Unit 4 - Notes

MTH005

Unit 4: Application of Differentiation and Partial Differentiation

1. Growth Rate and Rates of Change

Differentiation is fundamentally the study of how a function changes as its input changes. In applied mathematics and economics, the derivative represents the instantaneous rate of change.

1.1 Rate of Change

If $y = f(t)$ , the rate of change of $y$ with respect to time $t$ is given by the first derivative:

\text{Rate of Change} = \frac{dy}{dt} = f'(t)

1.2 Relative (Instantaneous) Growth Rate

While the derivative gives the absolute change, the growth rate usually refers to the percentage change or relative change at a specific instant.

Formula:

r = \frac{f'(t)}{f(t)} = \frac{d}{dt} [\ln f(t)]

Key Application (Exponential Growth):
If a population or investment grows according to $N(t) = N_0 e^{kt}$ :

The rate of change is $N'(t) = k \cdot N_0 e^{kt} = k \cdot N(t)$ .
The relative growth rate is .
- Here, $k$ is the constant proportional growth rate.

2. Economic Applications: Cost and Revenue

In business mathematics, derivatives are used to perform marginal analysis—analyzing the effect of small changes (usually increasing production by one unit).

Let $x$ represent the number of units produced/sold.

2.1 Cost Functions

Total Cost Function, $C(x)$ :
Represents the total cost to produce $x$ units.

C(x) = \text{Variable Cost} + \text{Fixed Cost}

Average Cost ( $AC$ ):
The cost per unit of production.

AC(x) = \bar{C}(x) = \frac{C(x)}{x}

Marginal Cost ( $MC$ ):
The approximate cost of producing one additional unit. Mathematically, it is the instantaneous rate of change of the total cost.

MC(x) = C'(x) = \frac{d}{dx} C(x)

2.2 Revenue Functions

Total Revenue Function, $R(x)$ :
The total income generated from selling $x$ units at price $p$ .

R(x) = p \cdot x

Note: If price $p$ is dependent on demand (i.e., $p = f(x)$ ), then $R(x) = x \cdot f(x)$ .

Marginal Revenue ( $MR$ ):
The additional revenue generated from selling one additional unit.

MR(x) = R'(x) = \frac{d}{dx} R(x)

2.3 Profit Functions

Total Profit, $\Pi(x)$ :

\Pi(x) = R(x) - C(x)

Marginal Profit:

\Pi'(x) = R'(x) - C'(x) = MR - MC

3. Maxima and Minima of Functions $y = f(x)$

Optimization involves finding the maximum or minimum values of a function. In economics, this is used to maximize profit or minimize cost.

3.1 Critical Points

A critical point occurs at $x = c$ if:

$f'(c) = 0$ (Stationary point), or
$f'(c)$ does not exist.

3.2 Method for Finding Local Maxima and Minima

To find local extrema for a function $y = f(x)$ :

Step 1: First Derivative Test (Find Critical Points)
Find the first derivative $f'(x)$ and solve for $x$ where $f'(x) = 0$ . Let the solution be $x_0$ .

Step 2: Second Derivative Test (Classify Points)
Find the second derivative $f''(x)$ and substitute $x_0$ :

Local Minimum: If $f''(x_0) > 0$ (Concave Up).
Local Maximum: If $f''(x_0) < 0$ (Concave Down).
Test Fails: If $f''(x_0) = 0$ (Requires higher-order derivatives or sign testing).

3.3 Economic Optimization Examples

Minimizing Average Cost:
Average cost is minimized where $AC'(x) = 0$ . Mathematically, this often occurs where Marginal Cost equals Average Cost ( $MC = AC$ ).
Maximizing Profit:
Profit is maximized when Marginal Profit equals zero:
$\Pi'(x) = 0 \implies R'(x) - C'(x) = 0 \implies MR = MC$
Additionally, the second derivative of profit must be negative ( $MR' < MC'$ ).

4. Partial Differentiation

When a dependent variable is a function of two or more independent variables, we use partial differentiation.

Let $y = f(x_1, x_2)$ . Here, $y$ depends on both $x_1$ and $x_2$ .

4.1 Definition

The partial derivative of $f$ with respect to $x_1$ is found by differentiating $f$ with respect to $x_1$ while treating $x_2$ as a constant.

Notation:

Partial wrt $x_1$ : $\frac{\partial y}{\partial x_1}$ or $f_{x_1}$
Partial wrt $x_2$ : $\frac{\partial y}{\partial x_2}$ or $f_{x_2}$

4.2 Higher Order Partial Derivatives

We can differentiate the first derivatives again:

Direct Second Partials:
- $f_{x_1 x_1} = \frac{\partial^2 f}{\partial x_1^2}$ (Differentiate wrt $x_1$ twice)
- $f_{x_2 x_2} = \frac{\partial^2 f}{\partial x_2^2}$ (Differentiate wrt $x_2$ twice)
Cross (Mixed) Partials:
- $f_{x_1 x_2} = \frac{\partial^2 f}{\partial x_2 \partial x_1}$ (Differentiate wrt $x_1$ , then $x_2$ )
- Young’s Theorem: For continuous functions, $f_{x_1 x_2} = f_{x_2 x_1}$ .

5. Maxima and Minima of Functions $y = f(x_1, x_2)$

Optimization for multivariable functions is crucial for scenarios like maximizing utility given two goods or maximizing output given inputs of Labor ( $L$ ) and Capital ( $K$ ).

5.1 Necessary Condition (First Order Condition)

For a function $y = f(x_1, x_2)$ to have a relative maximum or minimum at a point $(a, b)$ , the first partial derivatives must simultaneously equal zero:

\frac{\partial f}{\partial x_1} = 0 \quad \text{and} \quad \frac{\partial f}{\partial x_2} = 0

Solving this system of equations yields the critical point(s).

5.2 Sufficient Condition (Second Order Condition)

Once critical points are found, use the Hessian Determinant (or Discriminant) method to classify them.

Let:

A = f_{x_1 x_1} = \frac{\partial^2 f}{\partial x_1^2}

B = f_{x_1 x_2} = \frac{\partial^2 f}{\partial x_1 \partial x_2}

C = f_{x_2 x_2} = \frac{\partial^2 f}{\partial x_2^2}

Define the discriminant $D$ as:

D = AC - B^2

Classification Rules:
At the critical point $(a, b)$ :

Maximum:
If $D > 0$ and $A < 0$ (or $C < 0$ ), then the function has a Relative Maximum.
Minimum:
If $D > 0$ and $A > 0$ (or $C > 0$ ), then the function has a Relative Minimum.
Saddle Point:
If $D < 0$ , the point is a Saddle Point (neither max nor min; it is a min in one direction and a max in another).
Inconclusive:
If $D = 0$ , the test fails.

5.3 Summary of Multivariable Optimization Procedure

Find $f_{x_1}$ and $f_{x_2}$ .
Set both to zero and solve the system to find critical points $(x_1^*, x_2^*)$ .
Find second derivatives: $f_{x_1 x_1}$ , $f_{x_2 x_2}$ , and $f_{x_1 x_2}$ .
Calculate $D = (f_{x_1 x_1})(f_{x_2 x_2}) - (f_{x_1 x_2})^2$ at the critical points.
Check the signs of $D$ and $f_{x_1 x_1}$ to determine the nature of the stationary point.