BCA TU, Math II – Unit 3: Application of Differentiation

Differentiation is not just about finding derivatives—it is about applying them to analyze curves, optimize values, and solve real-world problems. This unit focuses on slope, monotonicity, maxima–minima, marginal analysis, and graph sketching.

1. Derivatives and Slope of a Curve

The derivative of a function at a point gives the slope of the tangent line at that point:

• If → slope is positive → curve rises
• If → slope is negative → curve falls

Example:
For ,

• At , slope = 2 → rising
• At , slope = -2 → falling

2. Increasing and Decreasing Functions

• Increasing Function:
• Decreasing Function:

Example:
For ,

So, the function is always increasing (never decreases).

3. Convexity and Concavity of Curves

The second derivative determines the shape of curves:

• If , curve is convex (concave upward)
• If , curve is concave (concave downward)

Point of Inflexion: Where , the curve changes from convex to concave or vice versa.

4. Maximization and Minimization

Extreme values occur at critical points where :

• If , point is minimum
• If , point is maximum

Example:

5. Differentiation and Marginal Analysis

In economics and business, derivatives are applied in marginal analysis:

• Marginal Cost (MC): (rate of change of cost with respect to output)
• Marginal Revenue (MR):
• Marginal Profit (MP):

Example:
If , then

6. Price and Output (Competitive Equilibrium)

In competitive markets, equilibrium is reached when:

This condition determines the optimal price and quantity for maximum profit.

7. Curve Sketching Using First and Second Derivatives

Steps to sketch an algebraic function:

1. Find domain of the function
2. Compute first derivative → find increasing/decreasing intervals
3. Compute second derivative → determine concavity
4. Locate critical points where
5. Find maxima, minima, and inflexion points
6. Plot key points and sketch the curve

Example:
For

At ,

At ,

Key Takeaways

• Derivatives give the slope of the tangent and indicate the increasing/decreasing nature of functions.
• First derivative test: finds where a function rises or falls.
• Second derivative test: determines maxima, minima, and concavity.
• Applications in business and economics: marginal cost, revenue, and profit.
• Competitive equilibrium: occurs when .
• Curve sketching requires analysis of both first and second derivatives.