- 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
- T5(x) for f(x) = 3 tan x, at point c = =4.
- T2(x) for f(x) = ecosx, at point c = 0.
- For f(x) = excos x at c = 0
- Find T2(x) by hand.
(b) Use Taylor’s theorem to give an estimate of error jf(0:5) T2(0:5)j. Compare it with the true
|(c) Approximate||Z0||1||f(x) dx by Z0||1||T2(x) dx. Find the true error.|
(d) Find the Taylor series for f(x) = x2 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?
- Find the Taylor series of x at c = 1. Determine the interval of convergence (be careful with the ending points.).
- Use the result you nd in question 2 to evaluate 2. Round your answer to 4 decimals.
- Convert (100010010111011)2, (10:11)2to decimal (base 10).
- Convert 37, 0:43, 10:11 to binary representation.
- Convert (1234)5to decimal, then base 8 representation.
- Write down the IEEE format of the following numbers
- 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:b1b2b32E 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.
- 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:
178212 + 184112 = 192212
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 xn + yn = zn. Did Homer dream up a
counterexample to Fermat’s last theorem?
- Compute 12178212 + 184112 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
398712 + 436512 = 447212:
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;
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 = x7 7x6 + 21x5 35x4 + 35x3 21x2 + 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).