Applications of Differential Calculus — Formula Sheet (Class 12)

Applications of Differential Calculus — Formula Sheet

Rates and motion

• Velocity \( v=\dfrac{ds}{dt} \); acceleration \( a=\dfrac{dv}{dt}=\dfrac{d^{2}s}{dt^{2}} \).
• At rest: \( v=0 \). Direction change: \( v \) changes sign.
• Average rate over \( [a,b] \): \( \dfrac{f(b)-f(a)}{b-a} \); instantaneous rate at \( x \): \( f'(x) \).

Solid-geometry rates

• Sphere: \( V=\dfrac{4}{3}\pi r^{3}\Rightarrow\dfrac{dV}{dt}=4\pi r^{2}\dfrac{dr}{dt} \); \( S=4\pi r^{2}\Rightarrow\dfrac{dS}{dt}=8\pi r\dfrac{dr}{dt} \).
• Cone: \( V=\dfrac{1}{3}\pi r^{2}h \); cylinder: \( V=\pi r^{2}h \).
• Cube of side \( x \): \( V=x^{3}\Rightarrow\dfrac{dV}{dx}=3x^{2} \).

Tangent, normal and angle

• Slope of tangent: \( m=f'(x_1)=\tan\theta \).
• Tangent: \( y-y_1=m(x-x_1) \). Normal: \( y-y_1=-\dfrac{1}{m}(x-x_1) \).
• Horizontal tangent: \( \dfrac{dy}{dx}=0 \). Vertical tangent: \( \dfrac{dy}{dx} \) undefined (denominator \( =0 \)).
• Angle between curves: \( \tan\varphi=\left|\dfrac{m_1-m_2}{1+m_1 m_2}\right| \); orthogonal if \( m_1 m_2=-1 \).

Mean Value Theorems

• Rolle: \( f(a)=f(b)\Rightarrow f'(c)=0 \) for some \( c\in(a,b) \).
• Lagrange: \( f'(c)=\dfrac{f(b)-f(a)}{b-a} \) for some \( c\in(a,b) \).

Series expansions

• Taylor about \( a \): \( f(x)=\displaystyle\sum_{n=0}^{\infty}\dfrac{f^{(n)}(a)}{n!}(x-a)^{n} \).
• Maclaurin: \( f(x)=\displaystyle\sum_{n=0}^{\infty}\dfrac{f^{(n)}(0)}{n!}x^{n} \).
• \( e^{x}=\displaystyle\sum_{n=0}^{\infty}\dfrac{x^{n}}{n!} \); \( \sin x=x-\dfrac{x^{3}}{3!}+\dfrac{x^{5}}{5!}-\cdots \); \( \cos x=1-\dfrac{x^{2}}{2!}+\dfrac{x^{4}}{4!}-\cdots \); \( \log(1+x)=x-\dfrac{x^{2}}{2}+\dfrac{x^{3}}{3}-\cdots \).

L'Hopital's rule (indeterminate forms)

• For \( \dfrac{0}{0} \) or \( \dfrac{\infty}{\infty} \): \( \displaystyle\lim_{x\to a}\dfrac{f(x)}{g(x)}=\lim_{x\to a}\dfrac{f'(x)}{g'(x)} \).
• \( 0\cdot\infty \) and \( \infty-\infty \): rewrite as a single fraction first.
• \( 0^{0},\,1^{\infty},\,\infty^{0} \): if \( y \) is the expression, find \( L=\lim\ln y \), then answer \( =e^{L} \).

First derivative (monotonicity / extrema)

• \( f'(x)>0\Rightarrow \) increasing; \( f'(x)<0\Rightarrow \) decreasing.
• Critical point: \( f'(x)=0 \) or \( f'(x) \) undefined.
• First Derivative Test at \( c \): \( -\to+ \) min, \( +\to- \) max, no change \( \Rightarrow \) neither.

Second derivative (concavity / extrema)

• \( f''(x)>0\Rightarrow \) concave up; \( f''(x)<0\Rightarrow \) concave down.
• Point of inflection: \( f''=0 \) (or undefined) with a sign change of \( f'' \).
• Second Derivative Test (where \( f'(c)=0 \)): \( f''(c)<0\Rightarrow \) max; \( f''(c)>0\Rightarrow \) min; \( f''(c)=0\Rightarrow \) inconclusive.

Asymptotes

• Vertical: \( x=a \) if \( \displaystyle\lim_{x\to a}y=\pm\infty \).
• Horizontal: \( y=b \) if \( \displaystyle\lim_{x\to\pm\infty}y=b \).
• Slant: when the numerator's degree exceeds the denominator's by exactly one (use division).