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    Big-Oh Notation

    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...
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    Combinatorics Class - Sum Question

    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...
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    Sum of Series

    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...
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    Comparison Test: Am I using a good comparison function?

    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...
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    Partial Frac. Decomp. Integral ( long div? )

    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...
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    Integral (Partial-Frac Decomp) SIMPLE?

    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...
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    Trig Identity Integral

    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...
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    Derivative of an Integral

    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?
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    Definite Integration by Substitution

    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...
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    Integration by Substition

    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?
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    Differential Equation Involving Natural Log.

    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...
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    Discrete Counting Question

    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...
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    Discrete Relations: can't understand relation definition

    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...
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    Integral of x(e^x)

    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...
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    Comparison Test on Interval 0 to 1

    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...
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    Comparison Test: converges or diverges?

    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...
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    Basis for kernel space

    Homework Statement Find a basis for the kernel space of the following matrix: -1 -2 -1 2 2 -2 -4 -4 10 2 1 2 2 -5 2 -1 -2 0 -1 0 row reduce to 1 2 0 1 0 0 0 1 -3 0 0 0 0 0 1 0 0 0 0 0 Somehow read the solution as { [-2 1 0 0 0]T, [-1 0 3 1 0]T } .. I don't...
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    Finding determinant with upper triangular matrix

    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...
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    Simple basis / spanning set question (T / F)

    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...
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    Calc Optimization Problem

    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...
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    Simple Anti-derivatives

    Homework Statement Find the anti-derivative of (2 + x^2)/(1 + x^2) Homework Equations f(x) = tan^-1(x) f'(x) 1/(1 + x^2) The Attempt at a Solution (2 + x^2) / (1 + x^2) = ( 2 / (1 + x^2) ) + ( x^2 / (1 + x^2) ) The anti-derivative of (2 / (1 + x^2) ) is 2tan^-1(x). I...
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    Orthogonal Bases

    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...
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    Orthagonal Sets

    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...
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    Linear independence

    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...
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    Vector Spaces Question

    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 de ned 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)...
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    Solving for x in equation (e^2x)-(3e^x)+2=0

    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
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    Intersection of Lines in R4

    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...
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