MATH 221

Arun Ram
Department of Mathematics and Statistics
University of Melbourne
Parkville, VIC 3010 Australia
aram@unimelb.edu.au

Last updated: 31 July 2014

Lecture 8

Review for exam

We covered:

(1)	Exponential function: $e^x$ $$\begin{array}{ccc}\begin{array}{ccc}{\displaystyle \frac{d{e}^x}{dx}}& =& {\displaystyle e^x,}\\ {\displaystyle e^{x+y}}& =& {\displaystyle e^x{e}^y,}\\ {\displaystyle {\left(e^x\right)}^y}& =& {\displaystyle e^{xy},}\end{array}& & \begin{array}{ccc}{\displaystyle e^0}& =& {\displaystyle 1,}\\ {\displaystyle e^{-x}}& =& {\displaystyle \frac{1}{e^x},}\\ {\displaystyle e^x}& =& {\displaystyle 1+x+\frac{x^2}{2!}+\frac{x^3}{3!}+\frac{x^4}{4!}+\cdots \text{.}}\end{array}\end{array}$$
(2)	The logarithm: $\text{ln}\hspace{0.17em}x$ $\text{ln}\hspace{0.17em}x$ is the function that undoes $e^x\text{.}$ $$\begin{array}{ccc}\begin{array}{ccc}{\displaystyle \text{ln}\hspace{0.17em}1}& =& {\displaystyle 0,}\\ {\displaystyle \text{ln}\left(\frac{1}{a}\right)}& =& {\displaystyle -\text{ln}\hspace{0.17em}a,}\\ {\displaystyle \frac{d\text{ln}\hspace{0.17em}x}{dx}}& =& {\displaystyle \frac{1}{x}\text{.}}\end{array}& & \begin{array}{ccc}{\displaystyle \text{ln}\left(ab\right)}& =& {\displaystyle \text{ln}\left(a\right)+\text{ln}\left(b\right),}\\ {\displaystyle \text{ln}\left(a^b\right)}& =& {\displaystyle b\text{ln}\hspace{0.17em}a,}\end{array}\end{array}$$
(3)	Trig functions: $$\begin{array}{ccc}{\displaystyle \text{sin}\hspace{0.17em}x}& =& {\displaystyle x-\frac{x^3}{3!}+\frac{x^5}{5!}-\frac{x^7}{7!}+\frac{x^9}{9!}-\cdots ,}\\ {\displaystyle \text{cos}\hspace{0.17em}x}& =& {\displaystyle 1-\frac{x^2}{2!}+\frac{x^4}{4!}-\frac{x^6}{6!}+\frac{x^8}{8!}-\frac{x^{10}}{10!}+\cdots ,}\\ {\displaystyle \text{tan}\hspace{0.17em}x}& =& {\displaystyle \frac{\text{sin}\hspace{0.17em}x}{\text{cos}\hspace{0.17em}x},}\\ {\displaystyle \text{cot}\hspace{0.17em}x}& =& {\displaystyle \frac{1}{\text{tan}\hspace{0.17em}x},}\\ {\displaystyle \text{sec}\hspace{0.17em}x}& =& {\displaystyle \frac{1}{\text{cos}\hspace{0.17em}x},}\\ {\displaystyle \text{csc}\hspace{0.17em}x}& =& {\displaystyle \frac{1}{\text{sin}\hspace{0.17em}x},}\\ {\displaystyle e^{ix}}& =& {\displaystyle \text{cos}\hspace{0.17em}x+i\text{sin}\hspace{0.17em}x,}\\ {\displaystyle {\text{sin}}^{2}x+{\text{cos}}^{2}x}& =& {\displaystyle 1,}\\ {\displaystyle \text{sin}(x+y)}& =& {\displaystyle \text{sin}\hspace{0.17em}x\hspace{0.17em}\text{cos}\hspace{0.17em}y+\text{cos}\hspace{0.17em}x\hspace{0.17em}\text{sin}\hspace{0.17em}y,}\\ {\displaystyle \text{cos}(x+y)}& =& {\displaystyle \text{cos}\hspace{0.17em}x\hspace{0.17em}\text{cos}\hspace{0.17em}y-\text{sin}\hspace{0.17em}x\hspace{0.17em}\text{sin}\hspace{0.17em}y,}\\ {\displaystyle \text{sin}(-x)}& =& {\displaystyle -\text{sin}\hspace{0.17em}x,}\\ {\displaystyle \text{cos}(-x)}& =& {\displaystyle \text{cos}\hspace{0.17em}x,}\\ \multicolumn{3}{c}{\begin{array}{c} 2 1 45 45 1 \end{array}}\\ \multicolumn{3}{c}{\begin{array}{c} 2 1 30 60 3 \end{array}}\\ {\displaystyle \frac{d\text{sin}\hspace{0.17em}x}{dx}}& =& {\displaystyle \text{cos}\hspace{0.17em}x,}\\ {\displaystyle \frac{d\text{cos}\hspace{0.17em}x}{dx}}& =& {\displaystyle -\text{sin}\hspace{0.17em}x,}\\ {\displaystyle \frac{d\text{tan}\hspace{0.17em}x}{dx}}& =& {\displaystyle {\text{sec}}^{2}x,}\\ {\displaystyle \frac{d\text{cot}\hspace{0.17em}x}{dx}}& =& {\displaystyle -{\text{csc}}^{2}x,}\\ {\displaystyle \frac{d\text{sec}\hspace{0.17em}x}{dx}}& =& {\displaystyle \text{tan}\hspace{0.17em}x\hspace{0.17em}\text{sec}\hspace{0.17em}x,}\\ {\displaystyle \frac{d\text{csc}\hspace{0.17em}x}{dx}}& =& {\displaystyle -\text{cot}\hspace{0.17em}x\hspace{0.17em}\text{csc}\hspace{0.17em}x\text{.}}\end{array}$$
(4)	Inverse trig functions: ${\text{sin}}^{-1}x$ is the function that undoes $\text{sin}\hspace{0.17em}x,$ ${\text{cos}}^{-1}x$ is the inverse function to $\text{cos}\hspace{0.17em}x,$ ${\text{tan}}^{-1}x$ is the inverse function to $\text{tan}\hspace{0.17em}x,$ ${\text{cot}}^{-1}x$ is the inverse function to $\text{cot}\hspace{0.17em}x,$ ${\text{sec}}^{-1}x$ is the inverse function to $\text{sec}\hspace{0.17em}x,$ ${\text{csc}}^{-1}x$ is the inverse function to $\text{csc}\hspace{0.17em}x\text{.}$ $$\begin{array}{c}{\displaystyle \frac{d{\text{sin}}^{-1}x}{dx}=\frac{1}{\sqrt{1-x^2}},\frac{d{\text{cos}}^{-1}x}{dx}=\frac{-1}{\sqrt{1-x^2}},}\\ {\displaystyle \frac{d{\text{tan}}^{-1}x}{dx}=\frac{1}{1+x^2},\frac{d{\text{cot}}^{-1}x}{dx}=\frac{-1}{1+x^2},}\\ {\displaystyle \frac{d{\text{sec}}^{-1}x}{dx}=\frac{1}{x\sqrt{x^2-1}},\frac{d{\text{csc}}^{-1}x}{dx}=\frac{-1}{x\sqrt{x^2-1}}\text{.}}\end{array}$$
(5)	Derivatives: $$f⟶\begin{array}{c} ddx \end{array}⟶\frac{df}{dx}$$ (a) $\frac{dx}{dx}=1,$ (b) $\frac{d(f+g)}{dx}=\frac{df}{dx}+\frac{dg}{dx},$ (c) $\frac{d\left(cf\right)}{dx}=c\frac{df}{dx},$ if $c$ is a constant, (d) $\frac{d\left(fg\right)}{dx}=f\frac{dg}{dx}+\frac{dg}{dx},$ (e) $\frac{d{x}^n}{dx}=n{x}^{n-1},$ (f) $\frac{dc}{dx}=0,$ if $c$ is a constant, (g) $\frac{d1}{dx}=0,$ (h) $\frac{df}{dx}=\frac{df}{dg}\frac{dg}{dx}\text{.}$

Notes and References

These are a typed copy of Lecture 8 from a series of handwritten lecture notes for the class MATH 221 given on September 22, 2000.

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