3 more functions
3.1 power functions
3.2 exponentials and logarithms
3.2.1 exponential functions
for \(a\in{}R^+\setminus\{1\}\), the graph \(y=a^x\) has the following properties:
- the y-axis intercept is 1
- the x-axis is an asymptote
| index laws | |
|---|---|
| \(x^a\times{}x^b=x^{a+b}\) | \(x^a\div{}x^b=x^{a-b}\) |
| \((xy)^a=x^ay^a\) | \((\frac{x}{y})^a=\frac{x^a}{y^a}\) |
| \((x^a)^b=x^{ab}\) | \(x^0=1\) |
3.2.2 logarithmic functions
for \(a\in{}R^+\setminus\{1\}\), the graph \(y=\log_ax\) has the following properties:
- the x-axis intercept is 1
- the y-axis is an asymptote
| log laws | |
|---|---|
| \(\log_a1=0\) | \(\log_aa=1\) |
| \(\log_am+\log_an=\log_a(mn)\) | \(\log_am-\log_an=\log_a(\frac{m}{n})\) |
| \(p\log_am=\log_a(m^p)\) | \(\log_am=\frac{\log_em}{\log_ea}=\frac{\log_{10}m}{\log_{10}a}\) |
3.2.3 exponential growth and decay
model for quantity growing/decaying at an exponential rate:
\(A=A_0b^t\)
- \(A_0\) is the initial quantity
- \(b=e^k\) is the rate constant (growth if \(b>1\), decay if \(b<1\))
- \(t\) is time
cell growth - doubles every \(T_D\) (unit of time):
\(N=N_02^{\frac{t}{T_D}}\)
- \(N_0\) is the initial quantity
- \(T_D\) is the generation time (how long it takes to double)
- \(t\) is time
3.3 circular functions
3.3.1 exact values
for \(x\in{[0,\frac{\pi}{2}]}\)
| function | \(0\) | \(\frac{\pi}{6}\) | \(\frac{\pi}{4}\) | \(\frac{\pi}{3}\) | \(\frac{\pi}{2}\) |
|---|---|---|---|---|---|
| \(\sin\) | \(0\) | \(\frac{1}{2}\) | \(\frac{1}{\sqrt{2}}\) | \(\frac{\sqrt{3}}{2}\) | \(1\) |
| \(\cos\) | \(1\) | \(\frac{\sqrt{3}}{2}\) | \(\frac{1}{\sqrt{2}}\) | \(\frac{1}{2}\) | \(0\) |
| \(\tan\) | \(0\) | \(\frac{1}{\sqrt{3}}\) | \(1\) | \(\sqrt{3}\) | \(\text{undefine}\) |
3.3.2 symmetric properties
| \(-x\) | \(\pi-x\) | \(\pi+x\) |
|---|---|---|
| \(\sin(-x)=-\sin(x)\) | \(\sin(\pi-x)=\sin(x)\) | \(\sin(\pi+x)=-\sin(x)\) |
| \(\cos(-x)=\cos(x)\) | \(\cos(\pi-x)=-\cos(x)\) | \(\cos(\pi+x)=-\cos(x)\) |
| \(\tan(-x)=-\tan(x)\) | \(\tan(\pi-x)=-\tan(x)\) | \(\tan(\pi+x)=\tan(x)\) |
3.3.3 complementary angles
| \(\frac{\pi}{2}-x\) | \(\frac{\pi}{2}+x\) |
|---|---|
| \(\sin(\frac{\pi}{2}-x)=\cos(x)\) | \(\sin(\frac{\pi}{2}+x)=\cos(-x)\) |
| \(\cos(\frac{\pi}{2}-x)=\sin(x)\) | \(\cos(\frac{\pi}{2}+x)=\sin(-x)\) |