# Xam Idea Class 12 Maths Chapter 7 Solutions: Applications Of Derivatives

Xam Idea Class 12 Maths Solutions Chapter 7 **‘**Applications Of Derivatives**’ **can be a great assist for you to prepare for CBSE Class 12 board exams. Since you know the whole calculus is revolving around the derivatives and its various applications, so it is a must for you to give proper time to it. In this chapter, you are going to learn about increasing & decreasing functions, rate of change of quantities, tangents & normals, approximations, maxima/ minima, and the minimum and maximum values of the function in a closed interval.

Xam Idea Class 12 Maths Solutions for Applications Of Derivatives has a total of 64 questions categorized into three sections. The sections are Very Short Answer Type Questions, Short Answer, Long Answer Type Questions. All of these questions covered the major topics of this chapter beautifully as per the latest CBSE Class 12 Maths syllabus. The questions are based on topics such as rate of change of quantities, the slope of the tangent to a curve and approximate values. Practising these questions regularly will help you revise everything regarding this chapter for your exams.

Instasolv solutions for Xam Idea Class 12 book can help for strengthening your foundation of Applications of Derivatives. We have provided solutions to all kinds of problems whether it is a basic level or advanced level question. All of these solutions have been drafted after thorough research of chapter, by our subject experts. The solutions are well-structured and easy for anyone to understand. Also, our solutions can also help deal with questions that usually come in competitive exams like JEE Mains and Advanced along with CBSE Class 12.

## Important Topics of Xam Idea Class 12 Maths Solutions Chapter 7: Applications Of Derivatives

**Rate of Change of Quantities-**

Let’s take a function –

Y = f(x)

then the rate of change of function is defined as ** dy/dx.**

And, dy/dx is also abbreviated as **f‘(x) **

This is for one variable.

Now, if we talk about two variables x and y that varies with a variable, let say x = f(t) , y = f(t)

Then, in this condition the derivative-

dy/dx = f ‘(x) = **(dy/dt)/(dx/dt)**

**Note- **dx / dt must not be equal to zero.

**Increasing Functions-**

Increasing functions are those types of functions, in which, when x increases then the corresponding y will also go up.

Let’s say, we have a function, y = f(x), a continuous function and are differentiable at open interval (a,b). So, the function, f is increasing when the derivative f ‘(x) > 0 in a closed interval [ a,b]

**Decreasing Functions-**

For, function, y = f(x) is a decreasing function

When f ‘(x) < 0 for each value of x in (a,b)

**Tangents and Normals-**

Let’s say we have a function

**Y = f(x)**

And, you know every function is a curve, and also you know a curve is made up of a set of points.

So, tangents are nothing, they are a line that touches a curve at some particular point for which the slope (gradient) of the tangent is equal to the slope of Curve at the point on which the tangent has been made.

**dy / dx (at point of curve) = Slope of Curve **

And, id we talk about the Normals to a curve. So, Normal is a line that is perpendicular to the tangent to a curve.

**Maxima and Minima**

In maths, Maxima and Minima are considered as Extremes. If we talk about what is Maxima and Minima, then see, the maxima are the point on the curve at which the function’s value is maximum whereas the point at which function’s value is minimum is called the point of Minima.

See, this graph for better understanding-

But how to find Maxima and Minima?

So, for finding Maxima and Minima, we have derivative tests-

**First-Derivative Test-**

Suppose, we have a function

Y = f(x)

You need to find the slope of function f(x).

Then, find value of x at f ‘(x) = 0

**Second-Derivative test-**

After we have a value of x at f ‘(x) = 0, then you need to put this value of x in second derivative f ‘’(x), of the original function-

And, if the value of the second derivative is-

- Less than 0, then this point is considered as a Maxima
- If greater than 0, then point of minima
- Equals to 0, the test fails.

**Maximum and Minimum Values of Function-**

The Maximum value of a function is the place on a curve where the function reaches its highest point whereas the minimum value is the function’s value at its lowest point.

### Exercise Discussion of Xam Idea Class 12 Maths Solutions Chapter 7: Applications Of Derivatives

Xam Idea Class 12 Maths Solutions Chapter 7 ‘Applications Of Derivatives’ have 64 questions that will help you make a strong grip over concepts associated with the chapter.

**Very Short Questions**

- This section includes 5 Xam Idea Textbook problems, generally based on the rate of change of quantities and tangents.
- In one of the questions, you just need to find the radius of the sphere when the rate of change of volume and radius are equal.

**Short Answer Type Questions**

- There are 8 questions based on the concepts of Approximations, Rate of change of quantities, Increasing/ Decreasing functions.
- Then, there are 5 questions. In one of the questions, you need to find a point on which the tangent of the curve has its slope 2 / 3.

**Long Answer Type Questions Part I **

- Long Answer Type I has a total of 13 questions.
- You have to find the intervals at which the function is increasing or decreasing, questions on approximations. Some of them are from tangents and normals as well.
- There is one question in the exercise where you need to find the equation of the tangent to a curve which is parallel to the line.

**Long Answer Type Questions Part II**

- In this section, you will have 33 questions to solve.

- You will have questions on the Maxima and Minima of functions. These questions are quite tricky to solve, you need to go-through the Maxima/ Minima part seriously to cope up with these questions.
- In one of the questions, you will be asked, Rectangle of Maximum Perimeter which can be inscribed in a circle of radius r is the square of side
**√**2. - Some questions from the tangents part as well.

### Why use Xam Idea Class 12 Maths Solutions Chapter 7: Applications Of Derivatives By Instasolv?

- Instasolv Xam Idea solutions help you boost your confidence regarding solving the questions of Applications Of Derivatives for Class 12.
- These solutions are prepared by the best maths experts of India and will clear your doubts and queries in an orderly fashion. You will be able to solve each question of this chapter on your own.
- The simple and easy language used in our Xam Idea Class 12 Maths solutions will help you read and understand the concepts of the chapter.
- If you are done with CBSE syllabus for Derivatives for Class 12, you can refer to our solutions, as it offers clear understanding and it has been made by our subject professionals following the CBSE syllabus.