June 2002:
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06/24/02)
Part I: The sum of coordinates of a function vary directly with a power n of its y-value at any point, n being a positive integer. Where does this function have extrema? Why?
Part II: The sum of coordinates of a function vary directly with a power n of its x-value at any point, n being a positive integer. For what values of n does this function have extrema?
Part I: Mathematical Language requested! The question leads to the result that x + y = kyn, by the defintion of direct relationships. Implicit differentiation would get us this:
dy/dx = 1/(nkyn-1-1), which is NEVER zero. Therefore, these functions do not have an extrema whatsoever.
Part II: Doing the same process would get us the following:
dy/dx = nkxn-1-1. This can be zero. In fact, it is zero where x equals the (n-1)th root of 1/(nk). This means that n has to be odd for the function to have extrema. Beautifully solved, right?
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06/17/02) It has been found that the acceleration and velocity of an object are related in the following equation: a = v2-t2-2bt where a is acceleration, v is velocity, t it time and b is a positive constant. Further information reveals that the object's velocity varies linearly where v = t+b. Calculate the displacement of the object from t = 0 to t = 20 and describe the motion.
First let's find b. Knowing that velocity is t+b, we can plug that back into our a equation. So, a = (t+b)2-t2-2bt. Yet we know that the slope of v = t+b is 1 and so is a. Hence, 1 = (t+b)2-t2-2bt = b2. Knowing that b is positive, we would get b = 1. Now, we know that v = t+1. The displacement would be the integral of this from 0 to 20, which is 220. The object is moving with a constant acceleration and increasing speed.
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06/10/02) A line is divided into equal parts, each being a diameter of a semicircle. If the process of dividing continues forever, to what value, if any, does the sum of all circumferences converge? What about the sum of areas? Going further, imagine we start adding the areas of the semicircles accumulatingly starting with 2 divisions and going on and on. Would the sum of all infinite areas converge to a value? Explain.
We begin by the circumference. Each semicircle has a diamter of L/n where L is the line's length and n is the number of divisions. The circumferences of the semicircles would be (pi(L/n)*n)/2, since there are n semicircles. As you can see, this is always equal to pi(L/2). The area in terms of L and n would be (pi*L2)/8n. The limit of this as n goes to infinity is zero.
The second part of this problem asks for the accumulative area, which can be noted as sum of (pi*L2)/8n from n=1 to infinity. In the study of series, this is just a multiple of harmonic series, which would always diverge.
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06/03/02) A rectangle is to be made in such a way that its base lays on the x-axis and its height is bound by a triangle of vertices (0,0), (6,0) and (4,2). Find the maximum area this rectangle could attain.
Let's start by finding the equations of the lines making the triangle. One is y = x/2 and the other is y = -x+6. The base of the rectangle has a length of x1-x2 where x1 is x-value of y = -x+6 and x2 is the x-value of the other line. These x-values are 6-3y and 2y respectively. So the base of the rectangle is 6-y-(2y) or 6-3y. The height is simply y. The area, hence, is y(6-3y). Optimization tells us that the derivative of the area function has to be zero. Doing that we get y = 1. So the height should be 1 and base has to be 3, using 6-3y = 1. Now we are left with the maximum area of 3(1) or 3. Kinda neat, wasn't it?
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