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(a) Graph several members of the family of the functionsforand look at the regions enclosed by these curves and the x-axis. Make a conjecture about how the areas of these regions are related.
(b) Prove your conjecture in part (a).
(c) Take another look at the graphs in part (a) and use them to sketch the curve traced out by the vertices (highest points) of the family of functions. Can you guess what kind of curve is this?
(d) Find an equation of the curve you sketched in part (c).
The blood vascular system consists of blood vessels (arteries, arterioles, capillaries, and veins) that convey blood from the heart to the organs and back to the heart. This system should work so as to minimize the energy expended by the heart in pumping the blood. In particular, this energy is reduced when the resistance of the blood is lowered. One of Poiseuille’s Laws gives the resistance R of the blood as
where L is the length of the blood vessel, r is the radius, and C is a positive constant determined by the viscosity of the
(a) Use Poiseuille’s Law to show that the total resistance of the blood along the path ABC is
where a and b are the distances shown in the figure.
(b) Prove that this resistance is minimized when
(c) Find the optimal branching angle (correct to the nearest degree) when the radius of the smaller blood vessel is two-thirds the radius of the larger vessel.
6. Consider the graph of the curve y = (a) Find the area under the curve from :=1 to r= 100. (b) What happens to the area under the curve as the right hand endpoint goes to co? (c) Find the volume of the solid obtained by rotating this curve around the c-axis from x=1 to r= 100. (d) What happens to the volume in part (c) as the right hand endpoint goes to co? (e) Find the volume of the solid whose base is this region bounded by the curves y = 1/2, y=0, x= 1 and 2 = 100 and whose cross-sections perpendicular to the z-axis are right triangles whose height is half their base. (f) What happens to the volume in part (e) as the right hand endpoint goes to co?
for
and look at the regions enclosed by these curves and the x-axis. Make a conjecture about how the areas of these regions are related.
