Mathematical Applications in Economics and Physics: Linear Equations, Demand, and Optimization

Linear Equations and Applications

Linear Regression and Estimation

Linear regression is a statistical method used to model the relationship between two variables by fitting a linear equation to observed data. The general form of a linear equation is:

• Equation: $y = mx + b$, where m is the slope and b is the y-intercept.

• Covariance: Measures how two variables change together. The formula is $\text{cov}(X, Y) = E[(X - E(X))(Y - E(Y))]$.

• Regression Slope (m): $m = \frac{\text{cov}(X, Y)}{\text{cov}(X)}$

• Regression Intercept (b): $b = E(Y) - mE(X)$

• Application: Used to estimate values (e.g., price for a given number of carats).

Example: Given data for carats and price, use the above formulas to estimate the price for a specific carat value.

Price-Demand Relationship

The price-demand relationship describes how the demand for a product changes as its price changes. This is often modeled linearly:

• Equation: $P = mD + b$, where P is price, D is demand, m is the slope, and b is the intercept.

• Estimation: Use regression to find m and b from data, then predict demand for given prices.

Example: Given demand and price data, estimate demand for specific price points using the regression equation.

Physics Applications: Speed of Sound

Linear Relationship Between Speed and Temperature

The speed of sound in air varies linearly with temperature. The relationship can be modeled as:

• Equation: $v = mt + b$, where v is speed, t is temperature, m is the rate of change, and b is the speed at 0 degrees.

• Constructing the Equation: Use two data points to solve for m and b:

$m = \frac{v_2 - v_1}{t_2 - t_1}$ $b = v_1 - m t_1$

Example: Given speeds at two temperatures, construct the linear equation for speed as a function of temperature.

Investment and Return

Investment Allocation for Desired Return

To achieve a target return by investing in two options with different returns, set up a weighted average equation:

• Equation: $r = x r_1 + (1 - x) r_2$, where r is the desired return, r_1 and r_2 are the returns of the two investments, and x is the fraction invested in the first option.

• Solving for x: Rearrange to find the proportion to invest in each option.

Example: If an investor wants an 8% return from investments yielding 5% and 9%, solve for the required allocation.

Calculus Applications: Derivatives and Optimization

Derivatives of Power Functions and Sums

The derivative of a function measures its rate of change. For power functions and sums:

• Power Rule: $\frac{d}{dx} x^n = n x^{n-1}$

• Sum Rule: The derivative of a sum is the sum of the derivatives: $\frac{d}{dx}(f(x) + g(x)) = f'(x) + g'(x)$

Example: For $C(x) = 20x^2 - 360x + 12000$, the derivative is $C'(x) = 40x - 360$.

Optimization: Minima and Maxima

To find the minimum or maximum of a function, set its derivative to zero and solve for the variable:

• Critical Points: Points where $f'(x) = 0$.

• Second Derivative Test: If $f''(x) > 0$, the point is a minimum; if $f''(x) < 0$, it is a maximum.

Example: For a polynomial with a positive leading coefficient, the function has a minimum where the derivative is zero.