Application of Derivatives (JEE)

Cheatsheet Content

1. Rate of Change of Quantities If $y = f(x)$, then $\frac{dy}{dx}$ represents the rate of change of $y$ with respect to $x$. If $x=f(t)$ and $y=g(t)$, then $\frac{dy}{dx} = \frac{dy/dt}{dx/dt}$, provided $\frac{dx}{dt} \neq 0$. Interpretation: $\frac{dy}{dx} > 0 \implies y$ increases as $x$ increases. $\frac{dy}{dx} Common Mistakes: Units of rate of change. E.g., area in $cm^2/s$, volume in $m^3/hr$. 2. Tangents and Normals Slope of Tangent: At point $(x_1, y_1)$ to curve $y=f(x)$ is $m_T = \left(\frac{dy}{dx}\right)_{(x_1, y_1)}$. Equation of Tangent: $y - y_1 = m_T(x - x_1)$. Slope of Normal: $m_N = -\frac{1}{m_T}$ (if $m_T \neq 0$). Equation of Normal: $y - y_1 = m_N(x - x_1)$. Special Cases: If $m_T = 0$, tangent is $y=y_1$ (horizontal). Normal is $x=x_1$ (vertical). If $m_T$ is undefined, tangent is $x=x_1$ (vertical). Normal is $y=y_1$ (horizontal). Angle between two curves: Angle between their tangents at the point of intersection. If $m_1, m_2$ are slopes, $\tan \theta = \left|\frac{m_1 - m_2}{1 + m_1 m_2}\right|$. Orthogonal Curves: If $m_1 m_2 = -1$ at intersection point. Length of Tangent: $PT = |y_1|\sqrt{1 + (1/m_T)^2}$ Length of Normal: $PN = |y_1|\sqrt{1 + m_T^2}$ Length of Subtangent: $ST = |\frac{y_1}{m_T}|$ Length of Subnormal: $SN = |y_1 m_T|$ 3. Monotonicity (Increasing/Decreasing Functions) Strictly Increasing (S.I.): $f'(x) > 0$ for all $x$ in interval $I$. Increasing: $f'(x) \ge 0$ for all $x$ in interval $I$, and $f'(x)=0$ at isolated points only. Strictly Decreasing (S.D.): $f'(x) Decreasing: $f'(x) \le 0$ for all $x$ in interval $I$, and $f'(x)=0$ at isolated points only. Algorithm: Find $f'(x)$. Set $f'(x) = 0$ to find critical points. Mark critical points on the number line and check the sign of $f'(x)$ in each interval. Trap: $f'(x) \ge 0$ does not guarantee strictly increasing. E.g., $f(x)=x^3$, $f'(x)=3x^2 \ge 0$. Must Remember: For a function to be monotonic over an interval, $f'(x)$ must maintain the same sign throughout that interval. 4. Maxima and Minima (Local Extrema) Critical Points: Points where $f'(x)=0$ or $f'(x)$ is undefined. These are candidates for local extrema. First Derivative Test: If $f'(x)$ changes sign from $+$ to $-$ at $x=a$, then $x=a$ is a point of local maxima. If $f'(x)$ changes sign from $-$ to $+$ at $x=a$, then $x=a$ is a point of local minima. If $f'(x)$ does not change sign at $x=a$, then $x=a$ is a point of inflection (not extremum). Second Derivative Test: Find $f'(x)$ and $f''(x)$. Set $f'(x)=0$ to find critical points, say $x=c$. If $f''(c) If $f''(c) > 0 \implies x=c$ is a point of local minima. If $f''(c) = 0$, test fails. Use First Derivative Test. Absolute Maxima/Minima (Global Extrema) on $[a, b]$: Find all local extrema in $(a, b)$. Evaluate $f(x)$ at these points and at endpoints $a, b$. The largest value is absolute maximum, smallest is absolute minimum. Common Mistake: Confusing local extrema with absolute extrema. Edge Case: Discontinuous functions may have extrema at points of discontinuity or boundary points. 5. Concavity and Point of Inflection Concave Upward (Convex): $f''(x) > 0$ for $x$ in an interval. Tangents lie below the curve. Concave Downward: $f''(x) Point of Inflection: A point where the concavity changes. Condition: $f''(x)=0$ or $f''(x)$ is undefined, AND $f''(x)$ changes sign around that point. If $f''(x)=0$ and $f'''(x) \neq 0$ at $x=c$, then $(c, f(c))$ is a point of inflection. 6. Mean Value Theorems Rolle's Theorem: If $f(x)$ is continuous on $[a, b]$, differentiable on $(a, b)$, and $f(a) = f(b)$, then there exists at least one $c \in (a, b)$ such that $f'(c) = 0$. Lagrange's Mean Value Theorem (LMVT): If $f(x)$ is continuous on $[a, b]$ and differentiable on $(a, b)$, then there exists at least one $c \in (a, b)$ such that $f'(c) = \frac{f(b) - f(a)}{b - a}$. Geometric Interpretation: There is a point $c$ where the tangent is parallel to the secant joining $(a, f(a))$ and $(b, f(b))$. Cauchy's Mean Value Theorem (CMVT): If $f(x)$ and $g(x)$ are continuous on $[a, b]$ and differentiable on $(a, b)$, and $g'(x) \neq 0$ for any $x \in (a, b)$, then there exists at least one $c \in (a, b)$ such that $\frac{f'(c)}{g'(c)} = \frac{f(b) - f(a)}{g(b) - g(a)}$. 7. Approximations and Errors Approximation: $y = f(x)$, then $\Delta y \approx dy = f'(x) \Delta x$. New Value: $f(x + \Delta x) \approx f(x) + f'(x) \Delta x$. Absolute Error: $\Delta x$. Relative Error: $\frac{\Delta x}{x}$. Percentage Error: $\frac{\Delta x}{x} \times 100\%$. Similarly for $y=f(x)$, $\Delta y \approx dy$. Absolute error in $y$: $\Delta y \approx f'(x) \Delta x$. Relative error in $y$: $\frac{\Delta y}{y} \approx \frac{f'(x) \Delta x}{f(x)}$. Percentage error in $y$: $\frac{\Delta y}{y} \times 100\%$. 8. Important JEE Shortcuts & Tips For polynomial $P(x)$, if $P(x) = (x-a)^k Q(x)$ where $Q(a) \neq 0$: If $k$ is even, $x=a$ is a local extremum. If $k$ is odd, $x=a$ is a point of inflection. For $f(x) = ax^2 + bx + c$, critical point is $x = -b/(2a)$. If $f(x)$ is continuous and differentiable on $[a, b]$, and $f'(x)$ has $k$ roots, then $f(x)$ can have at most $k+1$ local extrema. If $f'(x)$ changes sign but is undefined at a point, that point can still be a local extremum (e.g., $f(x)=|x|$ at $x=0$). When optimizing, always check domain constraints and boundary points. Many optimization problems involve maximizing/minimizing geometric shapes (area, volume) or sums/products. Convert to a single variable function. Common Mistake: Not checking if the extremum found is indeed within the valid domain.