Mathematics (Coursework Sample)

Question 1
Costs = C(n) = 220n + 202,500; Revenue = R(n) = -3n²+2020n
a) Break-even point (BEP)
At the BEP, costs are equal to the revenues such that:
C(n) = R(n)
220n + 202,500 = -3n²+2020n
3n2 – 2020n + 220n + 202500 = 0
3n2 – 1800n + 202500 = 0
We 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 conditioners
b) Maximize Profit
The profit is obtained as the difference between revenue and costs
Thus, P(n) = R(n) – C(n)
P(n) = (-3n² + 2020n) – (220n + 202,500)
P(n) = -3n² + 2020n – 220n – 202500
P(n) = -3n² + 1800n – 202500
P’(n) = -6n + 1800
The maximum profit is obtained at: -6n + 1800 = 0
6n = 1800
n = 300
Thus, the maximum profit is obtained at n = 300 air conditioners
c) The maximum profit
P(n) = -3n² + 1800n – 202500
P(n) = -3(300)² + 1800(300) – 202500 = $67,500
Question 2
A graph of f(x) = 3x – 7 if x ≥ and f(x) = - x – 1 if x < 2
Plot generated at: http://rechneronline.de/function-graphs/
Question 3
The difference quotient of g(x) = 2x²- 3x +1 is obtained as:
Where: and
Thus, the difference quotient is:
The difference quotient is: 4x + 2h – 3
Question 4
f(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 4
lim 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 5
The average rate of change of the function g(x) = 7x – 2 between x = 3 and x = 5 is obtained as:
We have:
x1 = 3
y1 = 7(3) – 2 = 19
Thus, (x1, y1) = (3, 19)
x2 = 5
y2 = 7(5) – 2 = 33
Thus, (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 = 3
y1 = 7(3) – 2 = 19
Thus, (x1, y1) = (3, 19)
x2 = 4
y2 = 7(4) – 2 = 26
Thus, (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 = 3
y1 = 7(3) – 2 = 19
Thus, (x1, y1) = (3, 19)
x2 = 3.5
y2 = 7(3.5) – 2 = 22.5
Thus, (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 = 3
y1 = 7(3) – 2 = 19
Thus, (x1, y1) = (3, 19)
x2 = 3.1
y2 = 7(3.1) – 2 = 19.7
Thus, (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 = 3
y1 = 7(3) – 2 = 19
Thus, (x1, y1) = (3, 19)
x2 = 3.01
y2 = 7(3.01) – 2 = 19.07
Thus, (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...