gibson:teaching:spring-2018:math445:lab2
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gibson:teaching:spring-2018:math445:lab2 [2018/01/30 03:05] gibson |
gibson:teaching:spring-2018:math445:lab2 [2018/01/31 16:07] (current) gibson [Part 3: Plots] |
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| - | ====== Math 445 lab 2: plotting and scripts====== | + | ====== Math 445 lab 2: vectors, plots, and scripts====== |
| - | ==== Part 1: vector manipulation ==== | + | Do the problems for parts 1 and 2 at the Matlab prompt, saving your work to a file with the ''diary'' command. |
| - | Do the following in Matlab, saving your work to a file with the ''diary'' command, and labeling each problem with a Matlab comment (e.g. ''% problem 1''). Reduce unnecessary whitespace with the ''format compact'' command. When you're done, edit the diary file to eliminate errors, so that your diary shows just the correct work. | + | Be kind to your teaching assistant! |
| + | * Label each problem with a Matlab comment (e.g. <nowiki>% ==== Problem 1 ====</nowiki>). | ||
| + | * For questions with free-form response, write your answer as a Matlab comment. Write in complete sentences! | ||
| + | * Reduce unnecessary whitespace with the ''format compact'' command. | ||
| + | * When you're done, edit the diary file to eliminate errors, so that your diary shows just the correct work. | ||
| + | * Add and subtract blank lines as necessary to make your work organized and easiy readable. | ||
| - | 1. Assign a row vector with elements 3, 4, 5, 9 to the variable u. | + | ==== Part 1: Vector manipulation ==== |
| - | 2. Assign the transpose of u to the variable w. | + | Relevant Matlab |
| + | <code Matlab> | ||
| + | [] % square brackets | ||
| + | , % row separator | ||
| + | ; % col separator | ||
| + | : % colon operator | ||
| + | ' % apostrophe | ||
| + | linspace | ||
| + | </code> | ||
| - | 3. Change the third element of w to 10. | + | **1.** Assign a row vector with elements 3, 4, 5, 9 to the variable u. |
| - | 4. Assign a column vector with the elements 7, 1, -2, 3 to the variable z. | + | **2.** Assign the transpose of u to the variable w. |
| - | 5. Add w and z. Does the result make sense? | + | **3.** Change the third element of w to 10. |
| - | 6. Add u and z. Does the result make sense? | + | **4.** Assign a column vector with the elements 7, 1, -2, 3 to the variable z. |
| - | 7. Create a vector of the even integers between 2 and 14, inclusive. | + | **5.** Add w and z. Does the result make sense? |
| - | 8. Create a vector of the odd integers between 7 and 19, inclusive. | + | **6.** Add u and z. Does the result make sense? |
| - | 9. What is the sum of every third number between 3 and 27, inclusive? | + | **7.** Create a vector of the even integers between 2 and 14, inclusive. |
| - | 10. What is the sum of the integers between 1 and 100, inclusive? | + | **8.** Create a vector of the odd integers between 7 and 19, inclusive. |
| - | 11. Make x a vector from 0 to $2\pi$ in increments of 0.02. | + | **9.** Make x a vector from 0 to $2\pi$ in increments of 0.02. |
| - | ==== Part 2: sums ==== | + | ==== Part 2: Computing sums ==== |
| + | Relevant Matlab syntax | ||
| + | <code matlab> | ||
| + | : % colon operator | ||
| + | sum % sum function | ||
| + | .* ./ .^ % dot syntax | ||
| + | </code> | ||
| - | ==== Part 3: plots ==== | ||
| - | 12. Make a plot of the polynomial $f(x) = x^3 -5x^2 + 2x + 3$ as a blue line. In the same plot, draw a line along the $x$ axis. (Hint: in Matlab create a zero vector of the same length as your $x$ vector with ''y=0*x'', then plot $y$ versus $x$ along with $f$ versus $x$.) Using this plot, estimate the zeros of the polynomial, i.e. the values of $x$ for which $f(x) = 0$. Make sure to find all the zeros of $f$ by adjusting limits of the plot until all intersectionf of $f$ with the $x$-axis are visible. | + | **10.** What is the sum of every third number between 3 and 27, inclusive? |
| + | **11.** What is the sum of the integers between 1 and 100, inclusive? | ||
| - | 13. Make a plot of a unit circle, i.e. a curve that satisfies $x^2 + y^2 = 1$. Draw the circle with a thick red line, label the axes, and give the plot a title. Make sure the circle looks like a circle and not an oval. Hint: don't try to draw the plot using the equation $x^2 + y^2 = 1$. Instead parameterize the curve in terms of the angle $\theta$, i.e. calculate $x$ and $y$ from $\theta$. | + | **12.** The infinite series |
| + | \begin{equation*} | ||
| + | \sum_{n=1}^{N} \frac{1}{n} | ||
| + | \end{equation*} | ||
| + | |||
| + | diverges as $ | ||
| + | N \rightarrow \infty$. Demonstrate this in Matlab by computing the sum for several values of $N$, e.g. $N=10$ to $N=10^5$ by powers of ten. | ||
| + | |||
| + | **13.** The $\sin$ function can be calculated from the infinite series | ||
| + | |||
| + | \begin{equation*} | ||
| + | \sin x = \sum_{n=0}^{\infty} (-1)^n \frac{x^{2n+1}}{(2n+1)!} | ||
| + | \end{equation*} | ||
| + | |||
| + | Of course we have to truncate this infinite sum to a finite number of terms in order to calculate it on a computer. How many terms do you need to keep in order to compute $\sin \: \pi/2 = 1$ to sixteen digits accuracy? | ||
| + | Hint: use ''format long'' to see all sixteen digits of the computation. | ||
| + | |||
| + | Are you amazed or what? | ||
| + | |||
| + | |||
| + | **14.** Now use the same infinite series for $\sin x$ to calculate $\sin \: \pi = 0$. How many terms do you need to keep in order get the correct answer to sixteen digits accuracy? | ||
| + | |||
| + | Are you perplexed or what? What is strange about this calculation? Can you explain what happened? | ||
| + | |||
| + | |||
| + | ==== Part 3: Plots ==== | ||
| + | |||
| + | Relevant Matlab | ||
| + | <code matlab> | ||
| + | .* ./ .^ % dot syntax | ||
| + | plot % plot function | ||
| + | axis % set limits on plot or aspect ratio | ||
| + | xlim % set x limits | ||
| + | ylim % set y limits | ||
| + | grid % turn grid on/off | ||
| + | help % help function | ||
| + | </code> | ||
| + | |||
| + | **15.** Make a plot of the polynomial $f(x) = x^3 -5x^2 + 2x + 3$ as a blue line. In the same plot, draw a line along the $x$ axis. (Hint: in Matlab create a zero vector of the same length as your $x$ vector with ''y=0*x'', then plot $y$ versus $x$ along with $f$ versus $x$.) Using this plot, estimate the zeros of the polynomial, i.e. the values of $x$ for which $f(x) = 0$. Make sure to find all the zeros of $f$ by adjusting limits of the plot until all intersection of $f$ with the $x$-axis are visible. | ||
| + | |||
| + | |||
| + | **16.** Make a plot of a unit circle, i.e. a curve that satisfies $x^2 + y^2 = 1$. Draw the circle with a thick red line, label the axes, and give the plot a title. Make sure the circle looks like a circle and not an oval. Hint: don't try to draw the plot using the equation $x^2 + y^2 = 1$. Instead parameterize the curve in terms of the angle $\theta$, i.e. calculate $x$ and $y$ from $\theta$. | ||
| + | |||
| + | (I previously had a problem 17, revisiting the sum of $1/n$...to be written. But on second thought, this problem belongs in lab 3. So expect it there.) | ||
gibson/teaching/spring-2018/math445/lab2.1517310313.txt.gz · Last modified: 2018/01/30 03:05 by gibson
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