### Figure 5-1 (p. ) a

```Section 9.1
Understanding Functions of
Two Variables
Section 9.1: Understanding Functions of Two
Variables
Applied Calculus, 3/E by Deborah Hughes-Hallet
Example: Wind Chill
If the actual temperature is -10°C and the wind
speed is 30 km/h, then what is the wind chill?
How would you write this using notation?
Is wind chill an increasing or decreasing
function of actual temperature? Of wind speed?
Applied Calculus, 3/E by Deborah Hughes-Hallet
Example: Ticket Sales
Find a formula for the revenue as a function of
the number of discount and full price tickets.
Applied Calculus, 3/E by Deborah Hughes-Hallet
Example: Auto Loan
The total amount, T, of dollars a buyer pays for an automobile is a function
of the down payment, P, and the interest rate, r, of the loan on the balance.
If the interest rate is held constant, is T an increasing or decreasing function
of P?
If the down payment is held constant, is T an increasing or decreasing
function of r?
Applied Calculus, 3/E by Deborah Hughes-Hallet
Example: Visualizing
f  x, y   x  2 y
1.
2.
3.
4.
Increasing or decreasing?
Table of values
Visualizing
Contour Plots
Applied Calculus, 3/E by Deborah Hughes-Hallet
The number of songbirds, B, on an island depends on the
number of hawks, H, and the number of insects, I.
Thinking about the possible relationship between these
three populations, decide which of the following depicts it.
(I) B increases as H increases and I remains constant.
(II) B decreases as H increases and I remains constant.
(III) B increases as I increases and H remains constant.
(IV) B decreases as I increases and H remains constant.
(a) I and III
(b) I and IV
(c) II and III
(d) II and IV
(e) None of these
ConcepTest • Section 9.1 • Question 4
(c). Under the assumption that hawks eat
songbirds, (II) is true. The answer (c) assumes that
the songbirds eat the insects. Some songbirds eat
only seeds, so in that case, if the insects eat the
seeds, the correct answer is (d).
COMMENT:
You might ask for a reason as well.
ConcepTest • Section 9.1 • Answer 4
For a certain function z = f(x, y), we know
that f(0, 0) = 50 and that z goes up by 3 units
for every unit increase in x and z goes down
by 2 units for every unit increase in y. What
is f(2, 5)?
(a) 51
(b) 46
(c) 1
(d) 55
(e) −4
(f) 16
ConcepTest • Section 9.1 • Question 5
(b). At the point (0, 0), the function value is 50.
When the x-value goes up by 2, the function value
goes up by 3·2 = 6. When the y-value goes up by 5,
the function value goes down by 5·2 = 10. We have
f(2, 5) = 50 + 6 − 10 = 46.
ConcepTest • Section 9.1 • Answer 5
Which of the graphs (a)–(f) shows a crosssection of f(x, y) = 50 − x2 + 5y with y held
fixed?
ConcepTest • Section 9.1 • Question 6