CBSE Ch 6 Important Questions Class 12 Maths 2025 PDF

Applications of Derivatives in Class 12 Mathematics is a very important chapter that connects abstract calculus concepts with practical problems. Derivatives are used to understand the rate of change and are applied largely in physics, engineering, economics, and biology. So, mastering the applications of derivatives will help the students solve a diverse range of problems, and thus, this chapter is an important part of the syllabus.

This article will explore some of the most important questions in the applications of derivatives with a focus on the most probable topics that are sure to be asked in the Class 12 examination. These topics are motion problems, optimisation curve sketching, and rates of change. Understanding these questions not only helps in getting good marks but also develops a deeper insight into the concept of calculus.

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Introduction to Applications of Derivatives

Before we go into the important questions on applications of derivatives, it is important to first revise the key concept. Derivatives describe how a function is changing concerning a variable. In everyday terms, derivatives help us grasp:

• Rates of change: how one quantity changes concerning another—for example, speed or population growth.
• Slopes of curves: The steepness or flatness of a curve at some point.
• Maxima and Minima: It is an important concept for finding a point that gives the maximum or minimum value of a function. Thus, optimisation problems can be solved.

Many applications of derivatives have significant applications in the real world, especially in physics, economics, and engineering. This is reviewed under the following topics of Class 12:

• Rate of Change of Quantities
• Tangents and Normals
• Increasing/Decreasing Functions and Extrema
• Optimisation Problems
• Motion and Speed
• Concavity and Points of Inflection

Now, let's talk about some important questions related to these applications.

Rate of Change of Quantities

Derivatives are useful in describing how one quantity varies concerning another. It is often used in applications that deal with speed, population growth, and many other physical processes.

Important Questions

1. If the radius of a circle is increasing at a rate of 2 cm/s, find the rate at which the area is changing when the radius is 4 cm.
2. If the radius of a sphere increases at a rate of 0.5 m/s, find the rate at which its volume is increasing when the radius is 3 m.

Tangents and Normals

A very common application of derivatives is to find the equation of a tangent or normal line to a curve at some point. The slope of the tangent line at that point is given by the derivative of the function at that location.

Important Questions 

1. Find the equation of the tangent and normal to the curve y=x²+2x at the point P(1,3).
2. Find the equation of the tangent to the curve y = x³ - 5x + 4 at the point (1,0).

3. Find the equation of the normal to the curve y = 2x² + 3x - 5 at x = 1.

Increasing/Decreasing Functions 

The derivative is useful in finding whether a function is increasing or decreasing. A positive derivative at a point shows that the function is increasing, whereas a negative derivative indicates that the function is decreasing.

Important Questions:

1. Find all intervals where f(x) = x³ - 3x² + 4 is either increasing or decreasing.
2. Find the intervals on which the function f(x) = x³ - 3x² + 2x is increasing and decreasing.

For the function f(x) = sin x + cos x, find the intervals where the function is increasing and decreasing.

Maxima and Minima (Optimisation Problems)

Optimisation problems deal with finding the maximum or minimum value of a function subject to certain constraints. These problems occur quite often in real life, like maximising profits, minimising costs, and reducing distance or time in physics and engineering.

Important Questions 

1. Find the dimensions of a rectangle with the maximum area that can be inscribed in a semicircle of radius 10 cm.
2. A person wants to build a rectangular fence. The cost of the fencing material is $5 per meter for the two sides and $3 per meter for the other two sides. What should the dimensions of the fence be to minimise the total cost, given that the perimeter is fixed?

Concavity and Points of Inflection

Concavity refers to the direction in which a function's graph curves, meaning whether it bends upward or downward. The places where these curvature changes are called points of inflexion. To find concavity, we use the second derivative.

Important Questions 

1. Find the points of inflexion for the function f(x) = x⁴ - 4x³ + 6x² - 2x.
2. For the function f(x) = x³ - 6x² + 9x + 5, determine the concavity of the function and the points of inflexion.

Curve Sketching

Curve sketching is the plotting of a graph based on information obtained from its first and second derivatives. Through these derivatives, it can indicate the existence of critical points and inflexion points, the increase or decrease in intervals, and the concavity of the graph.

Important Questions 

1. Sketch the curve of the function y = x³ + 6x² - 9x+ 1 using the information obtained from the first and second derivatives.
2. Sketch the curve of y = 1 / x and discuss its asymptotes, increasing/decreasing behaviour, and concavity.

Motion Problems

Motion problems include the use of derivatives to analyse topics such as velocity, acceleration, and displacement. Here, velocity is the derivative of position, and acceleration is the derivative of velocity.

Important Questions 

1. If the position of an object is given by s(t) = t³ - 6t² + 9t, find its velocity and acceleration at t = 2.
2. A car travels along a straight road with the displacement given by s(t) = 4t² - 2t + 7. Find the velocity and acceleration at t = 3.

Derivatives in Class 12 Mathematics equip students with essential skills for handling practical problems involving rate of change, motion, optimisation, and curve sketching. All these concepts are very important, as they occur not only in exams but also form the foundation for disciplines such as physics, economics, and engineering.

Through working with the Class 12 Applications of Derivatives Important questions, students can enhance their problem-solving skills and gain a deeper insight into how derivatives are used in modelling and solving real-life problems.