Mathematics Coursework - 825 Words

Mathematics (Coursework Sample) Content: Question 1Costs = C(n) = 220n + 202,500; Revenue = R(n) = -3n+2020na) Break-even point (BEP)At the BEP, costs are equal to the revenues such that:C(n) = R(n)220n + 202,500 = -3n+2020n3n2 2020n + 220n + 202500 = 03n2 1800n + 202500 = 0We use the following equation to solve for n:
Where a = 3, b = -1800, and c = 202500



Thus, the break-even points are at: n = 150 air conditioners and n = 450 air conditionersb) Maximize ProfitThe profit is obtained as the difference between revenue and costsThus, P(n) = R(n) C(n)P(n) = (-3n + 2020n) (220n + 202,500)P(n) = -3n + 2020n 220n 202500P(n) = -3n + 1800n 202500P(n) = -6n + 1800The maximum profit is obtained at: -6n + 1800 = 06n = 1800n = 300Thus, the maximum profit is obtained at n = 300 air conditionersc) The maximum profit P(n) = -3n + 1800n 202500P(n) = -3(300) + 1800(300) 202500 = $67,500Question 2A graph of f(x) = 3x 7 if x and f(x) = - x 1 if x 2Plot generated at: http://rechneronline.de/function-graphs/Ques tion 3The difference quotient of g(x) = 2x- 3x +1 is obtained as:Where: and Thus, the difference quotient is:The difference quotient is: 4x + 2h 3Question 4f(x) is a piecewise function given by: In order to check whether f(x) is continuous at x = 4, we proceed as follows:f is defined at 4lim x 4- = -2 and lim x 4+ = -2 Thus, lim x 4 f(x) = f(4)In this case, the function f(x) is continuous at x = 4.Question 5The average rate of change of the function g(x) = 7x 2 between x = 3 and x = 5 is obtained as:We have:x1 = 3y1 = 7(3) 2 = 19Thus, (x1, y1) = (3, 19)x2 = 5y2 = 7(5) 2 = 33Thus, (x2, y2) = (5, 33)The average rate of change is obtained as the slope in the following manner:
The average rate of change of g(x) = 7x 2 between x = 3 and x = 5 is 7.The average rate of change of the function g(x) = 7x 2 between x = 3 and x = 4 is obtained as:We have:x1 = 3y1 = 7(3) 2 = 19Thus, (x1, y1) = (3, 19)x2 = 4y2 = 7(4) 2 = 26Thus, (x2, y2) = (4, 26)The average rate of change is obtained as the slope in the following manner:
The average rate of change of g(x) = 7x 2 between x = 3 and x = 4 is 7.The average rate of change of the function g(x) = 7x 2 between x = 3 and x = 3.5 is obtained as:We have:x1 = 3y1 = 7(3) 2 = 19Thus, (x1, y1) = (3, 19)x2 = 3.5y2 = 7(3.5) 2 = 22.5Thus, (x2, y2) = (3.5, 22.5)The average rate of change is obtained as the slope in the following manner:
The average rate of change of g(x) = 7x 2 between x = 3 and x = 3.5 is 7.The average rate of change of the function g(x) = 7x 2 between x = 3 and x = 3.1 is obtained as:We have:x1 = 3y1 = 7(3) 2 = 19Thus, (x1, y1) = (3, 19)x2 = 3.1y2 = 7(3.1) 2 = 19.7Thus, (x2, y2) = (3.1, 19.7)The average rate of change is obtained as the slope in the following manner:
The average rate of change of g(x) = 7x 2 between x = 3 and x = 3.1 is 7.The average rate of change of the function g(x) = 7x 2 between x = 3 and x = 3.01 is obtained as:We have:x1 = 3y1 = 7(3) 2 = 19Thus, (x1, y1) = (3, 19)x2 = 3.01y2 = 7(3.01) 2 = 19.07Thus, (x2, y2) = (3.01, 19.07)The average rate of change is obtained as the slope in the following manner:
The average rate of change of g(x) = 7x 2 between x = 3 and x = 3.01 is 7.At all the given intervals of x, the average rate of chang...