Download Solutions to Graded Problems Math 200 Homework 1 September 10

Solutions to Graded Problems Math 200
Homework 1
September 10, 2010
Section 1.1
For the following functions, find the domain and sketch the graph.
38. F (x) = |2x + 1|
Solution. Note that for any number x in the set of real numbers (this is denoted x ∈ R),
|2x + 1| is again a real number. Thus, the domain of F is R. F can be thought of as the
positive part of the line y = 2x + 1 with the negative part of the line reflected into the upper
half plane. The graph looks like this:
y
F
x
42. f (x) =
3 − 12 x if x ≤ 2
2x − 5 if x > 2
Solution. Note that if x ≤ 2 then f has a well defined image for x. The same is true for
x > 2. Thus, f is well defined for all real numbers, so the domain is R. Here is the graph:
y
f
x
1
Solutions to Graded Problems Math 200
Homework 1
September 10, 2010
Section 1.2
12. The manager of a weekend flea market knows from past experience that if he charges
x dollars for a rental space at the market, then the number y of spaces he can rent is given
by the equation y = 200 − 4x.
(a) Sketch a graph of this linear function. (Remember that the rental charge per space and
the number of spaces rented can’t be negative quantities.)
Solution.
y
200
50
x
(b) What do the slope, the y-intercept and the x-intercept of the graph represent?
Solution. The slope is rise over run or xy . For this function, that represents the decrease
in the amount of spaces the manager can expect to rent for every dollar increase in price.
The y-intercept is the number of spaces (200) rented if each space is free.
The x-intercept is the price ($50) at which no one will rent a space from the manager.
Section 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 radius, find V ◦ r and interpret
it.
Solution. Since the radius increases by 2 cm each second, we can write r(t) = r0 + 2t.
This function shows that for each second, 2 cm is added to whatever the balloon’s initial
radius was.
(b) If V is the volume of the balloon as as function of the radius, find V ◦ r and interpret it.
Solution. Recall that the volume of a sphere is given by 34 πr3 . We may substitute r =
r0 + 2t into V to obtain
4
(V ◦ r)(t) = π(2t)3 .
3
2
Solutions to Graded Problems Math 200
Homework 1
September 10, 2010
We have thus written V as a function of t. That is, we have an expression for how the
sphere’s volume changes in time.
3