Math 370 practice problems

1. For f[x] = , use the McClaurin series for to give appropriate values for f[0], f ' [0] and f ''[0]. Check your answers by using Mathematica to evaluate expressions like Limit[f[x],x->0] that calculate these values exactly.

2. Consider f[x] = . Find the second degree Taylor polynomial of f[x] centered at a = 1, say . Use to approximate . What is the maximum possible error in using to approximate , for ?

3. Convert the hexadecimal number to both binary and decimal.

4. Imagine an IEEE machine similar to the Marc 32, except this one uses only 16 bit words (maybe the Marc 16). Recall that the Marc 32 encodes the number as the binary digits scf, where s has 1 bit, c has 8 bits and f uses 23 bits. In the Marc 16, we will still have s with 1 bit, c will have 5 bits, the exponent will be c - 15, and f will have 10 bits (it is still 1.f that is encoded). Find the Hexadecimal version of -26.5 in the Marc 16 machine, and find the decimal number stored as 4A10 in the Marc 16.

5. Consider f[x] = x*() for x = 100.0
a) Calculate x, , and , using 6 significant figures and rounding.
b) Calculate f[100.0] using your values from part (a) and rounding to 6 significant figures at each intermediate calculation.
c) Find an alternate version of the formula for f[x] (with no loss of precision near x = 100.0) and use it with 6 significant figures and rounding to calculate f[100.0].

6. Let T = e (the base of the natural log) and A = 19 / 7 be an approximation to T.
a) Calculate the absolute error in using A to approximate the true value T.
b) Calculate the relative error in using A to approximate T.
c) How many significant figures (correct digits) are there in using A to approximate T?

7. Calculate (exactly-or by hand) under 3-digit precision and chopping to 3 significant digits the value of . What is the relative error, what is the absolute error. Repeat but this time use 3-digit and rounding for all intermediate calculations.

8. Let f[x] = , a = 0, and b = 1. Use the bisection method twice to estimate a root of f[x]. Let be the result, and then use Newton's method twice (starting at ) to give a better estimate of the root.

9. Let f[x] = x^3 + 2x^2 -x -1.
a) Use FindRoot to estimate the roots of f[x] accurately.
b) Verify that Newton's method will converge to the largest positive root if you start at = -1.4.
c) Find as large of an interval as you can (don't waste too much time trying to make your interval as large as possible), containing x = -1.4, so that for each x in the interval, Newton's method, when started at x, will converge to the largest root of f[x].
d) On a graph of f[x], draw in a line from the most negative root, to the graph of f[x] near x = 0, so that your line is tangent to the graph of f[x], near x = 0. If you start Newton's method near x = 0, where your line is tangent to the graph of f[x], it should converge to the most negative root of f[x]. Find an x value near x = 0, so that Newton's method, when started at this x-value, will converge to the most negative root of f[x].
e) Looking at your graph, let r be the most neagtive root of f[x], and explain why finding such an x value (near 0) is related to finding a root of the equation f '[x] = f[x] / (x - r). Use the secant method (and Mathematica) to solve this last equation accurate to 3 decimal places.