CBSE Class 12 Mathematics Application of Derivatives Notes Set 02

BASIC CONCEPTS

1. Rate of Change: If a quantity \( y \) varies with another quantity \( x \), satisfying some rule \( y = f(x) \), then \( \frac{dy}{dx} \bigg|_{x = x_0} \) (or \( f'(x_0) \)) represents the rate of change of \( y \) with respect to \( x \) at \( x = x_0 \).

2. Differentials: Let \( y = f(x) \) be any function of \( x \) which is differentiable in \( (a, b) \). The derivative of this function at some point \( x \) of \( (a, b) \) is given by the relation \[ \frac{dy}{dx} = \lim_{\Delta x \to 0} \frac{\Delta y}{\Delta x} = \lim_{\Delta x \to 0} \frac{f(x + \Delta x) - f(x)}{\Delta x} = f'(x) \]
\( \implies \) \( \frac{dy}{dx} = f'(x) \)
\( \implies \) \( dy = f'(x) dx \), where \( dy \) is called the differential of the function.
Note: Formula \( dy = f'(x) dx \) or \( \Delta y = f'(x) \Delta x \) is very useful in measuring the errors in the dependent variable for given error in independent variable.

• (i) Absolute Error: The error \( \Delta x \) in \( x \) is called the absolute error.
• (ii) Relative Error: If \( \Delta x \) is error in \( x \) then ratio \( \frac{\Delta x}{x} \) is called the relative error.
• (iii) Percentage Error: If \( \frac{\Delta x}{x} \) is relative error, then \( \frac{\Delta x}{x} \times 100 \) is called percentage error in \( x \).

3. Tangents and Normals:

(i) Slope and equation of tangent:
Slope of chord \( AB = \frac{f(a + \Delta x) - f(a)}{a + \Delta x - a} = \frac{f(a + \Delta x) - f(a)}{\Delta x} \)
Obviously, if \( \Delta x \to 0 \) comes very close to A and then chord AB becomes tangent at A i.e., \( x = a \).
i.e., slope of tangent at \( x = a(a, f(a)) = \lim_{\Delta x \to 0} \frac{f(a + \Delta x) - f(a)}{\Delta x} = [f'(x)]_{(a, f(a))} = \left( \frac{dy}{dx} \right)_{(a, f(a))} \)
Hence, equation of tangent to the curve \( y = f(x) \) at the point \( (x_1, y_1) \) is given by
\( (y - y_1) = \left( \frac{dy}{dx} \right)_{(x_1, y_1)} (x - x_1) \) [\( \because \) Equation of line is \( y - y_1 = m(x - x_1) \), where \( m \) is slope]

(ii) If \( \frac{dy}{dx} = \infty \) at the point \( P(x_1, y_1) \), then the tangent at \( P \) is parallel to y-axis and its equation is given by
\( \infty = \frac{y - y_1}{x - x_1} \)
\( \implies \) \( x - x_1 = 0 \)
\( \implies \) \( x = x_1 \)
[Note : In this case, \( \frac{dx}{dy} \) at \( P(x_1, y_1) = 0 \)]

(iii) If \( \frac{dy}{dx} = 0 \), at the point \( P(x_1, y_1) \), then the tangent at \( P \) is parallel to x-axis and equation is given as
\( 0 = \frac{y - y_1}{x - x_1} \)
\( \implies \) \( y - y_1 = 0 \)
\( \implies \) \( y = y_1 \)

(iv) Obviously, normal to the curve \( y = f(x) \) at \( P(x_1, y_1) \) is perpendicular to the tangent at \( P(x_1, y_1) \).
\( \therefore \) Slope of Normal \( = \frac{-1}{\text{slope of tangent}} = \frac{-1}{\left( \frac{dy}{dx} \right)_{(x_1, y_1)}} \)
Hence, equation of normal to the curve \( y = f(x) \) at \( P(x_1, y_1) \) is
\( (y - y_1) = \frac{-1}{\left[ \frac{dy}{dx} \right]_{(x_1, y_1)}} (x - x_1) \)

(v) If \( \frac{dy}{dx} \) at the point \( P(x_1, y_1) \) is zero, then the equation of normal is \( x = x_1 \).

(vi) If \( \left( \frac{dy}{dx} \right) \) at the point \( (x_1, y_1) \) does not exist, then the equation of normal is \( y = y_1 \).

(vii) The angle \( \theta \) between two given curves \( y_1 = f_1(x) \) and \( y_2 = f_2(x) \) at a point \( (x_1, y_1) \) is given by
\( \tan \theta = \left| \frac{m_1 - m_2}{1 + m_1 m_2} \right| \) where \( (x_1, y_1) \) is the point of intersection and \( m_1, m_2 \) are slopes of their tangents at point \( (x_1, y_1) \).
Note: The gradient of a curve at a point is defined as the slope of tangent to the curve at that point.

4. Nature of Function: To know the behaviour of a function in an interval, we study the properties of increasing or decreasing functions, maximum and minimum of the functions.

5. Monotonic Function: A function is said to be monotonic in an interval, if it is either increasing or decreasing in the given interval.

6. Increasing Function: A function \( f(x) \) is said to be an increasing function in \( (a, b) \) if
\( x_1 < x_2 \)
\( \implies \) \( f(x_1) \leq f(x_2) \quad \forall x_1, x_2 \in (a, b) \)
In this way. We can say
\( f(x) \) is increasing in \( (a, b) \) if \( \forall x \in (a, b), f'(x) > 0 \)
Obviously, the angle \( \theta \) made by tangent with +ve direction of x-axis in interval \( (a, b) \) is acute.
\( \implies \) \( \tan \theta \) is +ve
\( \implies \) slope is +ve
\( \implies \) \( \frac{dy}{dx} = f'(x) > 0 \)

7. Decreasing Function: A function \( f(x) \) is said to be decreasing in the interval \( (a, b) \) if
\( x_1 < x_2 \)
\( \implies \) \( f(x_1) \geq f(x_2) \quad \forall x_1, x_2 \in (a, b) \)
In other way,
\( f(x) \) is decreasing in interval \( (a, b) \) if \( \forall x \in (a, b), f'(x) < 0 \)
Obviously, the angle \( \theta \) made by tangent with +ve direction of x-axis in interval \( (a, b) \) is obtuse.
\( \implies \) \( \tan \theta \) is -ve
\( \implies \) slope is -ve
\( \implies \) \( \frac{dy}{dx} = f'(x) < 0 \)
[Note: A function \( f(x) \) is said to be:
Strictly increasing if \( x_1 < x_2 \implies f(x_1) < f(x_2) \quad \forall x_1, x_2 \in (a, b) \)
Strictly decreasing if \( x_1 < x_2 \implies f(x_1) > f(x_2) \quad \forall x_1, x_2 \in (a, b) \)]

8. Maximum and Minimum Value of a Function (or Absolute Maximum or Minimum Value)
A function \( f \) is said to attain maximum value at a point \( a \in D_f \), if \( f(a) \geq f(x) \quad \forall x \in D_f \) then \( f(a) \) is called absolute maximum value of \( f \).
A function \( f \) attains minimum value at \( x = b \in D_f \), if \( f(b) \leq f(x) \quad \forall x \in D_f \) then \( f(b) \) is called absolute minimum value of \( f \).
Note that a function \( 'f' \) may have maximum (or minimum) values in some parts (intervals) of the domain. Such values may occur at more than one point. These are therefore, called local (or relative) maxima (or minima).

9. Local Maxima and Local Minima (or Relative Extrema)
Local Maxima: A function \( f(x) \) is said to attain a local maxima at \( x = a \), if there exists a neighbourhood \( (a - \delta, a + \delta) \) of \( 'a' \) such that \( f(x) < f(a) \quad \forall x \in (a - \delta, a + \delta), x \neq a \), then \( f(a) \) is the local maximum value of \( f(x) \) at \( x = a \).
Local Minima: A function \( f(x) \) is said to attain a local minima at \( x = a \), if there exists a neighbourhood \( (a - \delta, a + \delta) \) of \( 'a' \) such that \( f(x) > f(a) \quad \forall x \in (a - \delta, a + \delta), x \neq a \), then \( f(a) \) is called the local minimum value at \( x = a \).
Caution:
(i) A function defined in an interval can reach maximum or minimum values only for those values of \( x \) which lie within the given interval.
(ii) One should not think that the maximum and minimum of a function are its respective largest and smallest values over a given interval.

10. Test for Identifying Relative (Local) Maxima or Minima

(i) First Derivative Test
Step I: Find \( f'(x) \)
Step II: The equation \( f'(x) = 0 \) is solved to get critical points \( x = c_1, c_2, \dots \dots, c_n \).
Step III: The sign of \( f'(x) \) is studied in the neighbourhood of each critical points. Let one critical point be \( x = c \).
If the sign of \( f'(x) \) changes from +ve to -ve as \( x \) increases through \( c \) (from left to right of \( c \)), then \( x = c \) is relative maxima and \( f(c) \) is relative maximum value.
If the sign of \( f'(x) \) changes from -ve to +ve as \( x \) increases through \( c \) (from left to right of \( c \)), then \( x = c \) is relative minima and \( f(c) \) is relative minimum value.

(ii) Second order derivative test
Step I: Find \( f'(x) = 0 \)
Step II: The equation \( f'(x) = 0 \) is solved to get critical points \( x = c_1, c_2, \dots \dots, c_n \).
Step III: \( f''(x) \) is obtained and the sign of \( f''(x) \) is studied for all critical points \( x = c_1, c_2, \dots \dots, c_n \).
 

11. Critical point: A point \( x = c \) is called critical point of the function \( f(x) \), if \( f(c) \) exists and either \( f'(c) = 0 \) or \( f'(c) = \infty \) (does not exist).

12. Point of Inflexion: If \( f(x) \) is a function and \( x = c \) is critical point, then \( x = c \) is called point of inflexion if
(i) \( f'(c) = 0 \)
(ii) \( f''(c) = 0 \)
(iii) \( f'''(c) \neq 0 \)