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Applications of Derivatives

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Class 12 Maths Chapter 6
Application of Derivatives
Important Questions

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For important questions from Class 12 Maths, Chapter 6 Application of Derivatives,” you can focus on various key topics that are typically covered in this chapter. Conceptual Understanding of rate of Change,Increasing and Decreasing Functions,Tangents and Normals are referring to your textbook.application of derivatives class 12 most important questions or additional academic resources can be very helpful.Tackling these questions not only boosts confidence but also understanding the concepts.

Introduction

The “Application of Derivatives” in Class 12 Mathematics is a significant chapter that extends the concepts of differentiation learned in earlier classes to practical and real-world situations. This chapter essentially deals with how derivatives can be used to understand the behavior of functions and solve various types of problems. . Let’s delve into a brief introduction of these concepts.class 12 application of derivatives important questions and answers on the basis for many theorems and applications in both pure and applied mathematics.

What are Application of Derivatives?

The applications of derivatives in mathematics and various fields are extensive and diverse. Derivatives, which are fundamental to calculus, are used to understand and describe the rate at which things change.In mathematics, derivatives serve as a fundamental tool for analysis, offering deep insights into the properties and behaviors of functions. The application of derivatives in mathematics encompasses various areas, from basic algebra to advanced calculus.

Class 12 Application of Derivatives Important Questions and Answers

Q1. The equation of the normal to the curve y = sin x at (0, 0) is
Options
(a) x = 0
(b) y = 0
(c) x + y =0
(d) x - y =0

Ans. (c) x + y = 0
Explanation:
Given: y = sin x
\text{Hence,} \dfrac{dy}{dx} = \text{cos x} \\[4.5 bp] \text{So, }\left(\dfrac{dy}{dx}\right) \text{at (0, 0) = cos 0 = 1}\\ Thus, the slope of the normal = \dfrac{-1}{\left(\dfrac{dy}{dx}\right)} = -1/1 = -1.
Therefore, the equation of the normal at (0, 0) is
y-0 =-1(x-0)
y = -x
Hence, x + y = 0 is the correct answer.

Q2: Find the interval in which function, f(x) = sin x + cos x, 0 ≤ x ≤ 2π is decreasing:
Options
(a) \left( \dfrac{\pi}{4}, \dfrac{5\pi}{4}\right) \\[4.5 bp] (b) \left( \dfrac{\pi}{4}, \dfrac{-5\pi}{4}\right) \\[4.5 bp] (c) \left( \dfrac{-\pi}{4}, \dfrac{5\pi}{4}\right) \\[4.5 bp] (d) \left( \dfrac{-\pi}{4}, \dfrac{\pi}{4}\right) \\[4.5 bp]

Ans. (a) \left( \dfrac{\pi}{4}, \dfrac{5\pi}{4}\right) \\[2.5 bp]

Explanation:
f(x) = sin x + cos x
f’(x) = cos x - sin x
Now, f’(x) = 0 gives
sin x = cos x,
which gives that x =\dfrac{π}{4}, \dfrac{5π}{4} as 0 ≤ x ≤ 2π.

Q3. The absolute maximum value of y = x^3 -3x+ 2 in 0 ≤ x ≤ 2?

Explanation:
Given: \text{y = x}^3 \text{- 3x + 2}\\ Therefore, \text{y’ = 3x}^2\text{ - 3}\\ For a point of absolute maximum or minimum, y’ = 0
Hence, x = ±1
Let y = f(x)
Therefore, f(0) = 0^3-3(0) +2 = 2\\ \text{f(1) = 1}^3 \text{– 3(1)+2 = 0}\\ \text{f(2) =} 2^3\text{ – 3(2) +2 = 4}\\ Hence, f(x) achieves a maximum value of 4 when x = 2.
Hence, the correct answer is an option (c) 4.

Q4. The line y = x + 1 is a tangent to the curve y² = 4x at the point.

Explanation:
y = x+1…(1)
y² = 4x …(2)
Substitute (1) in (2), we get
(x+1)^2 = 4x\\ x^2 \text{ + 1 + 2x = 4x}\\ x^2 \text{- 2x+1 = 0}, which is equal to (x-1)^2 = 0\\ ⇒ x = 1
Now, substitute x = 1 in y = x+1, we get
y = 1+1 = 2.
Hence, the line y = x+1 is a tangent to the curve y² = 4x at the point (1, 2).

Q5: The function f(x) = x + cos x is.

Explanation:
f(x) = x+cos x
f’(x) = 1 – sin x
f’(x)>0 for all values of x.
Since sin x is lying between -1 and +1, f(x) is always increasing

CBSE Class 12 Maths Chapter wise Important Questions

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Conclusion

The application of derivatives is a central concept in calculus and plays a crucial role in various fields, bridging theoretical mathematics with practical real-world applications. Application of derivatives is a testament to the incredible power and utility of mathematical concepts in understanding and shaping the world around us. Exploring cbse class 12 maths Application of derivatives important questions, becomes essential for a comprehensive grasp of this fundamental aspect of mathematics.. To enhance understanding, platforms like Oswal.io provide a wealth of resources, including question-answer sets, comprehensive class 12 questions and answers.

Frequently Asked Questions

Ans: The derivative of a function at a point is the rate at which the function’s value changes at that point. Mathematically, it’s the limit of the average rate of change of the function over a small interval, as the interval approaches zero.
Ans: Derivatives have numerous real-life applications including in physics (to determine velocity and acceleration), in economics (to find marginal cost and revenue), and in engineering (for optimizing structures and systems).
Ans: The first derivative of a function represents the slope of the tangent line to the function’s graph at a point. It indicates the rate of change of the function’s value with respect to its input.
Ans: By setting the first derivative of a function to zero, one can find its critical points. Analyzing these points and the behavior of the derivative around these points helps in determining local maxima and minima.
Ans: The second derivative is the derivative of the first derivative. It represents the rate of change of the rate of change of a function. It’s used to assess the concavity of a function and to find points of inflection.
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