in lab

1. start mathematica; follow along with my mini-intro with fibonnaci numbers.
2. go through wolfram's fast introduction, up to the "function definitions" section, including the "check your understanding" quizzes. (don't just read the mathematica code, but type it in and try it out). (wolfram also has an elementary introduction which is more basic with more explanations if you feel like you need it, but it contains a lot of stuff we don't care about.)
3. graph the function y=e^x on [0,10] using mathematica's Exp and Plot functions. using the PlotStyle option in Plot, redo the plot in red, and then redo the plot again used a dashed line.
4. define the factorial function g(x) = x! for a nonnegative integer x recursively (without using the built-in factorial function), i.e., by g(0) = 1 and g(x) = x*g(x-1). make a table of g(0), ..., g(10) with the Table function. plot the table values with ListLinePlot
5. put the previous two plots in different styles (different colors or solid vs dashed) on the same picture with Show. see here for a way to do this in stages.

get mathematica for home: if you like, you can obtain a copy of mathematica for your own machine through the ou it store.

for homework

1. consider the IVP y'(t) = 2t, y(0) = 0 and the approximate solution f(t) formed as follows: let f(0)=0 and inductively define f(t) to yield the line through f(n) of slope y'(n) for t in [n,n+1], for n = 0, 1, 2, ... (this is euler's method and is essentially the construction described on pp 17-18 of the text. euler's method is also described in stewart's calculus book, and i will briefly describe it in the next lab.) define the function f(t) in mathematica at nonnegative integers t. using Table, output the values of f(0), ..., f(10). plot the graph of f(t) against the graph of y(t) = t^2 on [0, 10] (plot y as a dashed line, and f as a solid line to distinguish).
2. let f(t) now denote the approximate solution to the IVP in the previous problem but instead of using line segments with change in x (Delta x) equals 1, using line segments with change in x being some arbitrary step size denoted step. define a function g[n_,step_] which is 0 when n=0 and the y-value of the n-th endpoint of the line segment for f(t) for n=1, 2, 3, ... using this function, plot on the same graph the different f(t)'s for step sizes 2, 1, 0.5, 0.2, 0.1 against the graph for y(t)=t^2 (again using dashed lines for the latter, and you may want to use different colors for the different f(t)'s. all the graphs should go from 0 to 10 on the t-axis). for these graphs, it might be helpful to use the Table function to create a list of pairs of (x,y) coordinates as in one of the ListLinePlot examples.
3. say you know a guy who knows a guy, and you invest $1,000 in this fund, with a (nominal) annual return rate of 10% (meaning 10% annual return without compounding). let f(t) be the amount of money in your investiment assuming automatic reinvestments and no withdrawals after t years. graph f(t) for t in [0,10] in each of the following situations: (i) compounding (once) annually, i.e., your returns are reinvested once a year at the same return rate; (ii) compounding monthly; (iii) compounding weekly; (iv) compounding daily. (for ease of graphing, you may assume your returns are continuous at a uniform rate, so the graph for f(t) on [0,1] in case (i) is just the line from (0,1000) to (1,1010), rather than a straight line from (0,1000) to (1,1000) with a jump at 1 to 1010.) place all the graphs in a single plot. are they approaching the graph of a function you know? if so, determine what function, and explain why, and plot them against this function.

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