# MTH 132

#### University of Rhode Island

Exam #2
Kingston 3/25/98
1.   Consider the following functions:
(a)   y = x                                  (b)  y = 3                           (c)  y =
Which of them are particular solutions to the differential equation
dy/dx = 2xy ?
Which of them is a solution to the initial value problem
dy/dx = 2xy ,   y(0)=1  ?

2.   A population of bacteria, Q(t), increases at a rate proportional at each instant t to the size of the population, with the coefficient of proportionality  0.06. The time t is measured in hours.
(a)   Write a  differential equation for the function Q(t).
(b)   Find the general solution to this equation.
(c)   If the initial population is 400, how many bacteria will be present after 10 hours ?

3.   Match the differential equations
(a)   dy/dx = y                                 (b)  dy/dx = 1 + y2                             (c)  dy/dx  = x .
(I)                                                                         (II)

(III)

4.   (a)  Find the general solution to the differential equation
dy/dt = 0.3(y - 15).
(b)  Solve the initial value problem
dy/dt = 0.3(y - 15),   y(0) = 20.

5.   When a murder is committed, the body, originally at 37 o C, cools according to Newton' s Law of Cooling. Suppose that after two hours the temperature is 35 o C, and that the temperature of the surrounding air is a constant 20 o C.
(a)  Find the temperature, H, of the body as a function of t, the time in hours since the murder was committed.
(b)  If the body is found at 8 a.m. at the temperature of 30 o C, when was the murder committed ?

6.   Each day at noon a hospital patient receives 4 mg of a drug. This drug is gradually eliminated from the system in such a way that over each 24 hour period the amount of the drug from the original dose is reduced by 30 %. On the other hand, the patient is receiving a new 4 mg dose each day.
(a)  How much of  the drug is present in the patient's system immediately after noon on the 10th day of treatment ?
(b)  If the regime is continued indefinitely, at what level will the amount of the drug in the patient's system stabilize, measured immediately after noon every day ?

7.   Let  f(x) = cos x.
(a)  Find the fourth Taylor polynomial P4 (x) of  f(x) (at zero).
(b)  Use P4 (x)  to estimate the value cos 0.9 using only simple arithmetic. Show your calculations. Compare your estimate with the actual value cos 0.9.

8.   Match the surfaces (I), (II), (III) below with the following equations:
(a)   z = 5 - ( x2  + y2 )                                (b) z =  x2  - y2                               (c)  z = .

(I)                                                                                                             (II)

(III)

9.   Match the surfaces (I), (II), (III) in the previous problem with contour diagrams (A), (B), (C) below.  Explain briefly how you arrived at your answers.
(I)                                                                                     (II)                                                   (III)

10.  The figure below shows level curves of the function giving the species density of breeding birds each point of the US, Canada and Mexico.

Looking at the map, are the following statements true or false ? Explain your answers.
(a)  The species density at the tip of Baja California is above 100.
(a)  Moving south to north across Canada, the species density increases.
(a)  The greatest rate of change in species density with distance is on the Yucatan peninsula.