Homework Statement
Determine the complexity of the following code:
for (i = 0; i < 2*n; i += 2)
{
for (j=n; j > i; j--)
{
a++;
}
}
The Attempt at a Solution
Well.. The first for block is O( n ) because i is incremented by 2 each loop up to 2n. The second block...
Homework Statement
For any positive integer n determine:
\sum\limits^n_{i=0} \frac{1}{i!(n-i)!}
Homework Equations
I don't really know where to start.. Up until this point we've just been doing permutations, combinations, and determining the coefficient of a certain term in the expansion of...
Homework Statement
Evaluate the sum of the following:
\sum\limits^\inf_{n=1} \frac{6}{n(n+1)}
Homework Equations
The Attempt at a Solution
Well... The denominator is going to get infinitely large as n approaches infinity, so would the value of the sum not converge to zero? The...
Homework Statement
Does the following interval diverge?
\int^9_1 \frac{-4}{\sqrt[3]{x-9}}
Homework Equations
The Attempt at a Solution
Well.. I've used the following function that I think is always less than the above function to prove that the function above DOES NOT diverge (by...
Homework Statement
\int^1_0 \frac{-3x^3+12x+30}{x^2-x-6}dx
Homework Equations
The Attempt at a Solution
I've attempted long division, but the long division does not seem to come to an end.. I'm not sure what to make of this.. If the long division does not end, does this mean that...
Integral (Partial-Frac Decomp) **SIMPLE?
Homework Statement
\int^3_2 \frac{-dx}{x^2-1}
Homework Equations
The Attempt at a Solution
= \int^3_2 \frac{A}{x} + \frac{B}{x-1} dx For some integers A and B.
-1 = A(x-1) + B(x+0)
-1 = Ax - A + Bx
Split into two equations...
Homework Statement
I missed one class on trigonometric identities in integrals, and I feel that one is needed here:
\int^3_0\frac{1+arctan(\frac{x}{3})}{9+x^2}dx
Homework Equations
The Attempt at a Solution
Again, I'm unsure what to do. I think that it is a trig identity, but I could be...
Homework Statement
Find the derivative of the function
F(x) = \int^0_{x^2-1}\frac{sin(t+1)}{t+1}dt
Homework Equations
The Attempt at a Solution
F'(x) = -\frac{sin(x^2)}{x^2}
I'm just learning this and unsure if this is correct. It seems too easy?
Homework Statement
\int^2_1 6x\sqrt{x-1}dx
Homework Equations
The Attempt at a Solution
Let u=x-1.
Then, u+1=x,
and du=dx.
Continued from problem statement,
=6 \int^1_0 (u+1)u^{\frac{1}{2}}du
=6 \int^1_0 u^{\frac{3}{2}} + u^{\frac{1}{2}}du
=6(1^{\frac{3}{2}} + 1^{\frac{1}{2}})
=6(2)
=12...
Homework Statement
\int \frac{-2x}{\sqrt[4]{x+2}}dx
The Attempt at a Solution
=-2*\int x(x+2)^{\frac{1}{4}}dx
Let u=x+2.
Then, u-2=x,
and du = dx
.. Continued from above,
=-2*\int (u-2)u^{\frac{1}{4}}du
=-2*\int u^{5/4}-2u^{\frac{1}{4}}du
Is that last step allowed?
Homework Statement
Calculate the following:
\int^{-4}_{-6} (x^-1+5x)dx
Homework Equations
The Attempt at a Solution
I've worked this down to
ln(-4) + 5(-4) - ln(-6) - 5(-6)
=ln(-4)-ln(-6)+10
The answer to the left half (the ln parts) of the equation is undefined, and the...
Homework Statement
How many was can 3 identical prizes be awarded to 98 potential winners?
Homework Equations
The Attempt at a Solution
Well. I know that if the prizes were unique, the first prize would have 98 possible winners, the second prize would have 97 possible winners...
Homework Statement
Let Z be the set of all integers.
Then, S is a relation on the set Z x Z defined by:
for (a1, a2), (b1, b2) belong to Z x Z,
(a1, a2)S(b1, b2) <-> a1b2 = a2b1.
Homework Equations
The Attempt at a Solution
The actual problem is about symmetry...
Homework Statement
What is the integral of x(e^x)?
The Attempt at a Solution
It's part of a larger question. I've got a midterm tomorrow and just realized I don't know the principles of integrating e^x other than that the integral of e^x is e^x. I've scoured my textbook and cannot find...
Homework Statement
Determine whether or not the integral from 0 to 1 of (5ln(x)) / ( x^(3/2) ) converges or not.
Homework Equations
The Attempt at a Solution
I just need to know which end of the integral they are talking about. As x=>0, y=>-infinity. As x=>1, y=>1. I'm assuming...
Homework Statement
Determine whether or not the improper integral from 0 to infinite of (e^x)/[(e^2x)+4] converges and if it does, find it's definite value.
Homework Equations
The Attempt at a Solution
I missed the lecture on the Comparison Test, so I'm essentially useless.
I...
So. I've been told by my prof that the best way to find the determinant of a matrix is to row reduce it to upper triangular and then take the product of the numbers on the diagonal. That's fine, BUT, how do you know how to reduce it?
Depending on what row operations you do, you get...
Homework Statement
Let T: R4 --> R7 be a linear map whose kernel has the basis of v = [1 0 1 2]T. What is the dimension of the image of T?
The Attempt at a Solution
I have a very loose understanding of kernel and image, and am trying very hard to get this question. From my understanding, the...
Homework Statement
A right circular cylinder is inscribed in a cone with height h and base radius r. Find the largest possible volume of such a cylinder.
Homework Equations
Vcone = (1/3)(pi)(r2)(h)
Vcylinder = (pi)(r2)(h)
The Attempt at a Solution
I've been trying to relate the...
Homework Statement
Let U be a subspace of R4 and let S={x1, x2, x3} be an orthogonal basis of U.
Given x1, x2, x3, find a basis for Uperp (the subspace containing all vectors orthogonal to all vectors in U).
I am actually given three vectors x1, x2, x3, but I am looking more to...
Homework Statement
Use the Gram-Schmidt algorithm to convert the set S={x1, x2, x3} to an orthagonal set, given x1 = [1 1 1 1]T, x[SUB]2 = [6 0 0 2]T, x[SUB]3 = [-1 -1 2 4]T.
Homework Equations
The Attempt at a Solution
I've used the algorithm to come up with the set of vectors...
Homework Statement
Let {X, Y, Z} be linearly independent in Rn. If {X, Y, Z, W} is linearly dependent, show that W \epsilon span{X, Y, Z}. NB: You must SHOW this.
Homework Equations
The Attempt at a Solution
For W to belong to the span of {X,Y,Z}, W = aX + bY + cZ where a, b, c...
Homework Statement
Is U = {f E F([a, b]) | f(a) = f(b)} a subspace of F([a, b]), where F([a, b]) is the vector space of real-valued functions dened on the interval [a, b]? (keep in mind that in the definition of U, the E means belonging to.. I couldn't find an epsilon character)...
Homework Statement
Solve for x.
e2x-3ex+2=0
The Attempt at a Solution
e2x-3ex+2=0
-e2x+3ex=2
ex(3-ex)=2
lnex(3-ex)=ln2
x(3-ex)=ln2
..not very sure where to go here. Any direction would be appreciated! Thanks
To find the intersection of two lines in R3, you set the lines equal, right?
[a,b,c] + d[e,f,g] = [h,i,j] + k[l,m,n]
Then split these into three equations,
1. a + d(e) = h + k(l)
2. b + d(f) = i + k(m)
3. c + d(g) = j + k(n)
And solve for k and d, correct?
If k and d are consistent...