Maharashtra State Board · Class 12 · Mathematics · Part II Chapter 2

Applications of Derivatives — Formula Sheet

Board Formulas

12 formulas

1.Rate of Change★
$Rate = \frac{d y}{d t} = \frac{d y}{d x} \cdot \frac{d x}{d t}$
Applies to related-rates: chain rule links two changing quantities.
2.Approximation (Differential)★
$f (x + Δ x) \approx f (x) + f^{'} (x) Δ x$
First-order Taylor expansion. dy = f'(x) dx is the differential of y.
3.Slope of Tangent and Normal
$m_{t an} = f^{'} (x_{0}), m_{n or ma l} = - \frac{1}{f ^{'} ( x _{0} )}$
Normal is perpendicular to tangent. Slope product = −1.
4.Equation of Tangent Line
$y - y_{0} = f^{'} (x_{0}) (x - x_{0})$
Point-slope form. Verify the point (x₀, y₀) lies on the curve first.
5.Increasing / Decreasing on Interval★
$f^{'} (x) > 0 \Rightarrow increasing; f^{'} (x) < 0 \Rightarrow decreasing$
Establish sign of f'(x) on the interval. Zero at endpoints doesn't spoil monotonicity.
6.First Derivative Test (extrema)★
$f^{'} (x_{0}) = 0, f^{'} (x) changes sign at x_{0}$
Sign change + to − ⇒ local maximum. Sign change − to + ⇒ local minimum.
7.Second Derivative Test★
$f^{'} (x_{0}) = 0, f^{''} (x_{0}) < 0 \Rightarrow local max; f^{''} (x_{0}) > 0 \Rightarrow local min$
Inconclusive if f''(x₀) = 0; use higher-order test or first-derivative test.
8.Global Maximum on Closed Interval
$max_{[a, b]} f = max {f (a), f (b), f (c_{i})}$
c_i are critical points inside [a,b]. Compare all these values.
9.Rolle's Theorem★
$f continuous [a, b], differentiable (a, b), f (a) = f (b) \Rightarrow \exists c \in (a, b) : f^{'} (c) = 0$
State all three conditions explicitly for full marks.
10.Lagrange's Mean Value Theorem
$\exists c \in (a, b) : f^{'} (c) = \frac{f ( b ) - f ( a )}{b - a}$
Generalisation of Rolle's; no requirement f(a) = f(b).
11.Percentage Error
$% error = \frac{Δ y}{y} \times 100 \approx \frac{f ^{'} ( x ) Δ x}{f ( x )} \times 100$
Useful for volume of a sphere, area of a circle, cube — small measurement errors.
12.Angle of Intersection of Curves
$tan θ = \frac{m _{1} - m _{2}}{1 + m _{1} m _{2}}$
m₁, m₂ are slopes of the two curves at the point of intersection.