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 ?

**6.**
**(a) **Find the indefinite integral:

(
3t^{4} - 2/ t^{2} + cos t - e^{2t} + 3) dt

**(b)**
Find the following integral by substitution. Show all steps

x
( x^{2} + 3) ^{50}dx.

**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 ^{3 }
**
(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 P_{4}**
**(x)
of
f(x)
(at zero).
**(b) **Use
P_{4}**
**(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 = x^{2} + 4 y^{2}
**(b)** z =
1 - 2x - y
**(c)** z
= xy.
**(I)
(II)**

**
(III)**

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

**(C)**

**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 |

**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^{2 } - y^{3 }. 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...