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Linear functions

Linear functions

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Linear functions

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  1. Linear functions Functions in general Linear functions C. Linear (in)equalities Handbook: Haeussler E., Paul R., Wood R.(2011). Introductory Mathematical Analysis for business, economics and life and social sciences. Pearson education

  2. Functions in general1. definition

  3. A. Functions in general Introduction In every day speech we often hear economists say things like “ interest rates are a function of oil prices”, “pension income is a function of years worked” Sometimes such usage agrees with mathematical usage, but not always. (Handbook: Section 2.1 p80, paragraph 1-2)

  4. A. Functions in general Example Taxidriver What does a taxi ride cost me with company A? • Base price: 5 Euro • Per kilometer: 2 Euro Price of a 7 km ride?

  5. A. Functions in general Example Taxidriver What does a taxi ride cost me with company A? • Base price: 5 Euro • Per kilometer: 2 Euro Price of an x km ride?

  6. A. Functions in general Definition • x and y : VARIABLES (length of ride in km) (price of ride in euro) • y depends on x: INPUT  OUTPUT x y y: DEPENDENT VARIABLE x:INDEPENDENT VARIABLE Function: rule that assigns to each input at most 1 output (Section 2.1 p81, last 4 paragraphs)

  7. A. Functions in general Definition • We say: y is FUNCTION of x, or in short f of x • We denote: y(x) or y=f(x) • Outputs are also calledfunction values (Handbook: Section 2.1 p82)

  8. Functions in general1. definition2. Three representations

  9. A. Functions in general Three representations First way: Most concrete form! Through a TABLE, e.g. for y = 2x + 5: But: limited number of values  no overall picture

  10. A. Functions in general Three representations Second way: Most concentrated form! Through the EQUATION, e.g. y = 2x + 5. formula y = 2x + 5: EQUATION OF THE FUNCTION

  11. A. Functions in general Three representations y 7 6 5 4 3 2 1 -1 x y 0 5 1 7 -4 -3 -2 -1 0 1 2 3 4 x Third way: Most visual form! Through the GRAPH rectangular coordinate system: x-coordinate, y-coordinate (Handbook: Section 2.5 p99)

  12. A. Functions in general Three representations Third way: Most visual form! Through the GRAPH e.g. for y = 2x + 5: STRAIGHT LINE! Note: In this example, the graph is a only a part of a straight line

  13. A. Functions in general Exercises p q 10 640 12 560 14 480 The demand qof a product depends on the price p. For a local pizza parlor some weekly demands and prices are given Remark: this table is called a demand schedule (a) What is the input variable? What is the output variable ? (b) Indicate the points in the table on a graph (Handbook: Section 2.1 p85 – example 5)

  14. A. Functions in general Exercises Suppose a 180-pound man drinks four beers in quick succession. The graph shows the blood alcohol concentration (BAC) as a function of the time. (a) Input ? Output ? (b) How much BAC is in the blood after 5 hours ? (c) What will be the maximal BAC ? After how much time, will this maximum be attained ? (d) What’s the behavior of the BAC as a function of time ? (Section 2.1 p79)

  15. A. Functions in general Summary - Definition input x, output y - 3 representations : table equation y=f(x) graph in rectangular coordinate system Extra: Handbook - Problems 2.1: Ex 17, 48, 50

  16. B. Linear Functions1. equation

  17. B. Linear functions Example Taxidriver y = 5 + 2x FIXED PART + VARIABLE PART FIXED PART + MULTIPLE OF INDEPENDENT VARIABLE FIXED PART + PART PROPORTIONAL TO THE INDEPENDENT VARIABLE

  18. B. Linear functions Example Taxidriver • Examples: cost of a ride with company B, C? B  base price: 4.5 euro, price per km: 2.1 euro C  base price 8 euro, price per km: 0.5 euro y = 4.50 + 2.10x; y = 8 + 0.5x; • In general: y = base price + price per km x y = b + mx

  19. B. Linear functions Equation A function f is a linear function if and only if f(x) can be written in the form f(x)=y=mx + b where m, b are constants. Caution: m and bFIXED: parameters x and y: VARIABLES! (Section 3.1 p138)

  20. B. Linear functions Applications • Cost y to purchase a car of 20 000 Euro and drive it for x km, if the costs amount to 0.8 Euro per km? y = 20 000 + 0.8x hence … y = mx + b! • Production cost c to produce q units, if the fixed cost is 3 and the production cost is 0.2 per unit? c = 3 + 0.2q hence y = mx + b!

  21. B. Linear functions Applications • The demand qof a product depends on the price p and vice versa. For a local pizza parlor the function is given by • p=26-q/40 • Note: The function p(q) is called the • demand function by economists

  22. B. Linear functions Exersises Rachel has saved $7250 for college expenses. She plans to spend $600 a month from this account. Write an equation to represent the situation.

  23. B. Linear functions Exersises For a local pizza parlor the weekly demand function Is given by p=26-q/40. (a) What will be the revenue for the pizza parlor if 400 pizza’s are ordered ? (b) Express the revenue as a function of the demand q. Note: Demand functions are not always linear ! !! Not all functions are first degree functions

  24. B. Linear Functions1. Equation2. Graph

  25. B. Linear functions Example Taxidriver y = 2x + 5 • The graph of a linear function with equation y=mx +b is • a STRAIGHT LINE

  26. B. Linear Functions1. Equation2. Graph3. Significance parameters b, m

  27. B. Linear functions Example Taxidriver A: y = 2x + 5 B: y = 4.5x + 2.1 C: y = 8x + 0.5 What’s the effect of the different values for m ? For b ?

  28. B. Linear functions Significance of the parameter b • Taxi company A: y = 2x + 5. Here b = 5: the base price. • Numerically: b can be considered as the VALUE OF y WHEN x = 0. • graphically: b shows where the graph cuts the Y-axis: Y-INTERCEPT

  29. B. Linear functions Significance of the parameter m • Taxi company A: y = 2x + 5, m = 2: the price per km. • Numerically: m is CHANGE OF y WHEN x IS INCREASED BY 1 INPUT OUTPUT x y 3 11 4 13 x= 1 y= 2 m is the RATE OF CHANGE of the linear function

  30. B. Linear functions Significance of the parameter m • Graphically: if x is increased by 1 unit, y is increased by m units m is the SLOPE of the straight line

  31. B. Linear functions Significance of the parameter m • Taxi company A: y = 2x + 5, m = 2: the price per km. • If x is increased by e.g. 3 (the ride is 3 km longer), y will be increased by 2  3 = 6 (we have to pay 6 Euro more). INPUT OUTPUT x y 3 11 6 17 • Always: x= 3 y= 3x2=6 y= mx (INCREASE FORMULA)

  32. B. Linear functions Significance of the parameter m if x is increased by x units, y is increased by m x units Increase formula:

  33. B. Linear functions Significance of the parameters b and m • The graph of a linear function with equation y=mx +b is • a STRAIGHT LINE • with y-intercept b • and slope m The equation y=mx +b is called the slope-intercept form of the line with slope m and intercept b. It is also called an explicit equation of the line.

  34. B. Linear functions Exercises • The cost c in terms of the quantity q produced of a good is given by c = 200 + 15 q. • Give a formula for the change of cost Δc. • Use this formula to determine how the cost changes when the production of the good is increased by 12 units. • Use this formula to determine how the cost changes when the production of the good is decreased by 2 units.

  35. B. Linear functions Supplementary exercises • Exercise 1 • Exercise 2 A, B, D (only the indicated points are to be used!)

  36. B. Linear functions Exercises Exercise 2

  37. B. Linear functions Slope of the line m Consider again supplementary Exercise 2 - Compare the slopes of lines A and D - What is the slope of line C ? - Compare the slopes of line A and B - Compare the slopes of lines D and E (Section 3.1)

  38. B. Linear functions Slope of the line m (Section 3.1 p128-129) Sign of m determines whether the linear function is - increasing / constant(!!) / decreasing • Note: what about a vertical line ? (Section 3.1 p131- Example 6)

  39. B. Linear functions Slope of the line m Size of m determines how steep the line is Note: the slope and thus the steepness of the line depends on the scale of the axes (Section 3.1 p128-129)

  40. B. Linear functions Parallel lines Perpendicular lines (Section 3.1 p128-129) Parallel lines have the same slope (Section 3.1 p133-134) Two lines with slopes m1 and m2 are perpendicular to each other if and only if Note: any horizontal line and any vertical line are perpendicular to each other

  41. B. Linear functions Slope of the line m Remember: y= mx (INCREASE FORMULA). Therefore: (Section 3.1 p128)

  42. B. Linear Functions1. Equation2. Graph3. Significance parameters b, m4. Determining a line based on the slope and a point / two points

  43. B. Linear functions Slope of the line m Slope of a straight line given by two points: (Section 3.1 p128)

  44. B. Linear functions Exercises John purchased a new car in 2001 for $32000. In 2004, he sold it to a friend for $26000. Find and interpret the slope.

  45. B. Linear functions Equation of lines A straight line through a given point (x0, y0)and with a given slope m satisfies the equation: This equation is called the point- slope form of the line Remember: The equation y=mx+b is called the slope-intercept form of the line (Section 3.1 p129-131)

  46. B. Linear functions Exercises John purchased a new car in 2001 for $32000. In 2004, he sold it to a friend for $26000. Find the equation that expresses the price as a function of time. You may assume that the price is a linear function of time Supplementary exercises • Exercise 3 • Exercise 4

  47. B. Linear Functions1. Equation2. Graph3. Significance parameters b, m4. Determining a line based on the slope and a point / two points5. Implicit equation

  48. B. Linear functions Equation of lines Note that e.g. the vertical line with equation x=2 can not be written in the slope-intercept form nor in the slope-point form The equation of a straight line can always be written using the general linear form Ax+By+C=0 (A and B not both 0). This is also called an implicit equation. (Section 3.1 p129-131)

  49. B. Linear functions Equation of lines Remember : Point-slope form Slope intercept form y=m x + b General linear from Ax + By + C = 0 note: vertical line: x=a horizontal line: y=b (Section 3.1 p129-131)

  50. B. Linear functions Exercises Find an equation of the line that has slope 2 and passes through (1, -3) using the - Point-slope form - Slope-intercept form - General linear form Supplementary exercises: • Exercise 5