October 2002:
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10/14/02) Derive a formula for the nth derivative of sin2x when n is in the form of 2k and 2k-1 separately, k belonging to the counting numbers.
Taking the derivatives of this function leads to a pattern. The first derivative is sin2x, second 2cos2x, third -4sin2x, fourth -8cos2x and etc. The coefficients are powers of 2, the trigonometric function alternates for n being even or odd (2k or 2k-1). However, one other problem exists, and that is the alternation of positive and negative apart from odd and even n. This can be formulated using cos(n*pi/2) or cos (n*pi). Thus, the formulas would be as follows:
-cos(n*pi/2)*2n-1*cos2x for n in form of 2k.
-cos(n*pi)*2n-1*sin2x for n iin form of 2k-1.
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10/21/02) Two similar containers are built. The first one's shape is resulted from rotating y=x1/2 about the y-axis from x=0 to x=m. The other is resulted from the same graph and about the same axis, but from x=0 to x=n where n>m. If the ratio of the first one's volume to the second one's is 1/32, and n-m=3, find n, m and the difference of volumes.
Using shells, the volume can be found by integrating 2pi*x*x1/2 from x=0 to x=k where k can be n or m. Using the fact that the ratio is 1/32 and n-m=3, we can conclude that m is equal to 1 and n is equal to 4. Thus, the difference of volumes would be 124*pi/5
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10/28/02) A sinusoidal model has a formula of y = cos3x*cos7x-cos2x*cos6x. Find the formula associating to the sum of its y-values at any point from x=0 to x=t.
The question simply asks for the integral of the function from x=0 to x=t. A trigonometric identity states that
cosA*cosB = (1/2)*cos(A+B)+(1/2)*cos(A-B). Using this, one can integrate y = (1/2)*(cos10x-cos8x), which is equal to (1/2)((sin10x)/10-(sin8x)/8). In this case, x would change to t since it is the integral from 0 to t and at x=0 the integral is zero.
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