MTH 132

University of Rhode Island

Exam #1
                                                          Kingston 02/11/98
1.   Find the average value of the function f(x) = x2 + x between x = 0 and x = 6.

2.   Find  x e3x  dx   using integration by parts.

3.   Let  C(t) denote the concentration of a drug in blood stream, measured in  mg/cm3   , 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/ t2 + e3t + tt + 3) dt,
(b)    Find the following integral by substitution. Show your work step-by-step
  2x x2 + 3 dx.

6.   Which of the following functions

 (a)   y = t                                                            (b)  y = e2t

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( t2 -  0.001339 t4 + 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.