## Description

- For each of the ve important series mentioned in the slides of section 1.2, write down the detailed derivation of its formula and the interval of convergence.

- Compute by hand

- T
_{5}(x) for f(x) = 3 tan x, at point c = =4.

- T
_{2}(x) for f(x) = e^{cos}^{x}, at point c = 0.

- For f(x) = e
^{x}cos x at c = 0

- Find T
_{2}(x) by hand.

(b) Use Taylor’s theorem to give an estimate of error jf(0:5) T_{2}(0:5)j. Compare it with the true

error jf(0:5) | T_{2} |
(0:5)j | |||

(c) Approximate | ^{Z}0 |
1 | f(x) dx by ^{Z}_{0} |
1 | T_{2}(x) dx. Find the true error. |

(d) Find the Taylor series for f(x) = x^{2} 1 at c = 1 and c = 2. Compare your result with the original function. Can you conclude what will the Taylor series of a general polynomial look like?

p

- Find the Taylor series of x at c = 1. Determine the interval of convergence (be careful with the ending points.).

p

- Use the result you nd in question 2 to evaluate 2. Round your answer to 4 decimals.

- Convert (100010010111011)
_{2}, (10:11)_{2}to decimal (base 10).

- Convert 37, 0:43, 10:11 to binary representation.

- Convert (1234)
_{5}to decimal, then base 8 representation.

- Write down the IEEE format of the following numbers

- 75

- 5, using rounding up, rounding down and rounding to the nearest

- 1, using rounding up, rounding down and rounding to the nearest

- What is the gap between 2 and the next larger Single-precision number?

- What is the gap between 201 and the next larger double-precision number?

- How many di erent normalized double-precision numbers are there?

- Consider a very limited system in which numbers are only of the form 1:b
_{1}b_{2}b_{3}2^{E}and the only

exponents are E = 1; 0; 1.

- What is the machine precision ” for this system?

- What are the smallest and largest representable positive number in this system?

- Consider the sequence mentioned in the last page of slides 2.1, starting with 1=3.

- Explain why the computed value eventually become 1.

1

- Determine (by hand computing) how many iterations is needed for the sequence to reach 1 for the rst time.

- In the 7th season episode Treehouse of Horrors VI of The Simpsons, Homer has a nightmare in which the following equation ies past him:

1782^{12} + 1841^{12} = 1922^{12}

^{ }

If this equation were true, this would contradict Fermat’s last theorem which states for n 3, there do not exist any natural numbers x; y and z such that x^{n} + y^{n} = z^{n}. Did Homer dream up a

counterexample to Fermat’s last theorem?

p

- Compute
^{12}1782^{12}+ 1841^{12}in Matlab. What does Matlab report?

- Try again by typing ’format long’ before your code in (1). What does Matlab report?

- Prove that the equation cannot hold. Such an example is called a Fermat near miss. (Hint: think about even and odd number on both sides.)

- In a later episode The Wizard of Evergreen Terrace, Homer writes the equation

3987^{12} + 4365^{12} = 4472^{12}:

Can you debunk this equation?

- (Challenge) Consider the following polynomial

p(x) = (x 1)^{7}

^{ }

(a) In Matlab, do the following and attach your graph.

x = 0.988:0.0001:1;

y = (x-1).^7;

plot(x,y);

This will plot the graph of p(x) on the interval [0:988; 1] using points with 0.0001 between each. (b) Notice

(x 1)^{7} = x^{7} 7x^{6} + 21x^{5} 35x^{4} + 35x^{3} 21x^{2} + 7x 1 This time do the following and attach your graph.

x = 0.988:0.0001:1;

y = x.^7-7*x.^6+21*x.^5-35*x.^4+35*x.^3-21*x.^2+7*x-1; plot(x,y);

Mathematically this should produce the same result as in (1).

(c) Explain why Matlab has no issue with (1) but large error is seen in (2).

2