#
MTH 132

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University of Rhode Island

Exam #1
Kingston 02/11/98
**1. **Find the average value of the
function f(x) = x^{2} + x between
x = 0 and x
= 6.
**2.** Find
x e^{3x } dx
using integration by parts.

**3. **Let C(t) denote
the concentration of a drug in blood stream, measured in mg/cm^{3 }
, t hours after
an injection. Using the graph of C(t) below
estimate the bioavailability of the drug over the 12-hour
period following the injection. Give units with your answer.

^{ }

**4. **The relative rate of growth P'(t)/P(t)
of a population P(t)
over a 50-year
period is given below. By what factor has the population increased
during the period ?
**5. **Find the following integrals:

(a)
(2 sin t - 1/ t^{2}
+ e^{3t} + t^{2 }t
+ 3) dt,

(b) Find the following
integral by substitution. Show your work step-by-step

2x x^{2}
+ 3 dx.
**6. **Which of the following functions

(a)
y = t^{2 }
(b) y = e^{2t}

are solutions to the differential equation t dy/dy
= 2y ? Explain your reasoning !

**7. **The length of a snake in inches, in an adult population
of a certain species is normally distributed with mean 46
and standard deviation 2.

(a) Write
a formula for the density of the distribution.

(b) Write
the definite integral which represents the fraction of the population with
the length between 44
and 48 inches.
Without evaluating the integral, what do you expect it to be ? Why ?

**8. **The graph below gives the
density function for the amount of time spent waiting for a bus at a bus
stop.

(a) What
is the maximum time that anyone will have to wait ?
(b) What
fraction of people will wait less than an hour ? Between 1.5
and 2 hours
?

(c) In
what half-hour interval is the waiting time most likely to fall ?

(d) Estimate
the median waiting time.

**9. **A possible model for the
density function for the US age distribution is

p(t) =
0.0000001( t^{2} - 0.001339 t^{4} + 123450),
for t
in years between 0 and
100.
(a) Write the
definite integral which represents the mean age in the US.

(b) Evaluate the
integral using the Fundamental Theorem of Calculus.

**10. **Find the cumulative distribution
function p(t) for
the age density in Problem 9. Explain its meaning.