lim x 3 x+3 x 2 +x 2 b) lim x 2 x+2 45

46 Continuity A function f is continuous at the point x = a if all of the following are true: 1. f(a) is defined 2. lim x a f(x) exists. 3. lim x a f(x) = f(a) If one or more of the conditions are not met, we say f is discontinuous at x = a. Ex41) For what values of x is the function discontinous? Explain. Definition. A function is continous on an interval if it is continuous at each point on the interval. 46

47 Ex42) Find the intervals on which the following functions are continuous. a) f(x) = 3x 9 +4x 5 7 b) g(x) = 2 3x +4 c) h(x) = log 8 (x 3) 2 d) k(x) = x+2 x 2 3x 10 e) m(x) = x 5 x+2 if x 3 2x 2 if 3 < x 0 x x 4 if x > 0 47

48 Ex43) Find the value(s) of k that make f(x) continuous everywhere. 2x 5 if x 1 f(x) = x 2 +k if x > 1 48

49 Section 3.2 Average Rate of Change The average rate of change of y = f(x) with respect to x from a to b is the quotient change in y change in x = y x = f(b) f(a). b a For a linear function, the average rate of change is the slope of line For a nonlinear function, the average rate of change is the slope of the secant line from (a,f(a)) to (b,f(b)) 49

50 Or Then we let (b,f(b)) approach (a,f(a)) along the curve by letting b approach a. by or Definition. Instantaneous rate of change Given a function y = f(x), the instantaneous rate of change of y with respect to x at x = a is given if these limits exist. f(b) f(a) lim b a b a f(a+h) f(a) lim h 0 h 50

51 Definition. Average and Instantaneous Velocity Suppose s = s(t) describes the position of an object at time t. The average velocity from a to a+h is average velocity = s(a+h) s(a) h The instantaneous velocity (or simply velocity) v(a) at time a is if this limit exists. v(a) = lim h 0 (average velocity) = lim h 0 s(a+h) s(a) h Ex44) Find the slope of the tangent line to the graph of f(x) = x 2 +2x at the point (1,3). 51

52 Ex45) If a rock is thrown upward on the planet Mars with a velocity of 10 m/s, its height in meters t seconds later is given by y = 10t 1.86t 2. a) Find the average velocity over the given time intervals: i) [1,1.1] ii) [1, 1.01] iii) [1, 1.001] b) Use part a) to estimate the instantaneous velocity when t = 1. c) Find the exact value of the instantaneous velocity when t = 1. 52

53 Ex46) Estimate the instantaneous rate of change of f(x) = x 2 at x = 1, and find exactly. 53

54 Ex47) Find the equation of the tangent line of f(x) = 5+x at x = 20. Ex48) For the function shown below, at what labeled points is the instantaneous rate of change a) positive b) negative c) zero d) most negative y x 54

55 Section 3.3 The Derivative Definition. Derivative The derivative of a function f at a number a, denoted by f (a), is or if this limit exists. f (a) = lim h 0 f(a+h) f(a) h f (a) = lim x a f(x) f(a) x a Interpretations of the Derivative The derivative has various applications and interpretations, including the following; 1. Slope of the tangent line: f (a) is the slope of the line tangent to the graph of f at the point (a,f(a)). 2. Instantaneous rate of change: f (a) is the instantaneous rate of change of y = f(x) at x = a. 3. Velocity: If f(x) is the position of a moving object at time x, then v = f (a) is the velocity of the object at time x = a. 55

56 Ex49) Given f(x) = x 2 8x + 9, find the derivative of f at x = 2 (use the definition) and the equation of tangent line at x = 2. 56

57 Definition. Derivative of function The derivative of f is defined as f (x) = lim h 0 f(x+h) f(x) h Other Notations for the derivative of y = f(x): f (x) = y = dy dx = df dx = d dx f(x) Ex50) Find f (x) if f(x) = 1 x 2+x 57

58 Definition. Differentiable A function f is differentiable at a if f (a) exists. Theorem If f is differentiable at a, then f is continuous at a. How can a function fail to be differentiable? 1. If the graph of a function f has a corner or kink in it, then the graph of f has no tangent at that point and f is not differentiable there. 2. If f is not continuous at a, then f is not differentiable at a. 3. The curve has a vertical tangent line when x = a, f is not differentiable at a. Ex51) The graph of f is given. State, with reasons, the numbers at which f is not differentiable. 58

59 Ex52) Where is f(x) = x 2 4 not differentiable. 1. When f(x) is decreasing, f (x) < When f(x) is increasing, f (x) > When f(x) is constant, f (x) = 0. 59

60 Ex53) Given the graph of f(x) below, sketch the graph of the derivative. a) b) 60

61 Ex54) Find f (x) if f(x) = 4 x 61