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# Mathematics (Coursework Sample)

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I was solving problem sets.

source..Content:

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 â‰¥ and f(x) = - x â€“ 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 â€“ 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 â†’ 4- = -2 and lim x â†’ 4+ = -2

Thus, lim x â†’ 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...

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 â‰¥ and f(x) = - x â€“ 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 â€“ 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 â†’ 4- = -2 and lim x â†’ 4+ = -2

Thus, lim x â†’ 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...

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