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MATHEMATICS OF BUSINESS By Purwanto Pur71wanto@yahoo 081380619254

MATHEMATICS OF BUSINESS By Purwanto Pur71wanto@yahoo.com 081380619254. References :. Barnett, Ziegler, Byleen, COLLEGE MATHEMATICS For Business, Economics, Life Sciences, and Social Sciences, 9 th Edition, Prentice-Hall, Inc.

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MATHEMATICS OF BUSINESS By Purwanto Pur71wanto@yahoo 081380619254

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  1. MATHEMATICS OF BUSINESSBy PurwantoPur71wanto@yahoo.com081380619254

  2. 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

  3. A Beginning Library of Elementary Functions

  4. 1-1 Functions • Cartesian Coordinate System • Graphing : Point By Point Sketch a graph of y = 9 – x2

  5. Guidelines for graphing functions

  6. 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

  7. 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

  8. 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

  9. -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

  10. 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 10

  11. Exercises Table 1. Price - Demand Table 2. Revenue 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)

  12. Plot the data in table 1. Then sketch a graph of the price-demand function in the same coordinates system. • What is the company’s revenue function for this camera, and what is the domain of this function ? • Complete table 2, computing revenues to the nearest million dollars. • Plot the data in table 2.Then sketch a graph of the revenue function using these points.

  13. Table 3. Cost Data 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)

  14. Plot the data in table 3. Then sketch a graph of equation (2) in the same coordinate system. • What is the company’s profit function for this camera, and what is its domain ? • Complete table 4, computing profits to the nearest million dollars. • Plot the points from part (c). Then sketch a graph of the profit function through these points.

  15. Applications - Cost function: C = (fixed costs) + (variable costs) • C = a + bx • E. g. C = 3000 + 200x • This is an example of a function defined by an equation, with • independent variable x • dependent variable C

  16. 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.

  17. 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.

  18. Some common business functions • Cost: • C = a + bx , the cost of producing x items • The “parameters” (numbers that are specific to a particular business situation), are a and b. • Example: C = 100 + 0.50x • fixed 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

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

  20. 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:

  21. Note: lowered prices  greater demand  increased sales • BUT the lower prices eventually overtake increased sales, ultimately decreasing revenues. • Profit: • Profit is, of course, Revenue – Cost: P = R – C • For 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 - 100 • Here, Profit is written in terms of Demand (x).

  22. 1-2 Transformations of Graphs Horizontal shifts reference: f(x) = x2

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

  24. shift right 2: f(x - 2) = (x - 2)2

  25. Vertical shifts reference: f(x) = x2

  26. shift up 2: f(x) + 2 = x2 + 2

  27. shift down 2: f(x) - 2 = x2 - 2

  28. Linear Functions and Straight Lines • Linear functions • are of the form f(x) = mx + b • e.g f(x) = -3x + 4 (m = -3, b = 4) • called linear because they graph as straight lines • sometimes written y = mx + b (slope-intercept form)

  29. Graphing a linear function using intercept method Example: 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:

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

  31. 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:

  32. Slope of a line

  33. Computing slope, given two points:

  34. In general, given two points (x1, y1) and (x2, y2), the slope of the line passing through them is m = (slope formula)

  35. Kinds of slope

  36. When the slope is zero, we have a constant function. When a function is written in slope-intercept form f(x) = mx + b or y = mx + b the coefficient m of x will be the slope the constant term b will be the y-intercept e.g. the graph of f(x) = - ¾ x + 12 has slope -3/4, and y-intercept 12.

  37. Interpretation of slope The following graph represents the value of an investment (in $’s) over time (in years):

  38. 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 units X is the x-axis unit

  39. Point-slope form of a line This 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 points Example: points: (1, 3) (3, 6)

  40. 1-3 Quadratic Functions and Their Graphs • Quadratic function: has a squared term, but none of higher degree • standard form of the quadratic function: • f(x) = ax2 + bx + c (a  0) • vertex = • vertex form of the quadratic function: • f(x) = a(x - h)2 + k • vertex = (h, k) • parabola: the graph of a quadratic function

  41. The anatomy of a parabola The role of a: if a > 0, parabola opens upward if a < 0, parabola opens downward

  42. Finding the vertex Example: f(x) = (x + 1)2 - 3 It is already in vertex form f(x)= (x - h)2 + k so vertex = (h, k) = (-1, -3) Example: f(x) = x2 + 2x – 2 x-coordinate of vertex = -b/2a = -2/2 = -1 y-coordinate of vertex = f( -1 ) = (-1)2 + 2(-1) –2 = -3 so 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

  43. Sketching the graph of a quadratic function • Example: R(x) = x(2000 – 60x) 1 ≤ x ≤ 25 • 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)

  44. (4) graph points and draw

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

  46. 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

  47. 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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