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Formula Sheets / Maths

Chapter 18

Application of Derivatives

Revise with flashcards Download this chapter (PDF) All Maths formulas (PDF)

Tangent and normal Rate of change and approximation Monotonicity Maxima and minima Mean value theorems

Tangent and normal

Tangent
$y - y_{1} = m (x - x_{1}),$ $m = (\frac{d y}{d x})_{(x_{1}, y_{1})}$
Normal
$y - y_{1} = - \frac{1}{m} (x - x_{1})$
Angle between curves
$tan θ = \frac{m _{1} - m _{2}}{1 + m _{1} m _{2}};$ orthogonal if $m_{1} m_{2} = - 1$

Rate of change and approximation

$\frac{d V}{d t} = \frac{d V}{d r} \cdot \frac{d r}{d t},$ $Δ y \approx f^{'} (x) Δ x$

Monotonicity

$f^{'} (x) > 0$ on $I \Rightarrow f$ increasing $;$ $f^{'} (x) < 0 \Rightarrow$ decreasing

Maxima and minima

First derivative test
$f^{'}$ changes $+ \to - :$ local max $;$ $- \to + :$ local min
Second derivative test
$f^{'} (c) = 0, f^{''} (c) < 0 :$ max $;$ $f^{''} (c) > 0 :$ min
Closed interval
absolute extrema on $[a, b] :$ compare $f$ at critical points and at $a, b$

Mean value theorems

Rolle's theorem
$f$ continuous on $[a, b],$ differentiable on $(a, b), f (a) = f (b) \Rightarrow f^{'} (c) = 0$
Lagrange's MVT
$f^{'} (c) = \frac{f ( b ) - f ( a )}{b - a},$ $c \in (a, b)$

Differentiation Indefinite Integration