HW #1
Covers material on §1.1–3
Do the following problems. Please make sure you write out the solutions neatly. Please show all your
work. If you only write the answer with no work you will not be given any credit. Also, do not forget to
• Write your name and your recitation section number on top and
• Staple your homework if you have multiple pages!!!
Set1-1
(1) 1Section 1.1 Problem 21
Solution:
(2) Set1-1
Section 1.1 Problem 30 (hint: the graph of y =
Solution:
√
4 − x2 is a very familiar geometric object)
!
(3) Section 1.1 Problem 51
main Sheet: 1 Page: 1 (January 18, 2012 13 : 42) [1-miscw58-main]
Solution:
1
opped from a window and falls to the ground below. The height, s (in meters), of the rock above
function of the time, t (in
since
the rock
wasadropped,
s =falls
f (t).to the ground below. The height, s (in meters), of the
(4)seconds),
A rock is
dropped
from
windowsoand
rock
above
ground
is
a
function
of
the
time,
t (in seconds), since the rock was dropped, so s = f (t).
a possible graph of s as a function of t.
(a)
Sketch
a
possible
graph
of
s
as
a
function
of t.
what the statement f (7) = 12 tells us about the rock’s fall.
(b)
Explain
what
the
statement
f
(7)
=
12
would
us about
ph drawn as the answer for part (a) should have a horizontal and vertical intercept. tell
Interpret
each the rock’s fall.
(c)
The
graph
drawn
as
the
answer
for
part
(a)
should
have an x-intercept and a y-intercept.
t in terms of the rock’s fall.
Interpret each intercept in terms of the rock’s fall.
ER:
Solution:
ght of the rock decreases as(a)
timeThe
passes,
so the
graph
fallsdecreases
as you move
from passes,
left to right.
Onegraph falls as you move from left to
height
of the
rock
as time
so the
ity is shown in Figure ??.
right. One possibility is shown in Figure below.
s (meters)
1-miscw58ans
t (sec)
!
Figure 1
(b) The statement f (7) = 12 tells us that 7 seconds after the rock is dropped, it is 12 meters above
the ground.
tement f (7) = 12 tells us that 7 seconds after the rock is dropped, it is 12
1 meters above the
tical intercept is the value of s when t = 0; that is, the height from which the rock is dropped. The
tal intercept is the value of t when s = 0; that is, the time it takes for the rock to hit the ground.
T ANSWER:
s (meters)
!
(c) The vertical intercept is the value of s when t = 0; that is, the height from which the rock is
dropped. The horizontal intercept is the value of t when s = 0; that is, the time it takes for the
rock to hit the ground.
(5) Set1-2
Section 1.2 Problem 58
Solution:
!
HW1 Solutions
(6) Section 1.2 Problem 61
Solution:
(7) Residents of the town of Maple Grove who are connected to the municipal water supply are billed a
fixed amount monthly plus a charge for each cubic foot of water used.
(therefore, the bill is a linear function of the number of cubic feet used!)
A household using 1000 cubic feet was billed $40, while one using 1600 cubic feet was billed $55.
(a) What is the charge per cubic foot?
(b) Write an equation for the total cost of a resident’s water as a function of cubic feet of water
used.
(c) How many cubic feet of water used would lead to a bill of $100?
Solution:
55−40
(a) Charge per cubic foot = 1600−1000
= $0.025/cubic ft.
(b) The equation is c = b + 0.025w, so 40 = b + 0.025(1000), which yields b = 15. Thus the equation
is c = 15 + 0.025w.
(c) We need to solve the equation 100 = 15 + 0.025w, which yields w = 3400. It costs $100 to use
3400 cubic feet of water
Solution:
Page 1
2
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0.333345
0.333340
0.333335
0.333330
0.333325
-0.010
0.005
-0.005
0.010
(8) Section 1.3 Problem 22 on page 34.
(Graphs are to be sketched on your HW by hand - based on what you see on your calculator or
graphing software like www.wolframalpha.com. Do not attach printouts of graphs.)
Page 1
(9) Let H(x) be the Heaviside function (defined on p. 29, example 6 of the text). In your own words,
explain why
lim H(x)
x→0
does not exist according to definition 1 on p. 25.
Solution:
We show there is no real number L such that
lim H(x) = L.
x→0
If L 6= 0, then for all x < 0, H(x) = 0 and thus cannot get within |L|
2 of L and thus not arbitrarily close.
If L = 0, then for all x > 0, H(x) = 1 and thus the values are never within 12 of L and thus not arbitrarily
close.
Thus, the limit does not exist.
3