Revision: 12th Std >> Applications of Derivatives MAH-MHT CET (PCM/PCB) Applications of Derivatives

The derivative \[\frac{dy}{dx}\] represents the rate of change of a variable (y) with respect to another variable (x).

In general, if a quantity depends on another, then its derivative gives the instantaneous rate of change of that quantity.

A function ( f(x) ) is said to be a decreasing function on (a, b) if x₁ < x₂ ⇒ f(x₁) ≥ f(x₂)

Strictly Decreasing Function:

• If x₁ < x₂ ⇒ f(x₁) > f(x₂)

A function ( f ) is said to be monotonic in an interval if it is either increasing or decreasing in that interval.

A function ( f(x) ) is said to be an increasing function on ((a, b)) if x₁ < x₂ ⇒ f(x₁) ≤ f(x₂)

Strictly Increasing Function:

• If x₁ < x₂ ⇒ f(x₁) < f(x₂)

Slope of tangent at a point:\[\frac{dy}{dx}\]

Slope of normal: \[-\frac{1}{\frac{dy}{dx}}\]

\[\mathrm{f(a+h)\approx f(a)+h~f^{\prime}(a)}\]

Statement:

If a function f(x):

1. Is continuous on the closed interval [a,b]

2. Is differentiable on the open interval (a,b)

3. Satisfies f(a) = f(b)

Then there exists at least one $c\in (a,b)$ such that:

\[f^{\prime}(c)=0\]

Statement: 

If a function f(x):

1. Is continuous on the closed interval [a,b]

2. Is differentiable on the open interval (a,b)

Then there exists at least one number c ∈ (a,b) such that:

\[f^{\prime}(c)=\frac{f(b)-f(a)}{b-a}\]

Equation of tangent at (x₁,y₁): \[y-y_1=\left(\frac{dy}{dx}\right)_{x_1}(x-x_1)\]

Equation of normal:

\[y-y_1=-\frac{1}{\left(\frac{dy}{dx}\right)_{x_1}}(x-x_1)\]

Tangent parallel to X-axis: \[\frac{dy}{dx}=0\]

Tangent parallel to Y-axis: \[\frac{dy}{dx}=\infty\quad(\mathrm{or~}\frac{dx}{dy}=0)\]

1. Basic Meaning:

• Derivative represents the rate of change of y with respect to x

2. Velocity:

• Velocity = rate of change of displacement
• v = \[\frac{ds}{dt}\]
• At t = 0 → initial velocity
• v > 0 → motion forward
• v < 0 → motion backward
• v = 0 → particle at rest

3. Acceleration:

• Acceleration = rate of change of velocity
• a = \[\frac{dv}{dt}\] = \[\frac{d^2s}{dt^2}\]
• a > 0 → velocity increasing
• a < 0 → velocity decreasing
• a = 0 → constant velocity

• f′(x) > 0 ⇒ function increasing
• f′(x) < 0 ⇒ function decreasing
• f′(x) = 0 ⇒ function constant

First Derivative Test:

Maxima at x = c:

• f′(c) = 0
• f′(c − h) > 0 and f′(c + h) < 0

Minima at x = c:

• f′(c) = 0
• f′(c − h) < 0 and f′(c + h) > 0

Second Derivative Test:

• If f′(a) = 0 and f″(a) < 0 → Maximum
• If f′(a) = 0 and f″(a) > 0 → Minimum
• If f″(a) = 0 → Test fails (use first derivative)