MATH 1320 : Spring 2014
Lab 2
Lab Instructor : Kurt VanNess
Name:
Score:
Write all your solutions on a separate sheet of paper.
1. Determine the average value of each of the following functions on the given interval.
(a) f (t) = t2 + sin2 t
on [−1, 4].
− cos 2t
(b) f (t) = 2 sin(2t)e
on [−π, π].
2. Determine the number c that satisfies the Mean Value Theorem for Integrals for the function f (x) =
x2 + x + 1 on the interval [1,4].
3. (a) The velocity of blood that flows in a blood vessel with radius R and length l at distance r from
central axis is
P
(R2 − r2 )
v(r) =
4ηl
where P is the pressure difference between the ends of the vessel and η is the viscosity of the
blood. Find the average velocity over the interval [0, R].
(b) The blood flux is the rate of the blood flow, volume per unit time. The blood flux F can be
computed by the formula
Z R
F =
2πrv(r)dr
0
Using the formula for the velocity given in part (a), evaluate the integral computing the flux F .
In addition, find the flux in a small human artery where η = 0.027, R = 0.0008 cm, l = 2 cm, and
P = 4000 dynes/cm2 .
4. (Hooke’s Law) If 6 J (or Newton-meters) of work is needed to stretch a spring from 10 cm to 12 cm
and another 10 J is needed to stretch it from 12 cm to 14 cm, find both the spring constant and the
natural length of the spring? [Hint: convert from cm to m.]
5. A 3 cm piece of wire has a density given by the function δ(x) = 1 + x2 grams per centimeter at the
point x cm from the left end of the wire. Where is the center of mass of the wire?
6. (a) Verify that the solution to the logistic differential equation
P
dP
= kP (1 −
)
dt
M
is
P (t) =
where
A=
M
1 + Ae−kt
M − P0
P0
and P0 is the initial value of P .
(b) Find the limit of P (t) as t → ∞.
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