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MATHEMATICS OF

BUSINESS By

PurwantoPur71wanto@yahoo.com

081380619254

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References :

• Barnett, Ziegler, Byleen, COLLEGE MATHEMATICS For Business, Economics, Life Sciences, and Social Sciences, 9th Edition, Prentice-Hall, Inc.

• Haeussler, JR, Paul, Wood, Introductory MATHEMATICAL ANALYSIS For Business, Economics, Life Sciences, and Social Sciences, 12th Edition, Pearson Education, Inc.

• Dumairy, Matematika Terapan untuk Bisnis dan Ekonomi, Edisi Kedua, BPFE Yogyakarta

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A Beginning Library of Elementary Functions

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1-1 Functions

• Cartesian Coordinate System

• Graphing : Point By Point

Sketch a graph of y = 9 – x2

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Guidelines for graphing functions

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Both axes must be labeled according to the names of the variables given by the problem: independent variable (often “x”) on the horizontal axis dependent variable (often “y”) on the vertical axis if the graph is the graph of a named function, there is no dependent variable; instead, the name of the function is used; e.g. "f(x)", as shown above

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• Function Notation

The symbol f(x)Exercises :

Using function notation for f(x) = x2 – 2x + 7, find :

a. f(a)

b. f(a + h)

c. f(a + h) - f(a)

h

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• Applications

- Cost function: C = (fixed costs) + (variable costs) C = a + bx

- Price-demand: p = m - nx

x is the number of items that can be sold for a price of $p per item

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- Revenue: Revenue will be (#items sold) x (price per item) R = xp where x = #items sold p = price per item

- Profit:Profit is, of course, Revenue – Cost: P = R – C

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• Exercises

A manufacturer of a popular automatic camera wholesales the camera to retail outlets throughout the United States. Using statistical methods, the financial department in the company produced the price-demand data in Table 1, where p is the wholesale price per camera at which x million cameras are sold. Notice that as the price goes down, the number sold goes up

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x (millions) R(x)

(millions)1369

1215

90

Table 2. Revenue

x (millions)

p($)

2

5

8

12

87

68

53

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Table 1. Price - Demand

Using special analytical techniques, an analyst arrived at the following price-demand function that models the table 1 data :

p(x) = 94.8 – 5x 1 ≤ x ≤ 15 (1)

Exercises

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a. Plot the data in table 1. Then sketch a graph of the price-demand function in the same coordinates system.

b. What is the company’s revenue function for this camera, and what is the domain of this function ?

c. Complete table 2, computing revenues to the nearest million dollars.

d. Plot the data in table 2.Then sketch a graph of the revenue function using these points.

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x (millions)

C(x)

(millions)

1

5

8

12

175

260

305

395

Table 3. Cost Data

x (millions)

P(x)(millions)

1369

1215

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Table 4. Profit

Using special analytical techniques (regression analysis), an analyst produced the following cost function the model the data :

C(x) = 156 + 19.7x 1 ≤ x ≤ 15 (2)

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a. Plot the data in table 3. Then sketch a graph of equation (2) in the same coordinate system.

b. What is the company’s profit function for this camera, and what is its domain ?

c. Complete table 4, computing profits to the nearest million dollars.

d. Plot the points from part (c). Then sketch a graph of the profit function through these points.

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• Applications

- Cost function: C = (fixed costs) + (variable costs) C = a + bx

E. g. C = 3000 + 200xThis is an example of a function defined by an equation, with independent variable x dependent variable C

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Alternatively, the Cost function could have been given as

C(x) = 3000 + 200x

This is a function defined using functional notation. Here, the “C” is the name of a function, not the name of a variable, as in the prior example.

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The independent variable is still x, but there is no dependent variable, so you can label the vertical axis with “C(x)”.

Using this notations, we can write:

C(1000) = 3000 + 200(1000) = 203000

That is, the cost of producing 1000 units is $203,000.00

The 1000 is called an input, with corresponding output 203000.

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Some common business functions • Cost: C = a + bx , the cost of producing x itemsThe “parameters” (numbers that are specific to a particular business situation), are a and b.

Example: C = 100 + 0.50xfixed cost = $100 cost per item produced = $ 0.50

• Price-demand: p = m - nx x is the number of items that can be sold for a price of $p per item

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Example: p(x)= 1 - 0.0001x:

Demand

Price

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• Revenue: Revenue will be (#items sold) x (price per item) R = xp where x = #items sold p = price per item

For our example: R(x) = xp = x(1 - 0.0001x)Here’s the Revenue graph for our example:

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Demand

Revenue

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Note: lowered prices greater demand increased salesBUT the lower prices eventually overtake increased sales, ultimately decreasing revenues.

• Profit:Profit is, of course, Revenue – Cost: P = R – CFor our example: C(x) = 100 + 0.50x R(x) = x(1 - 0.0001x)P(x) = x(1 - 0.0001x) – (100 + 0.50x) = -0.0001x2 + 0.50x - 100Here, Profit is written in terms of Demand (x).

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1-2 Transformations of Graphs

Horizontal shifts

reference:

f(x) = x2

x

y

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x

y

shift left 2: f(x + 2) = (x + 2)2

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shift right 2:

f(x - 2) = (x - 2)2

x

y

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Vertical shifts

reference:

f(x) = x2

x

y

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shift up 2:

f(x) + 2 = x2 + 2

x

y

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shift down 2:

f(x) - 2 = x2 - 2

x

y

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Linear Functions and Straight Lines

Linear functionsare of the form f(x) = mx + be.g f(x) = -3x + 4 (m = -3, b = 4)called linear because they graph as straight linessometimes written y = mx + b (slope-intercept form)

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Graphing a linear function using intercept methodExample: or f(x) = 2x + 4(1) convert to equation form: y = 2x + 4(2) Find intercepts: set x = 0, solve to get y-intercept = 4

set y = 0, solve to get x-intercept = -2(3) Plot the intercepts and draw the line:

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x

y

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Graphing a function having restricted domainMost real-world functions will have restricted domains, e.g.A = 6t + 10, 0 ≤ t ≤ 100The “0 ≤ t ≤ 100” is a domain restriction, meaning that the function is valid only for values of t between 0 and 100, inclusive.

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To graph it, calculate the points at the extreme left and right:if t = 0, A = 10 point (0, 10)if t = 100, A = 610 point (100, 610)Graph the points and draw the line:

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t

A

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Slope of a line

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Computing slope, given two points:

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In general, given two points (x1, y1)

and (x2, y2), the slope of the line

passing through them is

m = (slope formula)12

12

xx

yy

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Kinds of slope

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When the slope is zero, we have a constant function.When a function is written in slope-intercept formf(x) = mx + b or

y = mx + bthe coefficient m of x will be the slopethe constant term b will be the y-intercepte.g. the graph of f(x) = - ¾ x + 12 has slope -3/4, and y-intercept 12.

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Interpretation of slopeThe following graph represents the value of an investment (in $’s) over time (in years):

x

1

100

slope = rate = 100 $/yr

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As you can see, it is a linear function, and has slope = 100.By looking at the graph, you can see that the investment grows by $100/year, so the interpretation of “slope = 100” for this linear function is:“The investment increases by $100 per year ($100/year)”Notice the form of this statement: “Y per X” or “Y/X”Y is the slope expressed in y-axis unitsX is the x-axis unit

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Point-slope form of a lineThis form is used to find the equation of a line when you know a point (x1, y1) on the line, and its slope m:y – y1 = m(x – x1) (point-slope form)Finding the equation of a line, given two pointsExample: points: (1, 3) (3, 6)

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1-3 Quadratic Functions and Their GraphsQuadratic function: has a squared term, but none of higher degreestandard form of the quadratic function:•f(x) = ax2 + bx + c (a 0)•vertex =

2a

bf,

2a

b

vertex form of the quadratic function:•f(x) = a(x - h)2 + k•vertex = (h, k)parabola: the graph of a quadratic function

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The anatomy of a parabola

The role of a:if a > 0, parabola opens upwardif a < 0, parabola opens downward

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Finding the vertexExample: f(x) = (x + 1)2 - 3It is already in vertex form f(x)= (x - h)2 + kso vertex = (h, k) = (-1, -3)Example: f(x) = x2 + 2x – 2

x-coordinate of vertex = -b/2a = -2/2 = -1y-coordinate of vertex = f( -1 ) = (-1)2 + 2(-1) –2 = -3so vertex = (-1, -3)Note: this is not the way shown in the book (i.e. by completing the square), but is far superior to it

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Sketching the graph of a quadratic functionExample: R(x) = x(2000 – 60x) 1 ≤ x ≤ 25(1) write it in standard form: R(x) = 2000x – 60x2

(2) find the vertex: x-coordinate = -b/2a = -2000/-120 = 50/3 = 16.67 y-coordinate = 50/3(2000 – 60(50/3)) = 16667(3) find the points at extreme left and right of

range : R(1) = 1940 point on graph is (1, 1940) R(25) = 12500 point on graph is (25,12500)

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(4) graph points and draw

x

R(x)

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Maximum and minimum of a quadratic functionHere’s your familiar parabola (graph of a quadratic function):

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Maximum: • refers to the largest value that f(x) can ever

have for this example, it is 4• maximum is second coordinate of the vertex

point we say that "f attains its maximum of 4 for x = 1" f has a maximum value because it opens down

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The following parabola opens up, therefore doesn't have a maximum, but rather a minimum:

Minimum: this parabola attains its minimum of -3 for x = 5

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Finding maximum or minimumfor a quadratic function

Example: When a cannon is fired at a certain angle, the distance h (in meters) of the shell above the ground t seconds after firing is given by the formula

h(t) = -4.9t2 + 24t + 5Find the maximum height attained by the shell.

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Notes:h(t) is a quadratic function (parabola) has a vertex!for this parabola, a = - 4.9 and b = 24a < 0 the parabola opens downward has a max value

The answer: The shell reaches its maximum height of 34.4 meters 2.45 seconds after the cannon is fired.

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Break-even analysis

Example: R(x) = x(2000 – 60x) 1 ≤ x ≤ 25C(x) = 5000 + 500x 1 ≤ x ≤ 25

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Graphing them on the same system, we see:

x

R(x), C(x)

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A break-even point is a production level in response to demand (x) for which Revenue = CostGraphically, this means x-values where the graphs intersect.Algebraically, we solve the equation Revenue = Costx(2000 – 60x) = 5000 + 500xto get x = 4 or 21 as production levels where we break even.