About the final exam

You'll have two hours.

Equation sheet

These are the equations I'll make available to you during the final exam:

Logarithms and exponentials

• $\ln(a*b)=\ln a+\ln b$
• $\ln(a^t)=t\ln a.$
• $\ln(a/b)=\ln(a*b^{-1})=\ln a - \ln b$
  (These properties are shared by the $\log$ function as well.).
  
  
• $b^{x}b^y=b^{(x+y)};\ \ \ \ \frac{b^x}{b^y}=b^{(x-y)}; \ \ \ \ (b^x)^t=b^{xt}.$
• $10^{\log x}=x;\ \ \log(10^x)=x;\ \ \ \ e^{\ln x}=x;\ \ \ln(e^x)=x.$

Function Transformations

Consider $$g(x)=\color{blue}{A} f(\color{orange}{B}(x- \color{orange}{h}))+ \color{blue}{k}.$$ The function $g(x)$ is the result of the following transformations of $f(x)$:

• Stretch $f(x)$ vertically by a factor $\color{blue}{A}$ away from the $x$ axis,
• Then shift it up by $\color{blue}{k}$ units.
  
  
• Compress $f$ horizontally by a factor $\color{orange}{B}$,
• Then shift it right by $\color{orange}{h}$ units.

Functions that vary periodically

• Period, $P$, for that general form of a sinusoidal function and $B$ are related by $$B=\frac{2\pi}{P}$$
• Angle $\theta$ (in radians) is related to radius $r$ and arclength $s$ by: $$\theta = \frac{s}{r}$$
• Tangent: $\tan \theta=\sin\theta / \cos \theta$
• $\cot \theta = 1/\tan\theta; \sec \theta = 1/\cos\theta; \csc \theta = 1/\sin\theta$
• Law of Sines: $$\frac{\sin A}{a}=\frac{\sin B}{b}=\frac{\sin C}{c}$$ (but you should know how the sides and angles are labelled for this to work)
• Law of Cosines: $$c^2=a^2+b^2-2ab\cos C$$
• double-angle formulas: $$\sin 2\theta=2\sin\theta\cos\theta$$ $$\cos 2\theta=1-2\sin^2\theta$$