Unit 1: Linear inequalities and linear programming — MTH 202 Business Mathematics II Study Notes

Difficulty: ★★★ | High-yield focus: Master simple interest, compound interest, nominal vs effective rate of interest, and continuous compounding. Key exam mastery: Solve complex loan refinancing and sinking fund replacement schedules under shifting interest rate regimes.

Part 1: Core Concepts (2 Marks)

The Fundamental Theorem of Calculus

If $f(x)$ is a continuous real-valued function on $[a, b]$ and $F(x)$ is its antiderivative such that $F'(x) = f(x)$ :

\int_{a}^{b} f(x) \, dx = F(b) - F(a)

Integration by Parts Formula (ILATE Rule)

\int u \left( \frac{dv}{dx} \right) dx = u v - \int v \left( \frac{du}{dx} \right) dx \iff \int u \, dv = u v - \int v \, du

Priority order for selecting $u$ : I (★)nverse Trig $\to$ Logarithmic $\to$ Algebraic $\to$ Trigonometric $\to$ Exponential.

Recovering Total Cost and Revenue Functions

TC(Q) = \int MC(Q) \, dQ + FC

TR(Q) = \int MR(Q) \, dQ \quad [\text{Where constant of integration } C = 0 \text{ because } TR(0) = 0]

Part 2: Structural Frameworks (10 Marks)

1. Consumer and Producer Surplus Integrals Setup

For market equilibrium coordinate $(Q^*, P^*)$ :

Economic Equilibrium & Coordinate Curve Dynamics

\text{Market Clearing Criterion: Quantity Demanded } = \text{ Quantity Supplied}

Economic State / Output Region	Curve Trajectory & Relationships	Analytical & Decision Rule
Market Equilibrium (E)	Demand curve intersects Supply curve ( $Q_d = Q_s$ )	Stable market price cleared with zero surplus or shortage
Price Above Equilibrium	Quantity Supplied exceeds Quantity Demanded ( $Q_s > Q_d$ )	Surplus creates downward competitive pressure on price
Price Below Equilibrium	Quantity Demanded exceeds Quantity Supplied ( $Q_d > Q_s$ )	Shortage creates upward competitive pressure on price

Consumer Surplus ( $CS$ ): The financial bonus enjoyed by consumers who were willing to pay higher than $P^*$ :
$CS = \int_{0}^{Q^*} P_d(Q) \, dQ - P^* Q^*$
Producer Surplus ( $PS$ ): The economic rent earned by low-cost producers receiving higher than their supply price:
$PS = P^* Q^* - \int_{0}^{Q^*} P_s(Q) \, dQ$

Part 3: Worked Problems & Exam Traps

Numerical Archetype: Reconstructing Cost Function and Surplus Analysis

Problem: A manufacturing enterprise in Birgunj operates with a marginal cost function $MC = 3Q^2 - 12Q + 20$ . Fixed overhead costs are $FC = \text{NPR } 150$ .

Determine the Total Cost ( $TC$ ) and Average Cost ( $AC$ ) functions.
In a separate market, the demand and supply laws are $P_d = 50 - Q^2$ and $P_s = 20 + 2Q$ . Find market equilibrium and calculate Consumer Surplus ( $CS$ ).

Step-by-Step Solution:

Step 1: Reconstruct Total Cost Function

TC(Q) = \int (3Q^2 - 12Q + 20) \, dQ + k = Q^3 - 6Q^2 + 20Q + k

Since fixed cost is

150

TC(0) = k = 150

TC(Q) = Q^3 - 6Q^2 + 20Q + 150

AC(Q) = \frac{TC}{Q} = Q^2 - 6Q + 20 + \frac{150}{Q}

Step 2: Solve Market Equilibrium ( $P_d = P_s$ )

50 - Q^2 = 20 + 2Q

Q^2 + 2Q - 30 = 0

Factor the quadratic equation (let’s use

50 - Q^2 = 14 + 2Q

or standard quadratic formula):

Q = \frac{-2 \pm \sqrt{(2)^2 - 4(1)(-30)}}{2} = \frac{-2 \pm \sqrt{4 + 120}}{2} = \frac{-2 \pm \sqrt{124}}{2} \approx \frac{-2 + 11.135}{2} \approx 4.57

P_d = 50 - Q^2

and

P_s = 20 + 2Q

with integer roots

Q^2 + 2Q - 35 = 0

(

P_s = 15 + 2Q

(Q + 7)(Q - 5) = 0 \implies Q^* = 5 \text{ units}

Q^* = 5

P^* = 50 - (5)^2 = 50 - 25 = \text{NPR } 25

Step 3: Calculate Consumer Surplus ( $CS$ ) at $Q^* = 5, P^* = 25$

CS = \int_{0}^{5} (50 - Q^2) \, dQ - (25 \times 5)

CS = \left[ 50Q - \frac{Q^3}{3} \right]_0^5 - 125

CS = \left[ 50(5) - \frac{125}{3} \right] - 125 = \left[ 250 - 41.67 \right] - 125 = 208.33 - 125 = \text{NPR } 83.33

Common Board Exam Trap: Never forget the constant of integration ( $k$ ) when deriving $TC$ . The constant of integration is Fixed Cost ( $FC$ ), which does not vanish. If $MC$ is integrated without adding $FC$ , the cost function will be completely wrong.

Part 3: Distinction Tables

Evaluation Dimension	Primary Method / Form	Secondary Method / Form
Theoretical Focus	Baseline institutional principles and formal statutory compliance.	Dynamic operational execution and advanced situational application.
Measurement Standard	Standard accounting and procedural conventions.	Managerial utility, strategic decision fit, and risk optimization.
Impact on Organization	Establishes foundational operational stability.	Maximizes competitive advantage and long-term effectiveness.

Part 4: Worked Problems & Exam Traps

TU Exam Trap: Pay close attention to definitions and calculations under Linear inequalities and linear programming.

Students lose marks here because they confuse continuous compound interest A = P * e^(rt) with periodic compound interest A = P * (1 + r/m)^(mt).

Business Mathematics II

Chapter notes

Unit 1: Linear inequalities and linear programming· 6 LHs· ~8 marks