JEE Maths Notes: Chapter on Application of Derivatives

1. Introduction to the Application of Derivatives

This section introduces the concept of derivatives in the context of real-world applications. The derivative, which measures the rate of change of a function, plays a significant role in various mathematical and physical problems. The main focus of this chapter is on the application of derivatives to solve problems involving rates of change, optimization, tangents, and normals. By understanding how to apply differentiation in different contexts, students can solve practical problems encountered in JEE.

2. Rate of Change

The rate of change refers to how a quantity changes with respect to another quantity, often time. This subtopic explains how derivatives are used to compute the rate of change of physical quantities, such as velocity, acceleration, and population growth. It covers the general formula for rate of change and provides examples from physics, economics, and biology. Key concepts such as related rates of change and the use of the chain rule to relate different quantities are discussed.

3. Tangents and Normals

Tangents and normals are fundamental concepts in the study of curves. The equation of the tangent to a curve at a given point represents the line that touches the curve without crossing it at that point. The normal, on the other hand, is a line perpendicular to the tangent. This section covers:

• Equation of the Tangent Line: How to find the slope of the tangent to a curve at a specific point using the derivative.

• Equation of the Normal Line: Finding the equation of the normal by using the negative reciprocal of the tangent's slope. Students are guided through the process of finding both the tangent and normal at specific points on various types of curves.

4. Maxima and Minima

Maxima and minima are the highest and lowest points of a function, respectively, and are essential for solving optimization problems. In this subtopic, students learn how to:

• Identify Critical Points: Critical points occur where the derivative of a function is zero or undefined. These points are potential candidates for maxima, minima, or inflection points.

• First Derivative Test: The first derivative test is used to determine whether a critical point is a local maximum, minimum, or neither.

• Second Derivative Test: The second derivative test is used to confirm the nature of a critical point. If the second derivative is positive at a critical point, the function has a local minimum; if negative, it has a local maximum. This section includes examples to demonstrate how to find and classify critical points in functions.

5. Increasing and Decreasing Functions

Understanding whether a function is increasing or decreasing on a particular interval is crucial in the study of optimization and curve sketching. This subtopic covers:

• First Derivative Test for Increasing/Decreasing Functions: The sign of the first derivative determines whether a function is increasing or decreasing. If the derivative is positive, the function is increasing; if negative, the function is decreasing.

• Intervals of Increase/Decrease: How to find the intervals where the function is increasing or decreasing by analyzing the sign of the derivative.

6. Concavity and Points of Inflection

Concavity refers to the direction in which the curve bends. A function can be concave upwards (like a cup) or concave downwards (like a cap). This section covers:

• Second Derivative Test: The second derivative of a function provides information about its concavity. If the second derivative is positive, the function is concave upwards; if negative, the function is concave downwards.

• Points of Inflection: A point of inflection occurs where the function changes concavity. These points are identified by setting the second derivative equal to zero and solving for x.

7. Optimization Problems

Optimization problems involve finding the maximum or minimum values of a function, typically subject to certain constraints. In this section, students learn how to:

• Formulate Optimization Problems: Translating real-world scenarios, such as maximizing area or minimizing cost, into mathematical equations that can be solved using derivatives.

• Solving Optimization Problems: Using derivatives to find the critical points and then applying the first or second derivative tests to identify the maximum or minimum values. This subtopic includes practical problems related to geometry, economics, and physics, such as finding the maximum profit or the shortest distance.

8. Linear Approximation

Linear approximation involves using the tangent line at a point on a curve to approximate the value of the function near that point. This concept is especially useful when dealing with complex functions. The main ideas covered include:

• Tangent Line as a Linear Approximation: How the equation of the tangent line can be used to approximate the function’s value near a specific point.

• Differential as an Approximation Tool: Using the differential of a function to approximate small changes in the function’s value.

9. Mean Value Theorem

The Mean Value Theorem (MVT) states that for a continuous function that is differentiable on a closed interval, there exists at least one point in the interval where the derivative of the function equals the average rate of change over that interval. This section covers:

• Statement and Proof of MVT: The formal statement of the Mean Value Theorem and its proof.

• Applications of MVT: The MVT can be used to prove important results in calculus and to solve problems involving rates of change.

10. Curve Sketching

Curve sketching is the process of drawing the graph of a function based on the information obtained from its derivatives. This section covers:

• Identifying Key Features of a Function’s Graph: Using the first and second derivatives to find the function’s intercepts, critical points, inflection points, and asymptotes.

• Determining the Shape of the Curve: Analyzing the intervals of increase and decrease, concavity, and the location of maxima, minima, and points of inflection to sketch the graph of a function accurately.

This chapter on the application of derivatives is vital for mastering calculus and solving a variety of mathematical problems in JEE. The concepts covered provide the tools needed to approach real-world problems in physics, economics, and engineering, making it an essential chapter for students preparing for competitive exams.