Download Groupwork 1: Implicit Differentiation and Related Rates Applied

Groupwork 1: Implicit Differentiation and Related Rates
Applied Calculus II, MA 120 B
January 29, 2015
Work together with 2 or 3 students to answer the following questions.
1) Suppose x2/3 + y 2/3 = 2.
a) Find dy/dx by implicit differentiation.
b) Find the equation of the tangent line to the curve at the point (1, 1).
2) The relation between a demand of q units and price of p dollars for a certain product is given
by ln(q) = 7 − 0.63 ln(p).
a) Find dq/dp using implicit differentiation. What does this represent and what are its units?
b) Find dq/dp when the demand is 100 units and the price is $44.76 per unit.
3) Assume x and y are functions of t.
a) 4exy − y 2 = x. Find dy/dt when x = 0, y = 2 and dx/dt = 3.
x2 + y
b)
= 9. Find dx/dt when x = 4, y = 2 and dy/dt = 1/5.
x−y
4) Blood flows faster the closer it is to the center of a blood vessel. According to Poiseuille’s law,
the velocity v of blood is given by
v = 450(R2 − r2 ),
where R is the radius of the blood vessel and r is the distance of a layer of blood flow from the
center of the vessel.
a) Suppose R = .08 mm and the cold weather causes the vessel to contract at a rate of dR/dt = −.01
mm per minute. Treating r as a constant, find dv/dt.
b) Now suppose R is a constant (unchanging in time). Find dv/dt when r = .03 mm and dr/dt = .01
mm per minute.