James mainly sells confectionery items, newspapers, magazines and cigarettes in his convenience store. Noting his small business is not thriving, he thought of selling hot pies and rolls too.

Suppose the total cost function for rolls and pies is,

TC = 800 + 53𝑄, 𝑄 = 𝑄_{1 }+ 𝑄_{2}

where 𝑄_{1} and 𝑄_{2} denote the quantities of rolls and pies respectfully. If 𝑃_{1} and 𝑃_{2} denote the corresponding prices, then the inverse demand equations are:

𝑄_{1} = 73 − 𝑃_{1} and 0.5𝑄_{2 }= 100 − 𝑃_{2}

**REQUIRED: **

- If James decides to charge the same price for rolls and pies per day (that is, 𝑃
_{1 }= 𝑃_{2}), how many of rolls and pies in total should he make in order to maximize the profit of a particular day?

- If James decides to charge different prices as above for rolls and pies per day (that is, 𝑃
_{1 }≠ 𝑃_{2}), how many of rolls and pies should he make in order to maximize the profit of a particular day?

- Which of the above options (a) or (b) is more profitable? Provide the rationale for your answer.

- If James decides to make a total of 51rolls and pies per day and charges different prices as above (that is, 𝑃
_{1 }≠ 𝑃_{2}), how many of rolls and pies each should he make in order to maximize the profit of a particular day? Estimate the increase in maximum profit which results when the total number of rolls and pies per day (51) is increased to 52 [**note**: assume second-order conditions are satisfied].

- The COVID-19 pandemic saw the lockdown of many cities to reduce the spread of the virus. This unprecedented move can be viewed as a negative demand shock. Explain the impact of the lockdown of the city where James’ convenience store is located on the demand functions of rolls and pies (half a page maximum).

**(2.5 +3.5 + (2+1) + (3+1) + 5 = 18 marks) **

The supply and demand functions of a good are given by

𝑃_{𝑆 }= 32 + 𝑄_{𝑠}^{2}

and

𝑃_{𝐷 }= 140 − ^{𝑄}_{3}^{𝐷}^{2}

where 𝑃_{𝑆}, 𝑃_{𝐷}, 𝑄_{𝑆} and 𝑄_{𝐷} are the price and quantity supplied and demanded, respectively.

- Calculate the producer’s surplus and consumer’s surplus at the equilibrium

point.

- Explain the effect, if any, on producer’s surplus if the government imposes a fixed tax on this good (note: no calculation expected).

- A manufacturer’s marginal cost (MC) function is given by:

MC =

Find the equation of the total cost function if total costs are 5200 when 𝑄 = 8.

- A population of size 𝑥 is decreasing according to the law

^{𝑑𝑥 }= ^{−𝑥}

𝑑𝑡 250

where 𝑡 denotes the time in days. If initially the population is of size 𝑥_{0} (that is, when 𝑡 = 0, 𝑥 = 𝑥_{0}) find how long it takes for the size of the population to be halved.

- Use the inverse matrix method to solve the following system of equations (
**note**: use the Gauss-Jordan elimination method to determine the required inverse matrix. Be sure to show your workings. If any other method, other than the Gauss-Jordan, is used or your workings not shown, you will be awarded a zero mark).

3𝑥 + 2𝑦 − 𝑧 = 9

2𝑧 − 𝑥 + 4𝑦 = 5

−2𝑦 + 𝑥 + 𝑧 = 3

- 𝑥 2

- Let matrix A = (1 2 2 ). If |𝐴| = 0, find 𝑥
- −1 (𝑥 − 4)

** (5 + 3 = 8 marks) ** __FORMULA SHEET__

- 𝑎
^{0 }= 1 - 𝑎
^{1 }= 𝑎 - 𝑎
^{𝑚 }× 𝑎^{𝑛 }= 𝑎^{𝑚+𝑛} ^{𝑎}^{𝑚}_{𝑛 }= 𝑎𝑚−𝑛

𝑎

^{1}_{𝑛 }= 𝑎^{−𝑛}and_{𝑛 }= 𝑎^{𝑛}

𝑎 𝑎

- 𝑎
^{𝑚𝑘} - or 𝑎

- log (A) + log (B) = log (AB)
- log (A) – log (B) = log (
^{A})

B

- log (A
^{𝑘}) = 𝑘 log (A) - log (1) = 0 and ln (1) = 0
- log (10) = 1 and ln (e) = 1
- ln(e
^{k}) = k and e^{ln(k) }= k - log10
^{k }= k and 10^{logk }= k

- Power rule:

If 𝑦 = 𝑥^{𝑛} then ^{𝑑𝑦} = 𝑛𝑥^{𝑛−1}

𝑑𝑥

- Exponential rule

If 𝑦 = 𝑒^{𝑥} then ^{𝑑𝑦} = 𝑒^{𝑥}

𝑑𝑥

- Log rule If 𝑦 = ln (𝑥) then
^{𝑑𝑦}=^{1}

𝑑𝑥 𝑥

- Product or Multiplication rule

If 𝑦 = 𝑢(𝑥) × 𝑣(𝑥) (or 𝑦 = 𝑢𝑣) then ^{𝑑𝑦} = {𝑢 × ^{𝑑𝑣}} + {𝑣 × ^{𝑑𝑢}}

𝑑𝑥 𝑑𝑥 𝑑𝑥

- Division or Quotient Rule if 𝑦 = 𝑢(𝑥) (or 𝑢 𝑑𝑦 {𝑣×𝑑𝑢𝑑𝑥} - {𝑢×𝑑𝑥𝑑𝑣} 𝑣(𝑥) 𝑦 = 𝑣 ) then 𝑑𝑥 = 𝑣2

- Chain Rule or Function of Function Rule

If 𝑦 = 𝑢(𝑥) then ^{𝑑𝑦} = ^{𝑑𝑦} × ^{𝑑𝑢}

𝑑𝑥 𝑑𝑢 𝑑𝑥

- Power Rule

^{𝑥}𝑛+1

∫ 𝑥^{𝑛}𝑑𝑥 = + 𝑐

𝑛+1

- Constant rule:

∫ 𝑘𝑑𝑥 = 𝑘𝑥 + 𝑐, where 𝑘 is a constant

- Log rule

𝑐

𝑥

- d) Exponential Rule

∫ 𝑒^{𝑥 }= 𝑒^{𝑥 }+ 𝑐

**Evaluating definite integral: **

𝑥=𝑏

∫ 𝑓(𝑥)𝑑𝑥 = 𝐹(𝑏) − 𝐹(𝑎)

𝑥=𝑎

Linear/straight line: 𝑦 = 𝑚𝑥 + 𝑐

𝑚: slope, 𝑐: intercept

Quadratic: 𝑦 = 𝑎𝑥^{2 }+ 𝑏𝑥 + 𝑐

𝑐: intercept

To find the roots of any quadratic equation, 𝑎𝑥^{2 }+ 𝑏𝑥 + 𝑐 = 0,

−𝑏 ± √𝑏^{2 }− 4𝑎𝑐

𝑥 =

2𝑎

The n^{th} term of the arithmetic sequence is: 𝑇_{𝑛 }= 𝑎 + (𝑛 − 1)𝑑

𝑎: the first term of the sequence

𝑑: common difference between numbers

The sum of the first n terms,𝑆_{𝑛}, of an arithmetic series is given by the formula:

𝑆_{𝑛 } = ^{𝑛}_{2 }[2𝑎 + (𝑛 − 1)𝑑]

The n^{th} term of a geometric series is

𝑇𝑛 = 𝑎𝑟𝑛−1

𝑎: the first term of the sequence

𝑟: common multiple/ratio between numbers

The sum of the first 𝑛 terms of a geometric series is given by

𝑆_{𝑛} = 𝑎(_{1}1_{−}− _{𝑟}𝑟^{𝑛})

- (a + b)
^{2 }= a^{2 }+ 2ab + b^{2} - (a − b)
^{2 }= a^{2 }− 2ab + b^{2}

**Difference of two squares**

a^{2 }− b^{2 } = (a − b) × (a + b)

^{a}* = *(^{a}) *×* (^{d})

_{d }b c

- Total Revenue (TR) = P×Q
- Total Cost (TC) = Fixed cost (FC) + Variable cost (VC)
- Average Cost (AC) =
^{TC}Q - Average Revenue (AR) =
^{TR}Q - Marginal Cost (MC) =
^{d}^{(TC)}

dQ

- Marginal Revenue (MR) =
^{d}^{(TR)}

dQ

- Profit = TR – TC
- At the break-even, TR = TC or Profit = 0
- At the equilibrium, P
_{d}= P_{s}= P and Q_{d}= Q_{s}= Q - If price discrimination is not permitted, then P
_{1}= P_{2}= P. The overall demand is the sum of the two separate demands: Q = Q_{1}+ Q_{2 }

- Solve the equation 𝑓
^{′}(𝑥) = 0 to find the turning point(s), 𝑥 = 𝑎 - If 𝑓
^{′′}(𝑎) > 0, then the function has a minimum at 𝑥 = 𝑎

If 𝑓^{′′}(𝑎) < 0, then the function has a maximum at 𝑥 = 𝑎

- Solve the simultaneous equations,

𝑓_{𝑥}(𝑥, 𝑦) = 0

𝑓_{𝑦}(𝑥, 𝑦) = 0 to find the turning points, (𝑎, 𝑏).

- Let Δ = 𝑓
_{𝑥𝑥}𝑓_{𝑦𝑦 }− 𝑓_{𝑥𝑦}^{2}.- if 𝑓
_{𝑥𝑥}> 0 and 𝑓_{𝑦𝑦}> 0 and Δ > 0 at (𝑎, 𝑏), then the function has a minimum at (𝑎, 𝑏) - if 𝑓
_{𝑥𝑥}< 0 and 𝑓_{𝑦𝑦}< 0 and Δ > 0 at (𝑎, 𝑏), then the function has a maximum at (𝑎, 𝑏) - The point is a point of inflection if both second derivatives have the same sign but Δ < 0
- The point is a saddle point if the second derivatives have different signs and Δ < 0
- If Δ = 0 then there is no conclusion

- if 𝑓

- Profit is maximized when MR = MC and 𝑀𝑅
^{′ }< 𝑀𝐶^{′} - Profit is minimized when MR = MC and 𝑀𝑅
^{′ }> 𝑀𝐶^{′}

To find the optimum values of a function,𝑓(𝑥, 𝑦), subject to a constraint, 𝑎𝑥 + 𝑏𝑦 = 𝑀, define the Lagrangian function, 𝐿, where

𝐿 = 𝐿(𝑥, 𝑦, 𝜆) = 𝑓(𝑥, 𝑦) + 𝜆(𝑀 – 𝑎𝑥 – 𝑏𝑦)

where λ is called a Lagrange multiplier.

𝒅𝒙

- integrate both sides of the differential equation with respect to 𝑥. This gives the general solution.
- If conditions are given for 𝑥 and 𝑦, substitute these values into the general solution and solve for the arbitrary constant, 𝑐.
- Substitute this value of 𝑐 into the general solution to find the particular solution.

- Evaluating a 2×2 determinant:

|^{𝑎 𝑏}| = (𝑎 × 𝑑) – (𝑐 × 𝑏) = 𝑎𝑑 – 𝑐𝑏

𝑐 𝑑

- Evaluating a 3×3 determinant:

𝑎𝑎11_{21 }𝑎𝑎12_{22 }𝑎13 𝑎22 𝑎23 𝑎21 𝑎23

|𝑎31 𝑎32 𝑎23| =(𝑎11)× |𝑎32 𝑎33| - (𝑎12)×|𝑎31 𝑎33| +

𝑎33

(𝑎13)×|𝑎𝑎2131 𝑎𝑎2232|

- To write a system of equations in matrix form:

𝑎_{1}𝑥 + 𝑏_{1}𝑦 + 𝑐_{1}𝑧 = 𝑑_{1}

𝑎_{2}𝑥 + 𝑏_{2}𝑦 + 𝑐_{2}𝑧 = 𝑑_{2}

𝑎_{3}𝑥 + 𝑏_{3}𝑦 + 𝑐_{3}𝑧 = 𝑑_{3}

𝑎 (𝑎 |
𝑏 𝑏 𝑏 |
𝑐1 𝑥 𝑑1 𝑐 𝑐3 𝑧 𝑑3 |

This is known as 𝐴𝑋 = 𝐵 format,

𝑎 where 𝐴 = (𝑎 𝑎 |
𝑏 𝑏 𝑏 |
𝑐1 𝑥 𝑑1 𝑐 𝑐3 𝑧 𝑑3 |

- Inverse matrix method involves solving 𝑋 using 𝑋 = 𝐴
^{−1}𝐵

- Finding the inverse of matrix A using the Gauss-Jordan elimination method:

𝑎 Write down the augmented matrix as (𝑎 |
𝑏 𝑏 𝑏 |
𝑐 𝑐 𝑐 |
0 1 0 |
0 0) 1 |

Transform the above augmented matrix as

1 (0 0 |
0 1 0 |
0 𝑝 0|𝑝 1 𝑝 |
𝑞 𝑞 𝑞 |
𝑟 𝑟 𝑟 |

The original matrix 𝐴, is now reduced to the identity/unit matrix. The inverse of 𝐴 is given by the transformed unit matrix. That is,

𝑝1 𝑞1 𝑟1

𝐴^{−1} = (𝑝_{2 }𝑞_{2 }𝑟_{2})

𝑝3 𝑞3 𝑟3

There are three elementary row operations used to achieve the row echelon form:

- Swap the positions of two rows
- Multiply (or divide) each element of a row by a nonzero constant Replace a row by the sum of itself and a constant multiple of another row of the matrix.

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