NCERT Solutions for Class 12 Maths Chapter 6: Application of Derivatives

The NCERT Solutions for Class 12 Maths Chapter 6, "Application of Derivatives," is a vital part of the Mathematics syllabus. This chapter plays a crucial role in helping students understand the practical applications of derivatives in various real-life problems. Derivatives are a fundamental concept in calculus that has vast applications in fields like physics, engineering, economics, and biology. The chapter builds upon the concepts of differentiation introduced in previous classes and extends them to solve more complex problems.

In this comprehensive article, we will provide detailed insights into the topics covered in Chapter 6 of NCERT Class 12 Maths, along with an overview of the solutions provided for each exercise. We will also discuss the importance of understanding derivatives and their applications in various fields.

Understanding Derivatives and Their Applications

Derivatives represent the rate of change of a function with respect to a variable. In simpler terms, they help us understand how a function changes as its input changes. For example, if we consider the position of a moving object with respect to time, the derivative of the position function with respect to time gives us the velocity of the object. Similarly, the derivative of velocity with respect to time gives us acceleration.

Applications of derivatives are numerous and varied, and some of the key areas where derivatives are used include:

1. Rate of Change of Quantities: Understanding how one quantity changes concerning another, such as speed, population growth, etc.

2. Finding Tangents and Normals to Curves: Determining the slope of a tangent or normal to a curve at a given point.

3. Increasing and Decreasing Functions: Analyzing whether a function is increasing or decreasing in a particular interval.

4. Maxima and Minima of Functions: Determining the maximum or minimum value of a function, which is particularly useful in optimization problems.

5. Approximations and Errors: Using derivatives for estimating values of functions and determining errors in measurements.

Overview of NCERT Chapter 6: Application of Derivatives

NCERT Class 12 Maths Chapter 6 is divided into several sections that cover the different applications of derivatives. Each section focuses on a specific type of application and includes a variety of solved examples and exercises to help students understand the concepts better. Below is an overview of the topics covered in this chapter:

1. Rate of Change of Quantities
2. Increasing and Decreasing Functions
3. Tangents and Normals
4. Approximations
5. Maxima and Minima

1. Rate of Change of Quantities

In this section, students learn about the rate of change of various quantities concerning time or other variables. For instance, if $y = f(x)$ is a function, then the rate of change of $y$ with respect to $x$ is given by $\frac{dy}{dx}$. Problems related to velocity, acceleration, population growth, and other dynamic processes are discussed here. The NCERT solutions provide step-by-step methods to solve problems involving the rate of change, including real-life scenarios.

Example Problem:

Find the rate of change of the area of a circle with respect to its radius when the radius is 5 cm.

Solution:

Let the area of the circle be $A$. The area of a circle is given by $A = \pi r^2$. The rate of change of the area with respect to the radius is given by:

$$\frac{dA}{dr} = \frac{d}{dr}(\pi r^2) = 2\pi r$$

When $r = 5$ cm,

$$\frac{dA}{dr} = 2\pi \times 5 = 10\pi \, \text{cm}^2/\text{cm}$$

2. Increasing and Decreasing Functions

This section helps students understand how to determine whether a function is increasing or decreasing in a given interval. A function $f(x)$ is said to be increasing in an interval if for any two numbers $x_1$ and $x_2$ in the interval, $f(x_1) < f(x_2)$ whenever $x_1 < x_2$. Similarly, it is decreasing if $f(x_1) > f(x_2)$ whenever $x_1 < x_2$. The first derivative test is used to determine the nature of the function in a given interval.

Example Problem:

Determine the intervals in which the function $f(x) = 3x^3 - 9x^2 + 6x + 1$ is increasing or decreasing.

Solution:

First, find the derivative of $f(x)$:

$$f'(x) = 9x^2 - 18x + 6$$

Set $f'(x) = 0$ to find critical points:

$$9x^2 - 18x + 6 = 0 \implies x^2 - 2x + \frac{2}{3} = 0$$

Solving this quadratic equation, we find the critical points. By analyzing the sign of $f'(x)$ around these points, we can determine where the function is increasing or decreasing.

3. Tangents and Normals

In this section, students learn how to find the equation of tangents and normals to a curve at a given point. A tangent to a curve at a point is a straight line that touches the curve only at that point and has the same slope as the curve at that point. The normal is a line perpendicular to the tangent at the point of tangency.

Example Problem:

Find the equation of the tangent and normal to the curve $y = x^3 - 3x + 2$ at the point where $x = 1$.

Solution:

To find the slope of the tangent, we differentiate the curve:

$$\frac{dy}{dx} = 3x^2 - 3$$

At $x = 1$,

$$\frac{dy}{dx} = 3(1)^2 - 3 = 0$$

The tangent is horizontal. The equation of the tangent at $x = 1$ is $y = f(1) = 0$. The normal, being perpendicular to the tangent, would be vertical in this case.

4. Approximations

This section deals with using derivatives for approximations. Linear approximation is used to estimate the value of a function near a given point. The formula for approximation is:

$$f(x + \Delta x) \approx f(x) + f'(x) \Delta x$$

Example Problem:

Using linear approximation, estimate the value of $\sqrt{26}$.

Solution:

Let $f(x) = \sqrt{x}$. Then, $f'(x) = \frac{1}{2\sqrt{x}}$.

At $x = 25$, $f(x) = 5$ and $f'(x) = \frac{1}{10}$.

$$f(26) \approx f(25) + f'(25) \cdot 1 = 5 + \frac{1}{10} = 5.1$$

5. Maxima and Minima

The concept of maxima and minima is crucial in mathematical optimization. A function reaches its maximum or minimum value at critical points where its derivative is zero or undefined. The second derivative test helps to determine whether the critical point is a maximum or minimum.

Example Problem:

Find the maximum and minimum values of the function $f(x) = x^3 - 6x^2 + 9x + 15$.

Solution:

First, find $f'(x) = 3x^2 - 12x + 9$.

Set $f'(x) = 0$:

$$3x^2 - 12x + 9 = 0 \implies x^2 - 4x + 3 = 0$$

Solving this quadratic equation gives the critical points $x = 1$ and $x = 3$. The second derivative test is then used to determine whether these points are maxima or minima.

Conclusion

NCERT Solutions for Class 12 Maths Chapter 6, "Application of Derivatives," provides students with a deep understanding of how derivatives are used in various real-life scenarios. From calculating the rate of change to determining the maximum and minimum values of functions, derivatives are an essential tool in mathematics. The chapter's exercises and examples guide students through the practical application of derivatives, enhancing their problem-solving skills and preparing them for competitive exams and future studies in mathematics and related fields.

By mastering the concepts in this chapter, students can develop a strong foundation in calculus, which is crucial for higher education in science, engineering, economics, and other disciplines. The NCERT solutions provide a comprehensive approach to learning, ensuring that students can confidently tackle any problem related to the application of derivatives.