M408C: Review and Functions
September 2, 2008
1.
2.
3.
1
4. (1.1.8) A taxi company charges two dollars for the first mile (or part of a mile) and 20 cents
for each succeeding tenth of a mile (or part). Express the cost C (in dollars) of a ride as a function
of the distance x traveled (in miles) for 0 < x < 2, and sketch the graph of this function.
Solution:
C(x) =
2
0<x≤1
2 + .2 · 10 · (x − 1) 1 ≤ x < 2
Simplifying, we get
C(x) =
2 0<x≤1
2x 1 ≤ x < 2
The graph of this function is a piecewise continuous function that is constant at 2 for x between 0
and 1 and is a line with slope 2 for x between 1 and 2 connecting (1, 1) with (2, 4).
5. (1.2.10) Recent studies indicate that the average surface temperature of the earth has been
rising steadily. Some scientists have modeled the temperature by the linear function T = 0.02t+8.50
where T is ◦ C and t represents years since 1900, i.e. 8.4◦ C.
(a) What do the slope and T -intercept represent?
(b) Use the equation to predict the average global surface temperature in 2100.
Solution: (a) The slope represents the amount temperature increases as time increases, i.e. for
every 1 year, the temperature goes up by 0.02◦ C. The T -intercept represents the temperature (in
Celcius) in 1900.
(b) T = 0.02 · 200 + 8.50 = 4 + 8.50 = 12.50◦ C
6. (1.2.14) Jason leaves Detroit at 2:00pm and drives at a constant speed west along I-96. He
passes Ann Arbor, 40 mi from Detroit at 2:50pm.
(a) Express the distance traveled in terms of the time elapsed.
(b) Draw the graph of the equation in part (a).
(c) What is the slope of this line? What does it represent?
Solution: (a) One way to express the distance Jason travels in terms of the time elapsed is to
multiply his speed by the time that has elapsed. We determine his speed by noting that he traveled
40 mi in 50 minutes, thus his speed is 40/50 = 4/5 miles per minute. Thus d(t) = 4/5 ∗ t where t
is the time elapsed and d is the distance traveled.
(b) The graph is a line with slope 4/5 and d-intercept 0.
(c) The slope is 4/5 miles per minute, which means that for every minute Jason drives, he
travels 0.8 of a mile.
7. (1.3.22) Graph y = 41 tan x − π4 by starting with a graph of one of the standard functions
given in Section 1.2 and applying the appropriate transformations, not by plotting points.
Solution: Take the graph of tan(x), translate it it to the left by π/4 to get the graph of tan(x− π4 ),
and then scale it by 1/4 to get the final graph.
8. (1.3.54) A spherical balloon is being inflated and the radius of the balloon is increasing at a
rate of 2 cm/s.
(a) Express the radius r of the balloon as a function of the time t (in seconds).
(b) If V is the volume of the balloon as a function of the radius, find V ◦ r and interpret it.
Solution: (a) r(t) = 2t where t is the time that has elapsed in seconds and r is the radius in
centimeters. One way to see this is to do unit analysis: we look at the units of the numbers we have
2
and try to determine the formula we need. We are given the rate of inflation, 2 cm/s. Consider
just the unit cm/s. If we multiply this unit by seconds, we get
cm
· s = cm,
s
which suggests the formula we want to consider is r(t) = 2t.
(b) Recall the volume of a sphere: V (r) = 34 πr3 , where r is the radius of the sphere. We then
have
4
32
4
(V ◦ r)(t) = πr(t)3 = π · (2t)3 = πt3 .
3
3
3
Thus for every second that elapses, the volume increases by
32
3 π
cm which is about 33cm.
9. (1.3.63-66) Fill in the following table:
f +g
fg
f ◦g
f, g even
even
even
even
f even, g odd
?
odd
even
f odd, g even
?
odd
even
f, g odd
odd
even
odd
f even
?
?
?
f odd
?
?
?
g even
?
?
?
g odd
?
?
?
? denotes that the table entry cannot be determined to be odd or even.
3