Read sections 3.8, 4.1, and 4.3 of the text (Thomas/Finney).

Make sure you are able to solve all the core problems in these sections.

Write up the following problems and hand them in. Use maple to help with the plotting and the differentiation, if you'd like.

section 3.8: 2 (find x1, x2, x3, x4, x5, and x6), 10 (find x1, x2, x3, x4, x5, and x6), 18, 22, 24, 26, 28. In all of these, use the maple script I provided below, rather than your pocket calculator.

section 4.1: 4, 8, 11, 24, 30, 36, 45, 53, 56, 62, 65-68, 70

section 4.2: 2, 6, 8, 18, 20, 26, 30, 46, 50

section 4.3: 2, 4, 5, 9, 10, 12, 18, 20, 26, 28, 29, 44, 48, 59, 60.

Here's a sample worksheet where I coded up and used Newton's method: sample worksheet for help.

Problems with Maple? Here are the commands I used.

Still having problems? Try the following maple worksheet and commands. They're different from the one above and may work better for you.

Do the following problems in maple and hand in your print-outs.

• In section 3.8, #26. Use maple to find a point x so that x -> y -> x -> y -> x -> ... (hint: see the maple worksheet.) Question: is there some reason that we might know that y = -x? If so, why? And if this is true, is there an easier way to find the point x? (Easier than what I did in the worksheet.) Show that your idea works just as well as mine.

• Consider the following functions: f(x) = x, f(x) = x^2, f(x) = x^3, f(x) = x^4, and f(x) = x^5. All five of these have one root: x = 0. Do Newton's method for all five of these, using the initial guess of x0 = 1. Present the iterates x1, x2, x3, x4, x5, x6 for all five of the functions, as well as the graphs of the five functions.

  Now, look at f(x) = x^2. Find L0(x) the tangent line through (x0,f(x0)). Find L1(x) the tangent line through (x1,f(x1)). Find L2(x) the tangent line through (x2,f(x2)). Find L3(x) the tangent line through (x3,f(x3)). Plot f and L0 on the same plot. Plot f and L1 on the same plot. Plot f and L2 on the same plot. Plot f and L3 on the same plot.

  Now, look at f(x) = x^3. Find L0(x) the tangent line through (x0,f(x0)). Find L1(x) the tangent line through (x1,f(x1)). Find L2(x) the tangent line through (x2,f(x2)). Find L3(x) the tangent line through (x3,f(x3)). Plot f and L0 on the same plot. Plot f and L1 on the same plot. Plot f and L2 on the same plot. Plot f and L3 on the same plot. Discuss the plots from f(x) = x^2 and f(x) = x^3 and make a guess as to what's going on.

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