MTH 132

University of Rhode Island

Final Exam
                                                              Kingston 5/04/98

1.   The graph below shows plasma concentration curves for two pain relievers. Compare the two products in terms of level of peak concentration, time until peak concentration, and overall bioavailability.

2.   The graph below gives the density function for the amount of time spent waiting at a doctor's office.

(a)   What is the maximum amount of time that anyone will have to wait ?

(b)   Approximately what percentage of patients end up waiting between 1  and  2 hours ?

(c)   Approximately what percentage of patients wait less than an hour ?

3.   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.

4.   The graph below represents the density function for the shelf life of a brand of banana.

(a)   Estimate the median shelf life of a banana. Explain your reasoning.

(b)   Approximately what fraction of bananas last less than a week ?

5.   Test scores in a large class are normally distributed with mean  70 and standard deviation 8.

(a)   Write a formula for the density distribution p(x) of test scores.

(b)   Graph the density function showing units on the x axis.

(c)   Write the definite integral that represents the fraction of students with test scores between 62 and 78.

(d)   Without evaluating the integral, what do you expect it to be ? Why ?

(a)  Find the indefinite integral:
( 3t4 - 2/ t2 + cos t - e2t +  3) dt

(b)   Find the following integral by substitution. Show all steps
x ( x2 + 3) 50dx.

7.   The amount of a drug in a patient's bloodstream, Q(t),  in mg , decreases at a rate proportional at each instant to the amount of the drug present with the coefficient of proportionality -0.2. The time t is measured in hours.

(a)   Write a differential equation for the function Q(t).

(b)   Find the general solution to the equation.

(c)   If the initial amount  Q(0) = 200, how much drug is left after 10 hours ?

8.   Consider the following functions:

 (a)    y = x                       (b)    y =                      (c)   y = 2.

Which of them are particular solutions to the differential equation
                                                 dy/dx  = 3 x 2 y  ?
Which is the solution to the initial value problem

                                            dy/dx  = 3 x 2 y,  y(0) = 2  ?

9.   A detective finds the murder victim at 9 a.m. The temperature of the body is measured at 90.3 o F. One hour later, the temperature of the body is  89 o F.  The temperature of the room has been maintained at a constant 68 o F. Recall, the normal temperature of a live body is  98.6 o C.

(a)  Assuming that the temperature,  T, of the body obeys Newton's Law of Cooling, write a differential equation for T.

(b)  Solve the differential equation to estimate the time the murder occurred.

10.   Let  f(x) =  e x .
(a)  Find the fourth Taylor polynomial P4 (x) of f(x) (at zero).
(b)  Use P4 (x) to estimate e 0.3  using only simple arithmetic. Show your calculations. Compare your estimate with the calculator value of e 0.3 .

11.   Match the following equations (a), (b), (c) with the surfaces (I), (II), (III) below. Explain your answer.

(a)   z =  x2  + 4 y2                             (b) z =  1 - 2x - y                        (c)   z = xy.
(I)                                                                                                      (II)


12.   Match the contour diagrams (A), (B), (C) with the surfaces (I), (II), (III) of the previous problem.
(A)                                                                                      (B)


13.   Below you have a partial table of values for a linear function  f(x,y).
(a)   Fill in the blanks.
(b)   Find an equation for f(x,y).
x/y 10 20 30 40
100 3 9 12
200 2 5 11
300 1 10
400 0 3 6 9
14.   Find   fx fy for
 (a)   f(x,y) = 2 x - 4xy + 2 y                (b)   f(x,y) =  x sin(xy).

15.   Consider data points (1,2), (3,2.5), (2.5,4), (4,5). Suppose you want to find a least squares line  y = mx + b  for these data points without using a calculator program. What function  f(m,b)  would you have to minimize ? Write the function  f(m,b)  but do not perform actual calculations.

16.   Suppose demand for coffee, Q, in pounds sold per week, can be thought of as a function of the price of coffee, c , in dollars per pound, and the price of tea, t,   in dollars per pound. Thus, we have  Q = f(c,t). Using units, interpret in everyday terms the statements:

 (a) f(3,4) = 820              (b)  f c (3,4) = - 70                    (c)  f t (3,4) = 30 .

Estimate  f(3.5,4).

17.  Below you see a contour map of a function f(x,y).
(a)  Is  f x (9,76) positive or negative ? Why ?
(b)  Is  f y (9,76) positive or negative ? Why ?


18.   Let f(x,y) =  3xy -  x - y3 . Find all critical points, local minima, maxima and saddle points of  f(x,y).

19.   Let a function  f(x,y) and a point (a,b) be such that
                 f x (a,b) = f y (a,b) = 0,   f xx (a,b) > 0, f yy (a,b) > 0,   f xy (a,b) = 0.

(a)  What can you conclude about the behavior of   f(x,y) near (a,b) ?
(b)  Sketch a possible contour diagram for f(x,y) near (a,b).

20.   Use geometric series to find simple fractions that equal the following repeating decimals:

(a) A = 0.8888888...                             (b)   B = 0.30303030...