homework 8

## Homework 8, Due Wednesday 11/1 (for those in Tuesday recitations) and on Friday 11/3 (for those in Thursday recitations).

#### 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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