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 an increasing function on (a, b) if x₁ < x₂ ⇒ f(x₁) ≤ f(x₂)

Strictly Increasing Function:

• If x₁ < x₂ ⇒ f(x₁) < f(x₂)

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 is said to be constant on I if f(x) = c for every x ∈ I, where c is a constant.

Let x₀ be a point in the domain of a real-valued function f.

The function f is said to be increasing at x₀ if there exists an open interval containing x₀ in which f is increasing.

Similarly, f is said to be decreasing at x₀ if there exists an open interval containing x₀ in which f is decreasing.

A point in the domain of a function is called a critical point if either the derivative is zero there or the derivative does not exist there. Critical points are checked while locating possible maxima or minima.

The points where a function changes from decreasing to increasing or from increasing to decreasing are called turning points.

Let f be a function defined on an interval I.

Maximum value: f has a maximum value at c ∈ I if \[ \boxed{f(c) \geq f(x) \quad \text{for all } x \in I} \]

The value f(c) is called the maximum value and c is called a point of maximum.

Minimum value: f has a minimum value at c ∈ I if \[ \boxed{f(c) \leq f(x) \quad \text{for all } x \in I} \]

The value f(c) is called the minimum value and c is called a point of minimum.

Extreme value: A maximum or minimum value of f is called an extreme value.

 
           Maximum value                          Minimum value

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:

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

Assume f'(c) = 0 and the second derivative exists at c:

• Local Maximum: f''(c) < 0

• Local Minimum: f''(c) > 0

• Test Fails: f''(c) = 0. If this happens, you must go back and use the First Derivative Test to check if it is a maxima, minima, or point of inflection.

Let c be a critical point of a continuous function f:

• Local Maximum: If f'(x) changes sign from positive to negative as x passes through c, then cc is a point of local maximum.

• Local Minimum: If f'(x) changes sign from negative to positive then c is a point of local minimum.

• Point of Inflection: f'(x) does not change sign as x passes through c (it is neither a maxima nor a minima).

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

• Increasing means output does not decrease as input increases.
• Strictly increasing means output always increases.
• Decreasing means output does not increase as input increases.
• Monotonic means either increasing or decreasing on an interval.
• f′(x) > 0 implies increasing, f′(x) < 0 implies decreasing, and f′(x) = 0 on an interval implies constant behaviour.
• If \[ f'(x) = 0 \] throughout an interval, the function is constant on that interval.
• A single point where \[ f'(x) = 0 \] does not necessarily make the function constant.
• To find intervals of increase or decrease, find the zeros of f'(x), divide the domain into intervals, and check the sign of f'(x).
• A function that is increasing or decreasing on an interval is called monotonic on that interval.
• A function may be increasing on one interval and decreasing on another; in that case it is not monotonic on its entire domain.

• Maxima and minima are extreme values of a function.

• Critical points occur where \(f'(x)=0\) or \(f'(x)\) is not defined.

• If \(f'(x)\) changes from positive to negative, the function has a local maximum.

• If \(f'(x)\) changes from negative to positive, the function has a local minimum.

• If \(f''(c) < 0\), there is a local maximum at \(x=c\).

• If \(f''(c) > 0\), there is a local minimum at \(x=c\).

• For absolute extrema on \([a,b]\), compare values at critical points and endpoints.

• Not every critical point gives a maximum or minimum.

• The second derivative test is quick, but the first derivative test is often more reliable in detailed reasoning.