Question

The resistance of blood flowing through an artery is

R = C

L

r4

where L and r are the length and radius of the artery and C is a positive constant. Both L and r increase during growth. Suppose

r = 0.1 mm,

L = 1 mm,

and

C = 1.

(a) Suppose the length increases 10 mm for every mm increase in radius during growth. Use a directional derivative to determine the rate at which the resistance of blood flow changes with respect to a unit of growth in the r-L plane.

Cr4​

(b) Use a directional derivative to determine how much faster the length of the artery can change relative to that of its radius before the rate of change of resistance with respect to growth will be positive.


(c) Illustrate your answers to parts (a) and (b) with a sketch of the directional derivatives on a plot of the level curves of R. (Use u for the unit change described in part (a) and v for the unit change described in part (b).)

Answer 1

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