Review Problems for 324 Final

May 21, 2009 — koopakoo

Chapter 14 Review, pg 946. Q37, 39, 47, 48.

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Mid-Term 2 Solution

May 19, 2009 — koopakoo

1) (a) \int_{0}^{2 \pi}(\frac{1}{\cos{t} + 2} + \sin^2{t}, 2\cos{t}\sin{t}) \cdot (-\sin{t}, \cos{t}) dt.

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Math 324 Mid-Term 1 Solution

April 18, 2009 — koopakoo

1) (a) V(S) = \int_{-1}^{1}\int_{-\sqrt{1 - x^2}}^{\sqrt{1 - x^2}} \int_{x^2 + y^2}^{\sqrt{x^2 + y^2}}1 \, dz \, dy \, dx.

(b and c) \begin{aligned} V(S) &= \int_{0}^{2\pi}\int_{0}^{1}\int_{r^2}^{r}r \, dz \, dr \, d\theta\\ &= 2\pi \int_{0}^{1} r^2 - r^3 dr \\ &= 2\pi (\frac{1}{3} -\frac{1}{4}) = \frac{\pi}{6}.\end{aligned} Read the rest of this entry »

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Tricks of Integration (1) – The Koop Subs

April 8, 2009 — koopakoo

In this article, I will talk about some of my favorite substitutions for integration, which I will named them as the “Koop subs”. They are particularly useful in mathematics competitions such as the Putnam competition.

Example 1) Compute the integral: I = \int_{0}^{1} \frac{x^3}{1 - 3x + 3x^2} dx. Read the rest of this entry »

Posted in Calculus, Teaching. 6 Comments »

Solution to “Possible HKAL Pure Problems”

April 1, 2009 — koopakoo

1(a) Expanding along the last row, we have:

f(x) = 1 \det{\begin{pmatrix}x & x^3\\y & y^3 \end{pmatrix}} - z \det{\begin{pmatrix} 1 & x^3 \\1 & y^3 \end{pmatrix}} + z^3 \det{\begin{pmatrix} 1 & x \\1 & y \end{pmatrix}}.

Therefore, f(z) is a cubic in z with leading coefficient \det \begin{pmatrix} 1 & x \\1 & y \end{pmatrix} = y - x. Read the rest of this entry »

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Possible HKAL Pure Problem (2)

April 1, 2009 — koopakoo

Prove by contradiction or otherwise, show that \log(x) is not a polynomial with real coefficients.

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A possible HKAL Pure problem?

March 5, 2009 — koopakoo

Let A = \begin{pmatrix} 1 & x & x^3\\ 1 & y & y^3 \\1 & z & z^3 \end{pmatrix}

(a) By expanding the determinant along the last row (or otherwise), show that det(A) is a cubic polynomial in z, and we call det(A) = f(z). (1 point)

(b) Show that f(x) = f(y) = 0, and hence conclude that f(z) = (y - x)(z - x)(z - y)(z - r) for some r in terms of x and y. (5 points)

(c) Using the relationship between the roots and coefficients of a polynomial (or otherwise), show that r = -x - y. (3 points)

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Preparing for the Exam (2)

February 28, 2009 — koopakoo

Here are a collection of relatively easy problems:

1) Can you find a 4 by 4 matrix, that is NOT upper-triangular with charpoly being t^3(t - 3)?

2) When you have a matrix, always ask yourself the following questions: Read the rest of this entry »

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Answer to Basic Linear Algebra Questions (1)

February 26, 2009 — koopakoo

The following is a list of basic questions you should know how to answer (by heart)

(1) Suppose v_1, v_2, \dots,v_n are linearly independent and
a_1v_1 + \dots + a_nv_n = 0. What do you know about the a_i?

Ans: a_i = 0 for all i.

(2) If x \in \ker{A}, what is Ax?

Ans: Ax = 0. Read the rest of this entry »

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Hints to Selected Problems in HW 5

February 17, 2009 — koopakoo

For 3. Answer the following mini questions:
(a) Let A = B + I, what is rank(A)?
(b) What is \chi_{A}(t)?
(c) If \lambda \in \sigma(A), what should be in \sigma(B) = \sigma(A - I)? Read the rest of this entry »

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