The CBSE class 12 Term 2 Maths Chapter 6 Application of Derivatives is an extremely important chapter. Application of Derivatives is a part of Calculus which constitutes a lot of weightage in the term 2 CBSE class 12 Maths paper. A long answer-type question is confirmed to arrive in this chapter. The chapter includes six important topics. These include Decreasing and increasing functions, Newton’s method, Linear approximation, Rate of change of quantity, Maximum and minimum values, and Normal and tangent to a curve. Questions from one or more topics can be asked in three marks and four marks in the paper. We are just around the edge, and the preparation of the students has heated up.
A maximum of one or two questions will be asked from this segment of class 12 Maths. Since the question paper will be subjective, the students have to be well prepared to solve questions related to this chapter. In the article, we have covered all the important Maths concepts and made simple revision notes. Students can study our revision notes to ace this chapter.
What Are Derivatives
In the most basic sense, derivatives are the rate at which one quantity changes into another. This rate of change of function is represented in terms of functions. If a student has a function ‘y’ which changes concerning x such that y = kx then the derivative can be formulated as dy / dx = f (x) = y’.
The concept of derivatives is commonly employed in both large and small-scale companies. The idea of derivatives is crucial when evaluating or causing changes in temperature or the rate at which the water pumps out of a tank change. This latter element is dependent on several factors.
There are numerous applications of derivatives in the world. We shall focus only on the topics important for CBSE Class 12 Maths for this article.
What are Applications of Derivatives in Mathematics?
In previous chapters, students should have learned how to find the derivatives of various functions, such as implicit functions, trigonometric functions, and logarithmic functions. The derivatives of such functions have several uses. These applications may be found in mathematical principles as well as real-life circumstances.
Among these applications are:
- Decreasing and increasing functions
- Newton’s method
- Linear approximation
- A quantity’s rate of change
- Maximum and minimum values
- Normal and tangent to a curve
A derivative is defined as the rate of change of one quantity relative to another in its most basic form. In terms of functions, this rate of change of function is represented as dl / dm = f (l) = m’. The notion of derivatives is commonly employed in both large and small-sized companies.
This idea is crucial when determining or causing a change in temperature or the rate of change in an object’s size and form. This last part depends on various conditions. Before we jump to the next section of these revision notes, Class 12 Maths Chapter 6, let’s look at the important applications of derivatives in more depth.
Rate of Change of a Quantity
The most important and common use of derivatives is to figure out how quickly a number changes.
For example, if one wants to calculate the rate of change of a cube’s volume on the decreasing side, one can use the derivative form of ds / dt. The rate of change in the volume of a cube is represented by ds in this equation. dt, on the other hand, reflects the change in the cube’s edges.
Decreasing and Increasing Functions
Derivatives are used to determine whether a particular function decreases, grows, or remains constant. This may be accomplished with the use of a graph.
- F is a constant function in (m, n), if f’ (a) = 0 for each x belongs to (m, n)
- F is a decreasing function at (m, n), if f’ (a) < 0 for each x belongs to (m, n)
- F is an increasing function at (m, n), if f’ (a) > 0 for every x belongs to (m, n)
Learning about Tangents and Normal to a Curve
The application of tangent and normal to a curve is the next item we’ll study in these Maths Class 12 Chapter 6 Revision Notes. But first, let’s go through the fundamentals. A tangent is a line that intersects a curve at a specific location. This line does not intersect the curve. In addition, the normal is the line perpendicular to the tangent.
We can also write the straight-line equation that passes through a point, which has a slope m, as:
y- y1 = m (x – x1)
From this equation above, we can see that the slope of the tangent to the curve y = f (x) and at the point P (x1, y1), it is given as dy / dx at P (x1, y1) = f’ (x).
Hence, the equation of the tangent to the curve at P (x1, y1) can also be written as:
y – y1 = f’ (x1) (x – x1)
We can also write the equation of normal to the curve as:
y – y1 = [-1 / f’ (x1)] (x – x1)
The same can also be written as:
(y – y1) f’ (x1) + (x – x1) = 0
Concept of Maxima and Minima
The concept of minima and maxima is yet another key topic that we will be covering during these NCERT Class 12 Practice Notes Maths. According to the notes, the minima of a graph are the points over which the graph is at its lowest point.
Furthermore, the highest point on a graph is the maximum. A person can use a derivative function to compute the lowest and highest points of a curve in a graph or to discover the point at which the graph turns around.
- When p = k and if f (p) ≤ f (k), for every p in the domain, then f (p) has an absolute maximum value. Also, point a is the point of the maximum value.
- When p = k and if f (p) ≥ f (k), for each k in some open interval (a, b), then f (p) has a relative minimum value.
- When p = k and if f (p) ≥ f (k), for each p in the domain, then f (p) has an absolute minimum value. Also, point a is the point for the minimum value.
- When p = k and if f (p) ≤ f (k). for each p in some open interval (a, b), then f (p) has a maximum relative value.
Point of Inflection
In addition to the moment of inflection, there is another essential matter that we need to address.
A point of inflexion is defined as the point at which a tiny continuous function f (x) equals zero or where an infinitesimal continuous function f (x) does not exist at locations or points where an infinitesimal continuous function f’ (x0) exists, and where a tiny continuous function f (x) changes sign when it passes through x = x0, respectively.
There are also two important cases here. And these cases are:
- If f” (x) < 0, x ∈ (a, b), then the curve y = f (x) is concave downward
- If f” (x) > 0, x ∈ (a, b), then the curve y = f (x) is concave upward in the case of (a, b)
For example, if we have to solve the equation f (x) = sin x, we can solve it in the manner mentioned below.
F’ (x) = cos x
F” (x) = sin x = 0 x = n π, n ∈ z
Conclusion
Application of Derivatives is a crucial chapter for Class 12 Board Exams. Also, it plays a major role in competitions like JEE Main exam and JEE Advanced exam. Questions from ‘Application of Derivatives’ dominate the JEE 2022, CUCET NTA 2022, and other olympiads in India. The national engineering Olympiad and KVPY also contain questions from this chapter. The start of the term 2 class 12 board test is less than a month away. A student can easily obtain decent grades in every subject with an adequate approach and revision of crucial ideas. Students must have completed their course and must now concentrate solely on revisions and the completion of mock papers. Look up NCERT solutions for class 12 Maths if you are stuck on a problem.
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