November 2001:

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11/26/01) Find the following limit and show work.

Solution:

The key is to multiply the numinator and the denominator by (x2/3 + x1/3.h1/3 + h2/3). By the algebraic formula for difference of cubes, the numinator becomes x-h and the denominator is (x-h) times the multiplied expression. So the (x-h) cancels and as h approaches x, the denominator becomes 3x2/3. So the limit equals 1/3x2/3. Lovely, huh?

Topic:

Limits

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Find dy/dx:
11/19/01: (y-x)x2=cos(x-y)

Solution:

x2(y-x)x2-1(dy/dx-1)={-sin(x-y)}(1-dy/dx)
by algebra you would get: dy/dx=1, surprising, huh?

Topic:

Implicit Differentiation

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11/12/2001) Water is being poured into a conical container of height of 12 in. and radius of 5 in., with a speed of 2 in3/sec. How fast is the surface area of the water changing at the height of 6 in.?

Solution:

By the volume formula, we find dh/dt to be 0.10186 in/s. Using the ratio of h to r, we find the dr/dt to be 0.04244 in/s and by Mr. Pythagoras, we calculate s, the diagonal of h and r, to be 0.11035 in/s.

Now, with all these information, we use the surface area equation. Using the multiplication rule and all the fun of calculus we find dA/dt to be 2.59 in2/s. Interesting, huh?

Topic:

Related Rates of Change

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