Cost function
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This presentation introduces linear equations. It defines key terms like slope and y-intercept. Examples are provided to illustrate how to calculate the slope between two points and write the equation of a line given two points. Applications of linear equations in business are also discussed, including cost functions, break-even analysis and calculating mark-up. Homework problems are assigned from a Bowen textbook involving these concepts.
Cost function
- 1. Welcome to the presentation on LINEAR EQUATION Presented by Nasim Khan Assistant Professor Department of Mathematics Bangladesh University of Business and Technology, Dhaka.
- 2. July 12, 2014 NK,Assistant Professor of Mathematics,BUBT. 2 LINEAR EQUATION TOPICS Slope Equation of line Intercept Slope-intercept form Application of linear equation
- 3. July 12, 2014 NK,Assistant Professor of Mathematics,BUBT. 3 Linear equation Rule: If (x1,y1) and (x2 ,y2) be two points then Slope , m = y2 – y1 / x2 – x1 Example:- If (12,- 5 ) and (3,6) are two points then Slope, m = 6 – ( - 5 ) / 3 – 12 =11/(-8 ) Book- Bowen Page- 25 to26 Same Problems: 9 to22
- 4. July 12, 2014 NK,Assistant Professor of Mathematics,BUBT. 4 Linear equation Rule: If (x1,y1) and (x2 ,y2) be two points then the Line equation is 21 1 yy yy − − = 21 1 xx xx − −
- 5. July 12, 2014 NK,Assistant Professor of Mathematics,BUBT. 5 Linear equation Example:- If (-2, -4)and (-1, 5) are two points then, Find the equation of line. Solution: We know ,The equation of a line is = 21 1 yy yy − − 21 1 xx xx − −
- 6. July 12, 2014 NK,Assistant Professor of Mathematics,BUBT. 6 Linear equation ⇒ = ⇒ -9(x+ 2)= -1(y+ 4) ⇒ - 9x – 18= - y – 4 ⇒ - 9x + y = 14 Which is our required line equation. Book- Bowen Page- 36 Problem: 13 to 24 54 4 −− +y 12 2 +− +x
- 7. July 12, 2014 NK,Assistant Professor of Mathematics,BUBT. 7 Linear equation Rule: Slope-Intercept Form y = m x + b Where m is the slope and b is the y- intercept Example:-Write2y +3x =18 in slope-intercept Form and state the value of the slope and the y- intercept.
- 8. July 12, 2014 NK,Assistant Professor of Mathematics,BUBT. 8 Linear equation Solution: Given, the equationof the line is 2y + 3x=18 ⇒ 2y = -3x +18 ⇒ y = -3x/2 + 9 Which is our required form. Therefore, slope of the line is -3/2 and y-intercept is 9 . Book- Bowen Page - 37 Problems: 47 to 50
- 9. July 12, 2014 NK,Assistant Professor of Mathematics,BUBT. 9 Linear equation Problem: If 5y - 2x = 30 is a linear equation then find (a) Write the slope-intercept form (b) What is the slope of the line? (c) What is the y-intercept? (d) What is the x-intercept? (e) Write the co-ordinate of the points of the equation?
- 10. July 12, 2014 NK,Assistant Professor of Mathematics,BUBT. 10 Linear equation Solution: (a) Given, The line equation, 5y - 2x = 30 --- --- (1) ⇒ 5y = 2x + 30 ⇒ y = 2x/5 + 6 Which is slope-intercept form. (b) The slope of line is 2/5 (c) The y-intercept is 6 (d) Putting y = 0 in equation (1),then we have 5y – 2. 0 =30 ⇒ y = 30 So the x-intercept is 30. (e) The co-ordinates of the points is (0,6) and (30 ,0)
- 11. July 12, 2014 NK,Assistant Professor of Mathematics,BUBT. 11 Application of linear equation Cost function Average cost Marginal cost What is cost function? If x is the quantity produced of a certain goods by a firm at total cost c, we can write the total cost function C = c (x) There are two part of Total cost (i) Fixed cost which is independent of x (ii) Variable cost which is dependent of x Therefore, Total cost = Variable cost + Fixed cost ⇒ TC = VC + FC
- 12. July 12, 2014 NK,Assistant Professor of Mathematics,BUBT. 12 Application of linear equation What is average cost? If total cost (c) is represented as a function of output x, the average cost (AC) represents the cost per unit of production. i.e. Average cost = Total cost / Total quantity = c / x What is Marginal cost ? Marginal cost represents the change in the total cost for each additional unit of production. i.e. Marginal cost = dc / dx Marginal cost = Variable cost per unit = Slope
- 13. July 12, 2014 NK,Assistant Professor of Mathematics,BUBT. 13 Application of linear equation TOPICS Revenue function Break-even Mark Up Margin Break- even Chart
- 14. July 12, 2014 NK,Assistant Professor of Mathematics,BUBT. 14 Revenue function Revenue is the amount of money derived from the sale of a product and depends upon the price of a product and the quantity of the product that is actually sold. If q is the demand for the output of a firm at price p, then total revenue (R) collected by the firm is R(q) = pq Revenue function: R (q)=(Selling price per unit) (Number of units sold) = p q = Sales volumes (dollars)
- 15. July 12, 2014 NK,Assistant Professor of Mathematics,BUBT. 15 Application of linear equation Break-even: It means neither make a profit nor suffer a loss. It’s simply means that revenue must equal costs. So at break even, Revenue = Cost i.e. Profit = 0. Mark Up: It is the difference between the retail price and the cost. Therefore, Mark up = Retail price – cost
- 16. July 12, 2014 NK,Assistant Professor of Mathematics,BUBT. 16 Application of linear equation Example:- Suppose an stem costs $130 is price to sell at $200, Find Mark up ? Solution - We know, Mark up = Retail price –cost = $(200 - 130) = $70 Mark up percentage on cost = 100% = 54% Mark up percentage on retail price = 100% = 35%= Margin Margin: Mark up / Retail price = 70/200 = 0.35 =35/100 = 35% Margin 35% that means it is the markup percentage on retail price. So 35% margin means (100- 35)% = 65% cost 130 70 200 70
- 17. July 12, 2014 NK,Assistant Professor of Mathematics,BUBT. 17 Application of linear equation BREAK-EVEN CHART R&C R FCL Q NSO L T M I A FC VC FC VC FC TC (Number of units)
- 18. July 12, 2014 NK,Assistant Professor of Mathematics,BUBT. 18 Application of linear equation CR & C O q R 60,000 20,000 1,00,000 FCL BEP FC Profit Loss ( 20,000 , 100000 ) BREAK-EVEN CHART BOOK- BOWEN, Page- 64 to 65, Problems- 13,14,15,16,29,30,31,32.
- 19. July 12, 2014 NK,Assistant Professor of Mathematics,BUBT. 19 Home Work Problems: True/ False (1 to 12) & (19 to 28) 13,14,15,16,29,30,31,32 Page : 64 – 65 Book : Bowen