Summary

Rate of Change of Quantities

Derivative as a Rate

If $y = f(x)$, then $\frac{dy}{dx}$ represents the instantaneous rate of change of $y$ with respect to $x$.

Positive/Negative Rates

If $\frac{dy}{dx} > 0$, then $y$ increases as $x$ increases. If $\frac{dy}{dx} < 0$, then $y$ decreases as $x$ increases.

Related Rates (Chain Rule)

If two variables $x$ and $y$ both vary with respect to time $t$, and $y=f(x)$, then $$\frac{dy}{dt} = \frac{dy}{dx} \cdot \frac{dx}{dt}$$ This is the key formula for related rates problems.

Marginal Cost (MC) and Marginal Revenue (MR)

If $C(x)$ is total cost and $R(x)$ is total revenue, then $$MC = \frac{dC}{dx}, \qquad MR = \frac{dR}{dx}$$ where $x$ denotes the number of units produced or sold.

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Increasing and Decreasing Functions

Let $f$ be continuous on $[a,b]$ and differentiable on $(a,b)$.

• Strictly Increasing: If $f'(x) > 0$ for all $x \in (a,b)$, then $f$ is strictly increasing on $[a,b]$.
• Strictly Decreasing: If $f'(x) < 0$ for all $x \in (a,b)$, then $f$ is strictly decreasing on $[a,b]$.
• Constant Function: If $f'(x) = 0$ for all $x \in (a,b)$, then $f$ is constant on $[a,b]$.
• Critical Points: Points where $f'(x) = 0$ or $f'(x)$ does not exist are called critical points. These points help divide the domain into intervals on which the function may be increasing or decreasing.
• Important Note: A function can still be strictly increasing even if $f'(x)=0$ at isolated points, provided the function never decreases. For example, $f(x)=x^3$ is strictly increasing on $\mathbb{R}$.

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Maxima and Minima

First Derivative Test

Let $c$ be a critical point of $f$.

1. If $f'(x)$ changes sign from positive to negative as $x$ passes through $c$, then $f(c)$ is a local maximum.
2. If $f'(x)$ changes sign from negative to positive as $x$ passes through $c$, then $f(c)$ is a local minimum.
3. If $f'(x)$ does not change sign as $x$ passes through $c$, then $c$ is not a point of local extremum. In many such cases it may be a point of inflection, but that conclusion should be checked separately.

Second Derivative Test

Suppose $f'(c)=0$.

1. If $f''(c) < 0$, then $f(c)$ is a local maximum.
2. If $f''(c) > 0$, then $f(c)$ is a local minimum.
3. If $f''(c) = 0$, the test fails and we must use the First Derivative Test or some other method.

• Important Note: Every local maximum or minimum occurs at a critical point, but every critical point is not necessarily a maximum or minimum.

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Absolute Maxima and Minima

Extreme Value Theorem

A continuous function on a closed interval $[a,b]$ must attain both an absolute maximum and an absolute minimum.

Working Rule

1. Find all critical points in the open interval $(a,b)$.
2. Evaluate the function at each critical point.
3. Evaluate the function at the endpoints $a$ and $b$.
4. Compare all these values.
5. The greatest value is the absolute maximum, and the smallest value is the absolute minimum. Important Note: For open intervals or unbounded domains, absolute extrema may not exist.

Important Exam Tips for Board Exams

Declare Variables and Units Clearly

In rate of change problems, first define every variable clearly, such as "Let $r$ be the radius" or "Let $V$ be the volume." Write the final answer with proper units like $\text{cm}^2/\text{s}$ or $\text{m}^3/\text{min}$.

Differentiate with Respect to the Correct Variable

In related rates, the question usually involves time, so differentiate with respect to $t$, not with respect to a geometric variable like $r$ or $x$.

Show the Second Derivative Check in Optimization

In maximum-minimum word problems, after finding the critical point, verify whether it gives a maximum or minimum by checking the sign of the second derivative whenever possible.

Check Endpoints for Absolute Extrema

If the question asks for absolute maximum or minimum on a closed interval, always check the values at endpoints along with the critical points.

Organized Sign Analysis

In increasing-decreasing questions, factorize $f'(x)$ carefully, mark the critical points, and show the sign of $f'(x)$ in each interval before writing the conclusion.

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Important Exam Tips for JEE Main and Advanced

Shortest Distance Principle

For shortest distance between a curve and a line, the joining segment at the nearest point is along a normal to the curve. This usually leads to the condition that the tangent to the curve is parallel to the given line.

Monotonicity with Parameters

If a function is required to be increasing on all of $\mathbb{R}$, then usually we need $f'(x) \ge 0$ for all real $x$. For a quadratic derivative $Ax^2+Bx+C$, this requires $A>0$ and discriminant $D = B^2 - 4AC \le 0$.

Beware of Non-Differentiable Points

For modulus functions, the derivative may fail to exist where the expression inside modulus becomes zero. Such points must be checked separately.

Optimization Results Worth Remembering

1. The rectangle of maximum area inscribed in a circle is a square.
2. The altitude of the cone of maximum volume inscribed in a sphere of radius $R$ is $\frac{4R}{3}$.
3. The height of the cylinder of maximum volume inscribed in a sphere of radius $R$ is $\frac{2R}{\sqrt{3}}$.

Exact Language Matters

"Strictly increasing" and "non-decreasing" are not always identical statements. Read the question carefully and justify the conclusion using derivative sign analysis.