Math Is Fun Forum

If i let f(x) = sinx. Find the error E(x, pi/2) in equation:
E(x,a) = f'(a) - [(f(x)-f(a))/(x-a)]
as a function of x.  Graph E(x,a). Find a delta if epsilon = .01, .0001, 10^-10.

Honestly I can do the graphing part.  I will also be able to do the proof when epsilon = .0001, 10^-10 once I have seen a proof for one part.  I am having a hard time with the teacher - he has yet to do an example of how to do one of these proofs... and the book, needless to say, is not much better.

Any help would be great.

I'm just wondering if there is a good book or place to get self help on Analysis stuff?   I learned calc I and II through these books but I'm trying to learn more for Analysis but am not sure where to get help or find a good book about it

I need to:

Find the target of the series: 
1 - (3/4) + (9/16) - (27/64) + ... + (-1)^k*(3^k/4^k)
Then i need to find any value of n so that any partial sum with at least n terms is within 0.001 of the target value. Justify the answer.

So here's what I did. I made a program on Mathematica that gave me that the target value of the partial sum is .571429

I also found that at the sum from 0 to the 22nd term is equivalent to 0.572193

I suppose I have my answer but I cannot justify it - except using proof by mathematica program.  I am worried though that when I have an exam and am forced do this without mathematica I will not be able to.  Is there any advice? Or can someone teach me how to do this?

Let G be the set of all real-valued functions on the interval [0,1]. Define f+g for f,g in G by (f+g)(x)=f(x)+g(x).

1. Prove that G is a group
2. Define phi : G->R by phi(f)=phi(1/4) and prove that phi is a homomorphism
3. Let H={f in G such that f(1/4)=0}. Prove H is a subgroup of G.
4. What is G/H isomorphic to?

All in all - just stuck. Any ideas?

My work so far:
G is a group means that it is assosciative, closed, has an inverse and identity. (It's associative because addition is)

1. Given 3 non-collinear points, there is one and only one circle containing all three points.
2. If triangles ABC and DEF have right angles at A and D and have AB=DE and BC=EF show the triangles are congruent.

For 1: The three points are on the circle.  I'm not sure where to start. We've really only done work with triangles and lines in the class so far.
For 2: I now have angle-side-side which is not a valid form of proof. I've tried proving by contradiction with no success.  I'm not sure what step to take now.

ideas?

Do the odd permutations of S_n form a group? Explain.

I KNOW that they dont for the reason that the group of odd permutations does not include the identity. I'm not sure how to prove this in general.

Thanks.

1. What are the symmetries of a nonsquare rectangle?
2. What are the symmetries of a parallelogram that is neither a rectangle or a rhombus.
3. What are the symmetries of a rhombus that is not a rectangle?

My thoughts:
1. 4: The identity, 180 degree rotation, vertical reflection and horizontal refletion?
2. 4: The identity, 180 degree rotation, 2 reflections from opposite angles.
3. 4: same as above?

it doesn't seem right because 2 and 3 seems like they shouldn't be the same? THOUGHTS? Thanks.

Some things I'm confused about:
1. How to show that the union of countable sets is also countable. For example, let A1, A2, ... be a sequence of sets, each countable. How to prove the union is a countable set.

2. What the use of lattice paths is. I understand how to make/get one... by I'm not sure how, on the test for example, it could be used in a problem.

3. How to count something in 2 ways

4. Cardinality.

and
5. Binary coding / n-tuples. For example, our teacher emailed us today with a question he says we should be able to answer easily: The 51st State of the union is going to be the State of Tom.  I will be
issuing license plates for cars using 3 letters from the alphabet.  How many plates can I make?  In essence we are counting ?-ary ?-tuples.  Finally, what is the probability that a randomly made plate will spell Tom?

ANY ADVICE/HELP WOULD BE WONDERFUL.

1. Need to Use Pascal's Formula and induction on n to prove the Binomial Theorem:
(DONT UNDERSTAND THE PART IN BOLD) Correct otherwise??

Base Case:  (a+b)^0 = 1 = \sum_{k=0}^0 { 0 \choose k } a^{0-k}b^k.
For the inductive step, assume the theorem holds when the exponent is m. Then for n = m + 1
(a+b)^{m+1} = a(a+b)^m + b(a+b)^m \,

= a \sum_{k=0}^m { m \choose k } a^{m-k} b^k + b \sum_{j=0}^m { m \choose j } a^{m-j} b^j by the inductive hypothesis
= \sum_{k=0}^m { m \choose k } a^{m-k+1} b^k + \sum_{j=0}^m { m \choose j } a^{m-j} b^{j+1} by multiplying through by a and b
= a^{m+1} + \sum_{k=1}^m { m \choose k } a^{m-k+1} b^k + \sum_{j=0}^m { m \choose j } a^{m-j} b^{j+1} by pulling out the k = 0 term
= a^{m+1} + \sum_{k=1}^m { m \choose k } a^{m-k+1} b^k + \sum_{k=1}^{m+1} { m \choose k-1 }a^{m-k+1}b^{k} by letting j = k − 1
= a^{m+1} + \sum_{k=1}^m { m \choose k } a^{m-k+1}b^k + \sum_{k=1}^{m} { m \choose k-1 }a^{m+1-k}b^{k} + b^{m+1} by pulling out the k = m + 1 term from the RHS
= a^{m+1} + b^{m+1} + \sum_{k=1}^m \left[ { m \choose k } + { m \choose k-1 } \right] a^{m+1-k}b^k by combining the sums
= a^{m+1} + b^{m+1} + \sum_{k=1}^m { m+1 \choose k } a^{m+1-k}b^k from Pascal's rule
= \sum_{k=0}^{m+1} { m+1 \choose k } a^{m+1-k}b^k by adding in the m + 1 terms.

2. Summing the cubes?
a)Prove directly that m^3=6(m choose 3)+6(m choose 2)+m
b)Use part (a) to prove that the sum(from i=1 to n) i^3 = [(n(n+1)/2]^2.(without using induction)
c)Prove part (a) by counting a set in two ways (hint: Count the ordered triples that can be formed from [m].)

Really just not sure where to go here... maybe some hint may help? I think I can do it... not sure where to start?