unit 1.1 1.2 math review

Introduction

• Review of essentials from Math 1070 (MAT 1070).

  • Focus on major concepts: cost, revenue, profit, breakeven quantities, and mathematical processes.

  • Emphasis on mathematical operation with these notions using correct notation.

Cost, Revenue, and Profit

Understanding Cost

• Cost types: Fixed cost and Variable cost.

  • Fixed Cost: Costs that do not change with production level.

  • Example: Rent for a coffee shop.

    • Regardless of how many cups are sold, the rental cost remains constant.

  • Other examples include utilities and salaried wages.

  • Variable Cost: Costs that change with the production level.

  • Example: Costs for materials like cups, lids, condiments, and hourly wages for staff.

    • More production means more variable costs.

Total Cost

• Total Cost (TC) is the sum of fixed costs and variable costs: $TC = FC + VC$

  • Where:

  • $FC$ = Fixed Cost

  • $VC$ = Variable Cost

• Each product unit, like a cup of coffee, has its own cost and revenue dynamics depending on production.

Understanding Revenue

• Revenue (R): Amount of money generated from selling products.

  • Formula: $R = P imes Q$

  • Where:

    • $P$ = Selling price per unit

    • $Q$ = Quantity sold

• Example: Buying groceries where $P$ = 2 (price per item) and $Q$ = 10 (items bought).

Understanding Profit

• Profit (P): Difference between revenue and costs.

  • Formula: $P = R - TC$

  • Can also be expressed with functions as follows:

  • Profit Function: $P(q) = R(q) - C(q)$

• Where:

  • $( R(q) = P imes Q )$

  • Cost function depends on units produced.

Practical Examples and Calculations

Example 1: Coffee Shop

Given:

• Variable cost = $6 per unit.

• Selling price = $15 each.

• Fixed Cost = $250,000.

Find Total Profit for producing and selling $q$ units:

1. Cost Function (C):
   $C(q) = 250,000 + 6q$

2. Revenue Function (R):
   $R(q) = 15q$

3. Profit Function (P):
   $( P(q) = 15q - (250,000 + 6q) )$,
   This simplifies to:
   $( P(q) = 9q - 250,000 )$

Calculate Total Profit for $q = 40,000$:

• Plugging in: $P(40,000) = 9(40,000) - 250,000 = 110,000$

  • Total Profit is $110,000.

Example 2: Poster Production Costs

Given:

• Fixed costs = $700 (Designer + Licensing).

• Variable cost = $0.60 per poster.

Find:

• Cost function to produce $q$ posters:
  $C(q) = 700 + 0.6q$

Questions:

1. Cost to produce 300 posters?

   • Plugging in:
     $C(300) = 700 + 0.6(300) = 880$

   • Answer: $880.

Example 3: Breakeven Quantity

Given:

• Fixed Cost = $80 base + variable cost = $7.50 per shirt.

Breakeven means Revenue equals Cost.

1. Revenue Function: $R(q) = 20q$

2. Cost function: $C(q) = 80 + 7.50q$

Solve for Breakeven Quantity:

• Set $R(q) = C(q)$:
  $20q = 80 + 7.50q$

Resulting Equation:

• Solve to find $q$:
  $q = 32$ (breakeven quantity).

Functions and Notation

• Introduce mathematical notation for clarity:

  • $c(q)$ = Cost function, $r(q)$ = Revenue function, $p(q)$ = Profit function.

• This notation indicates functions based on quantity ( extbf{q}) produced.

• Discuss context sensitivity regarding $q$ in terms of production vs. sales.

Final Notes

1. Importance of understanding these functions for calculus, particularly for derivatives related to marginal costs, revenues, and profits.

2. Reminder around accuracy of calculations, especially with monetary values (always two decimal places for dollar amounts).