Problem 1. The concentration C of a drug in the blood (in mg per liter) following its
injection is a function of the dose x (in mg) and the time t (in hours) since the injection:
C = xte−t
(a) Find C|
(x,t)=(5,1). What does this tell you about the concentration of drug in the
blood stream?
(b) Find a formula for the section where x = 5 and use it to find
∂C
∂t

(x,t)=(5,1)
What are the units? What does this value tell you about the concentration of drug
in the blood stream?
(c) Find a formula for the section where t = 1 and use it to find
∂C
∂x

(x,t)=(5,1)
What are the units? What does this value tell you about the concentration of drug
in the blood stream?
Problem 2. The Cobb-Douglas function
P = f(N, V ) = 2N
0.6V
0.3
models the production of a small printing press P (in thousands of pages per day) as a
function of the number of workers N and the value of the equipment V (measured in units
of $25,000).
(a) What is the production level of the company if it currently employs 100 workers and
has 200 units (5 million dollars) worth of equipment?
(b) Find a formula for the section P = f(N, 200) and use it to find fN (100, 200). What
are the units? What does this tell you about production at this company?
(c) Find a formula for the section P = f(100, V ) and use it to find fV (100, 200). What
are the units? What does this tell you about production at this company?
Problem 3. The quantity Q = f(b, c) of beef that a community buys each week (in pounds)
is a function of the price of beef b (in dollars per pound) and the price of chicken c (in dollars
per pound).
(a) Do you expect fb(b, c) to be positive or negative? Explain.
(b) Do you expect fc(b, c) to be positive or negative? Explain.
(c) Suppose
f(b, c) = 70, 000, fb(10, 4) = −4, 000, and fc(10, 4) = 2, 000
Estimate f(10.5, 3.75).
Problem 4. The table of values below shows the weekly sales S (in dollars) of a company
that spends $x per week on online advertising and $y per week on tv advertising.
x
500 1000 1500 2000
500 3980 5250 6180 6930
y 1000 6930 9150 10800 12100
1500 9590 12700 14900 16700
2000 12100 15900 18700 21000
(a) Estimate ∂S
∂x

(x,y)=(1000,1500)
. What does this tell you about the company’s sales? In
particular, would it make sense for the company to increase the amount it spends on
online advertising?
(b) Estimate ∂S
∂y

(x,y)=(1000,1500)
. What does this tell you about the company’s sales? In
particular, would it make sense for the company to increase the amount it spends on
tv advertising?
(c) Estimate the company’s sales when x = 900 and y = 1700.
Problem 5. Shown below is a contour graph of a function z = f(x, y).
(a) Estimate the value of fx(1, 2).
(b) Estimate the value of fy(1, 2).
(c) Use your answers to estimate the value of f(1.5, 3).
Problem 6. Consider the surface z = f(x, y) shown below. Find the signs of each of the
following. (Note: the x-axis is pointing into the paper to the right and the y-axis is pointing
into the paper to the left.)
(a) fx(−3, 3)
(b) fy(−3, 3)
(c) fxx(−3, 3)
(d) fyy(−3, 3)
−5
0
5
−5
0
5
−30
0
30
x
y
z