CalculusVery High Yield (3-4 Qs in JEE)

Application of Derivatives (AOD) Formulas

Tangents, normals, monotonicity, maxima & minima, Rolle's theorem, and Lagrange Mean Value Theorem (LMVT).

8 Formulas•969+ Questions in Bank•

Concept Diagram

Definite Integral & Tangent Line

Geometric interpretation of ∫ f(x) dx as area under curve and derivative f'(x₀) as tangent slope.

y - y_1 = m(x - x_1) \quad \left(m = \left.\frac{dy}{dx}\right|_{(x_1, y_1)}\right), \quad \text{Normal: } y - y_1 = -\frac{1}{m}(x - x_1)

Point-slope equations for tangent and perpendicular normal lines to curve y = f(x).

Examiner Tip: If m = 0, tangent is horizontal (y = y_1). If m -> ∞, tangent is vertical (x = x_1).

\text{Subtangent} = \left|\frac{y}{m}\right|, \quad \text{Subnormal} = |y \cdot m|, \quad \text{Tangent length} = \left|\frac{y}{m}\right|\sqrt{1 + m^2}

Geometric lengths of projections on the x-axis for curve tangents and normals.

Examiner Tip: Subnormal is proportional to y for parabola y² = 4ax (constant subnormal = 2a).

f'(x) \ge 0 \; \forall x \in (a, b) \implies f(x) \text{ Increasing}; \quad f'(x) \le 0 \implies f(x) \text{ Decreasing}

Derivative test for strictly increasing and decreasing functions on open intervals.

Examiner Tip: f'(x) = 0 at discrete isolated points does NOT destroy strict monotonicity.

f'(c) = 0: \quad f''(c) < 0 \implies \text{Local Maxima}, \quad f''(c) > 0 \implies \text{Local Minima}

Stationary points and concavity curvature tests for extreme values.

Examiner Tip: If f''(c) = 0, test higher derivatives: odd order = inflection point, even order = extremum.

\frac{f(b) - f(a)}{b - a} = f'(c) \quad \text{for at least one } c \in (a, b)

Secant slope across [a, b] equals the tangent derivative slope at some interior point c.

Examiner Tip: Rolle's Theorem is the special case when f(a) = f(b) (giving f'(c) = 0).

\tan\theta = \left|\frac{m_1 - m_2}{1 + m_1 m_2}\right|, \quad \text{Orthogonal Condition: } m_1 m_2 = -1

Acute angle between tangents to curves y = f(x) and y = g(x) at their point of intersection.

Examiner Tip: Two intersecting confocal conics (e.g. ellipse and hyperbola with same foci) always intersect orthogonally.

f''(x) > 0 \implies \text{Concave Upward } (\cup), \quad f''(x) < 0 \implies \text{Concave Downward } (\cap), \quad f''(c) = 0 \; (\& \text{ changes sign}) \implies \text{Inflection}

Curvature direction and inflection point where tangent crosses the curve.

Examiner Tip: At a point of inflection, the tangent crosses the curve (e.g. y = x³ at the origin).

\frac{f(b) - f(a)}{g(b) - g(a)} = \frac{f'(c)}{g'(c)} \quad (c \in (a,b)), \quad \Delta y \approx f'(x)\,\Delta x, \quad \frac{\Delta y}{y} \approx \left[\frac{x f'(x)}{f(x)}\right]\frac{\Delta x}{x}

Generalized mean value theorem for two functions, and differential approximation for calculating percentage errors.

Examiner Tip: Percentage error in y is (% error in x) multiplied by the power index n when y = k x^n.

Frequently tested misconceptions in Application of Derivatives (AOD)

•Assuming f'(c) = 0 is sufficient for a local extremum (could be an inflection point like y = x³ at x = 0).
•Forgetting to verify hypotheses of Rolle's and LMVT (continuity on [a,b] AND differentiability on (a,b)).