====== Math 445 lab 4: dot syntax and plotting ======

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**Problem 1:** Create a plot that shows $\sin x$ and $\cos x$ over $-\pi \leq x \leq \pi$ with the sine in 
green and the cosine in blue. Choose enough gridpoints that the curve looks smooth. Label the $x$ axis and 
use a legend to label the curves. 

Helpful Matlab commands: ''linspace, plot, xlabel, legend, help.''

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**Problem 2:** Consider the two functions $f(x) = x^n$ and $g(x) = n^x$ for the fixed value $n=5$.
Which function increases faster as $x \rightarrow \infty$? (Hint: plot the two functions together 
on the range $0 \leq x \leq 100$, and plot the $y$ axis logarithmically.) Would the answer change 
if you increased the value of $n$?

Helpful Matlab commands: ''linspace, semilogy, xlabel, legend, help,'' plus dot-syntax. 

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**Problem 3:** How many real roots does the function $f(x) = x^3 - 5x^2 + 2x + 4$ have? 
Determine this by plotting the $f(x)$ and counting how many times it crosses zero.
Estimate the roots graphically to two digits by refining your plots in the neighborhood
of each zero crossing. 

Helpful Matlab commands: ''linspace, plot, xlabel, ylabel, axis,'' plus dot-syntax. In
particular you can use ''axis([xmin xmax ymin ymax])'' to zoom in on regions near zeros.
Or you can zoom in by refining the range of $x$ and replotting.

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**Problem 4:** The exponential function $e^x$ can be computed from the power series

\begin{eqnarray*}
e^x = \sum_{n=0}^\infty \frac{x^n}{n!}
\end{eqnarray*}

In practice one truncates the infinite sum to a finite number of terms, summing from $n=0$ to $n=N$ for
some fairly large $N$. How large does $N$ need to be to calculate $e^2$ to sixteen digits? You can answer 
this question very quickly by writing one line of Matlab code that evaluate the truncated sum for a fixed value of $N$, and then rerunning it with changing values of $N$. 

Helpful Matlab commands: ''sum, format, exp'' (for getting the correct value of $e$ to sixteen digits), colon syntax, dot syntax, and ''log10'' (for counting digits of precision). 

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**Problem 5:** The sine function can be computed from the power series

\begin{eqnarray*}
\sin x = \sum_{n=0}^\infty (-1)^n \frac{x^{2n+1}}{(2n+1)!}
\end{eqnarray*}

Like the exponential function, in practice one truncates this sum at $n=N$. How large does $n$ need to
be to calculate $\sin \pi/3$ to sixteen digits?

Helpful Matlab commands: same as problem 4.