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The resistance of blood flowing through an artery is R = C L r4 where L and r are...

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Advance Math

The resistance of blood flowing through an artery is

R = C

L
r4

where L and r are the length and radius of theartery and C is a positive constant. Both L andr 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 inradius during growth. Use a directional derivative to determine therate at which the resistance of blood flow changes with respect toa unit of growth in the r-L plane.

Cr4​


(b) Use a directional derivative to determine how much faster thelength of the artery can change relative to that of its radiusbefore the rate of change of resistance with respect to growth willbe positive.


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

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