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Calculus/Rate of Change

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Expert: - 10/29/2009


Question
Our prof is giving us some questions that he has never gone over in class so I was wondering if you'd be able to help me out.
1. A point moves along the parabola y=1/4x^2 in such a way that at x=2, the x-coordinate is increasing at the rate of 5 cm/s. Find the rate of change of y at this instant.
2. During early development, the limb of a fetus increases in size, but has a constant proportion. Suppose that the limb is roughly a circular cylinder with a radius r and length l in proportion where C is a positive constant (l/r = C#. It is noted that during the initial phase of growth, the radius increases at an approximately constant rate, ie. that dr/dt = a. At what rate does the mass of the limb change during this time? [Assume that the density of the limb is 1 gm/cm^3 and recall that the volume of a cylinder is V =Al where A is the base area #in this case of a circle) and l is length].

Thanks so much.

Answer
1. So we've got y = x²/4.
That makes the position (x, x²/4).
We are given dx/dt = 5cm/s.
Well, the movement equation is √((dx/dt)² + (d(x²/4)/dt)²).

That works into √(1² + (x/2)²)•(dx/dt), and we're given dx/dt.
That makes it be √(1+(x/2)²)•5.


2. If V = πlr², and l/r = c, so r = l/c.
The volume V(c) = πl(l/c)² = πl³/c².  Density is given as 1 gm/cm³.
Taking the derivative of volume gives dV/dl = 3π(l/c)².

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