Function, Vector Space

Exponential, Logarithmetic, Inverse Trigonometric

Polynomial function \(P(x)=a_{0}+ a_{1}x+ ...+a_{n}x^{n}, \ a_{n}\neq 0\)

Exponential Function \(f(x)=a^x, \ a>0\)

1. Predict Future Population P(t): population at time t

\(P_0\): present population

r: Malthus constant

\[P(t)=P_{0}e^{rt}\]

1. Radiocarbon Dating Model m(t): remaining density of radio carbon at future t

\(m_0\): present

k: negative decided by half-life

\[m(t)=m_{0}e^{kt}\]

e: Euler number For \(y=a^x\), when “slope of tangent line at (0,1)” = 1

e = a

Log Function

Inverse function of exponential function. \(y=a^x, a>0, a\neq 1\) \(x=log_{a}y\)

\(a^{log_{a}(y)}=y\): \(a^{log_{a}}\) and \(log_{a}(y)\) are inverse relation

• \(a^x\) is inverse function of \(log_{a}(y)\)

\(log_{a}a^{(x)}= x\): \(log_{a}a\) and \(a^{(x)}\) are inverse relation

• \(log_{a}(y)\) is inverse function of \(a^x\)

\[a^{x+y}=a^{x}\times a^{y}\] \[a^{x-y}=\frac{a^{x}}{a^{y}}\] \[(ab)^{x}=a^{x}\times b^{x}\] \[(a^{x})^{y}=a^{xy}\] \[log_{a}(xy)=log_{a}(x)+log_{a}(y)\] \[log_{a}(\frac{y}{x})=log_{a}(y)-log_{a}(x)\] \[log_{a}(b^{x})=x\log_{a}(b)\]

Trigonometry

1. \(\sin^{2}x+\cos^{2}x=1\) \(\sin^{2}x = (\sin x)^{2}, \sin x^{2}=\sin (x^2)\)
2. \[1+ \tan^{2}x= \sec^{2}x\]
3. \[1+ \cot^{2}x= \csc^{2}x\]
4. \(\sin (x\pm y)=\sin x\cos y\pm\cos x\sin y\) \(\cos(x\pm y)=\cos x\cos y\mp\sin x\sin y\) \(\tan(x\pm y)=\frac{\tan x\pm\tan y}{1\mp\tan x\tan y}\)
5. \(\sin^{2}\frac{x}{2}=\frac{1-\cos x}{2}\) \(\cos^{2}\frac{x}{2}=\frac{1+\cos x}{2}\)
6. Sine Formula \(\frac{a}{\sin A}=\frac{b}{\sin B}=\frac{c}{\sin C}\)
7. Cosine 1st Formula \(a= c\cos B+b\cos C\)
8. Cosine 2nd Formula \(a^{2}=b^{2}+c^{2}-2bc\cos \theta\)

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