Calculus 1
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The resistance of blood flow, R, in a blood vessel is dependent on the length of the blood
vessel, the radius of the blood vessel, and the viscosity of the blood. This relationship
is given by R = 8Lη/πr^4 where r is the radius, L is the length, and the positive value η is the viscosity (the larger the value of η the more viscous the blood).
(a) Find the derivative of R with respect to r. Is the derivative positive or negative? What is the physical interpretation of this? Does this result make sense?
(b) Find the derivative of R with respect to η. Is the derivative positive or negative? What is the physical interpretation of this? Does this result make sense?
(c) As body temperature increases, the blood vessel will widen (i.e. radius increases) and the blood becomes less viscous. Similarly as the body temperature lowers, the blood vessel will constrict and the blood will also become more viscous. We see then that the radius of the blood vessel and the viscosity of the blood are dependent on temperature. However, the length of the blood vessel essentially remains constant with temperature changes. Find the rate of change of resistance with respect to temperature, T. That is, find an expression for dR/dT
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