CBSE Class 12 Mathematics Notes Chapter 6 Applications of Derivatives

Applications of derivatives can be found in both mathematics and real life. For example, derivatives are used in mathematics to calculate the rate of change of a quantity, the approximation value, the equation of the tangent and normal to a curve, and the minimum and maximum values of algebraic expressions. The most common applications of derivatives can be observed in: 

• Finding the Rate of Change of a Quantity
• Finding the Approximation Value
• Finding the equation of a Tangent and Normal To a Curve
• Finding Maxima and Minima, and Point of Inflection
• Determining Increasing and Decreasing Functions

Derivatives are commonly used in areas such as science, engineering, and physics. In this article, we will look at how derivatives may be applied in real life. Application of derivatives is important for class 12 students, as well as engineering mathematics. CBSE Class 12 Mathematics Notes for Chapter 6 Application of Derivatives are given in the article below for easy preparation and understanding of the concepts involved.

Read More: 

---

Class 12 Mathematics Chapter 6 Notes – Applications of Derivatives

Rate of Change of Quantities

• If a variable quantity y is some function of time i.e. y = f(t), then a small change in time Δt has a corresponding change Δy in y.
• Thus the average rate of change = Δy/Δt.
• When limit Δt → 0 is applied, the rate of change becomes instantaneous and we get the rate of change with respect to t at that instant t i.e. lim _{Δt →0} \(\frac{\Delta y}{\Delta t} = \frac{dy}{dt}\).
• Hence it is clear that the rate of change of any variable with respect to some other variable is derivative of the first variable with respect to the other variable.

Increasing Function

• These functions are of two types
  • Strictly increasing function
  • Non-decreasing function

Strictly Increasing Function

• A function f (x) is known as a strictly increasing function in its domain if,

x₁ < x₂ ⇒ f(x₁) < f(x₂)

• Therefore, for the smaller input, we have a smaller output, and for a higher value of the input, we have a higher output.
• Graphically, it can be represented as

Strictly increasing function

Non-Decreasing Functions

• A function f(x) is said to be non-decreasing, if 

x₁ < x₂ ⇒ f(x₁) ≤ f(x₂)

• Graphically, it can be represented as

Non-decreasing function

Decreasing Functions

• These functions are also of two types
  • Strictly decreasing function
  • Non-increasing function

Strictly Decreasing Function

• A function f (x) is known as a strictly decreasing function in its domain if,

x₁ < x₂ ⇒ f(x₁) > f(x₂)

• Therefore, for the smaller input, we have a higher output, and for a higher value of the input, we have a smaller output.
• Graphically, it can be represented as

Strictly decreasing function

Non-Increasing Function

• A function f(x) is said to be non-decreasing, if 

x₁ < x₂ ⇒ f(x₁) ≥ f(x₂)

• Graphically, it can be represented as

Non-increasing function

Monotonic Function

• A function f is said to be monotonic in interval I if it is either increasing or decreasing in interval I.
• f(x) = ln x, f(x) = 2^x, f(x) = -2x + 3 are monotone functions.
• f(x) = x² is monotonic in (-∞, 0) or (0, ∞) but is not monotonic in R.

Properties of Monotonic Functions

• If f(x) is continuous on [a, b] such that f’(c) ≤ 0 (f’(c) < 0) for each c ϵ (a, b), then f(x) is monotonically decreasing function on [a, b].
• If f(x) is continuous on [a, b] such that f’(c) ≥ 0 [f’(c) > 0] for each c ϵ (a, b), then f(x) is monotonically increasing function on [a, b].

Tangent and Normal

• Slope of Tangent: let y = f(x) be a continuous curve and let P(x₁, y₁) be the point on it, then

(dy/dx) = tanθ = Slope of the tangent at P

• Equation of Tangent: The equation of the tangent to any curve at the point P(x₁, y₁) is

(y - y₁) = (dy/dx)(x - x₁)

• Slope of Normal: The normal to the curve at P(x₁, y₁) is a line perpendicular to the tangent at P(x₁, y₁) and passing through P.

Slope of normal at P = 1/(Slope of the tangent at P)

• Equation of Normal: The equation of normal to any curve at the point P(x₁, y₁) is

(y - y₁) = - [1/(dy/dx)](x - x₁)

Critical Points

• It is a collection of points for which
  • f(x) does not exists
  • f’(x) does not exist, or
  • f’(x) = 0
• All the values of x obtained from the above conditions are said to be critical points.
• It should be noted that critical points are the interior points of an interval.

Maxima

• Let y = f(x) be a function defined at x = a and also in the neighborhood of x = a.
• Then, f(x) is said to have a maximum or local maximum at x = a, if the value of the function at x = a is greater than the value of the function at the neighborhood points of x = a, i.e.

f(a) > f(a + h) and f(a) > f(a - h), where h > 0

Minima

• Let y = f(x) be a function defined at x = a and also in the neighborhood of x = a.
• Then, f(x) is said to have a minimum or local minimum at x = a if the value of the function at x = a is less than the value of the function at the neighborhood points of x = a, i.e.

f(a) < f(a + h) and f(a) < f(a - h), where h > 0

First Derivative Test

• Let f be a function defined on an open interval I that is continuous around a critical point c in I. Then,
  • If the sign of ff'(x) changes from positive to negative as x increases through c, then c is a local maxima.
  • If the sign of f'(x) changes from negative to positive as x increases through c, then c is a local minima.
  • If the sign of f'(x) does not change as x increases through c, then c is neither a local maxima nor a local minima. This is known as an inflection point.

Second Derivative Test

• Let f(x) be a function defined on the interval I, with c ∈ I. Let f be twice differentiable at c. Then,
  • x = c is a local maxima if f'(c) = 0 and f"(c) < 0.
  • x = c is a local minima if f'(c) = 0 and f"(c) > 0.
  • The test fails if both f'(c) and f"(c) is equal to 0.

Rolle’s Theorem

• If a function f(x)
  • is continuous in the closed interval [a, b]
  • is differentiable in an open interval (a, b) i.e. differentiable at each point in the open interval (a, b)
  • and, f(a) = f(b)
• Then, there will be at least one point c in the interval (a, b) such that f’(c) = 0.

Rolle’s Theorem

Lagrange’s Mean Value Theorem

• If a function f(x)
  • is continuous in the closed interval [a, b]
  • is differentiable in an open interval (a, b).
• Then, there exists at least one point c, where a < c < b such that

f’(c) = [f(b) - f(a)] / (b - a)

Lagrange’s Mean Value Theorem

There are Some important List Of Top Mathematics Questions On Applications Of Derivatives Asked In CBSE CLASS XII

---

Check-Out: