gibson:teaching:spring-2015:math445:lab4
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gibson:teaching:spring-2015:math445:lab4 [2015/02/04 20:52] gibson created |
gibson:teaching:spring-2015:math445:lab4 [2015/02/05 03:58] (current) gibson |
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| ====== Math 445 lab 4: dot syntax and plotting ====== | ====== 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 | **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 | 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. | use a legend to label the curves. | ||
| + | |||
| + | Helpful Matlab commands: ''linspace, plot, xlabel, legend, help.'' | ||
| + | |||
| + | ----- | ||
| **Problem 2:** Consider the two functions $f(x) = x^n$ and $g(x) = n^x$ for the fixed value $n=5$. | **Problem 2:** Consider the two functions $f(x) = x^n$ and $g(x) = n^x$ for the fixed value $n=5$. | ||
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| on the range $0 \leq x \leq 100$, and plot the $y$ axis logarithmically.) Would the answer change | on the range $0 \leq x \leq 100$, and plot the $y$ axis logarithmically.) Would the answer change | ||
| if you increased the value of $n$? | if you increased the value of $n$? | ||
| + | |||
| + | Helpful Matlab commands: ''linspace, semilogy, xlabel, legend, help,'' plus dot-syntax. | ||
| + | |||
| + | ----- | ||
| **Problem 3:** How many real roots does the function $f(x) = x^3 - 5x^2 + 2x + 4$ have? | **Problem 3:** How many real roots does the function $f(x) = x^3 - 5x^2 + 2x + 4$ have? | ||
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| Estimate the roots graphically to two digits by refining your plots in the neighborhood | Estimate the roots graphically to two digits by refining your plots in the neighborhood | ||
| of each zero crossing. | 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. | ||
| + | |||
| + | ----- | ||
| **Problem 4:** The exponential function $e^x$ can be computed from the power series | **Problem 4:** The exponential function $e^x$ can be computed from the power series | ||
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| In practice one truncates the infinite sum to a finite number of terms, summing from $n=0$ to $n=N$ for | 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 | 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 combining Matlab's dot-syntax and ''sum'' function. | + | 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). | ||
| + | |||
| + | ----- | ||
| **Problem 5:** The sine function can be computed from the power series | **Problem 5:** The sine function can be computed from the power series | ||
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| \end{eqnarray*} | \end{eqnarray*} | ||
| - | Like the expontential function, in practice one truncates this sum at $n=N$. How large does $n$ need to | + | 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? | be to calculate $\sin \pi/3$ to sixteen digits? | ||
| + | |||
| + | Helpful Matlab commands: same as problem 4. | ||
gibson/teaching/spring-2015/math445/lab4.1423111924.txt.gz · Last modified: 2015/02/04 20:52 by gibson
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