MATH 205 Quiz: MATH 205 Louisville Quiz 1 110902 Solutions
MATH 205–02H Quiz #1 Solutions
Any answers which require logarithms to be expressed should be put in terms of natural or common
logarithms. Show all work.
1. (3 points) If f(x) = 2x2−3x, simplify the expression f(a+h)−f(a).
f(a+h)−f(a) = 2(a+h)2−3(a+h)−2a2−3a
= (2a2+ 4ah + 2h2−3a−3h)−(2a2−3a)
= 4ah + 2h2−3h
2. (3 points) Find the equation of the line through the points (−2,8) and (2,2).
The slope of this line is given by m=2−8
2−(−2) =−6
4=−3
2. Thus the equation of the line is
y=−3
2x+bfor some b. Plugging in either of the known points on the line lets us solve for b:
8 = −3
2(−2) + b
8 = 3 + b
5 = b
so the resulting equation is y=−3
2x+ 5.
3. (6 points) Identify the domains of the following functions:
(a) (3 points) g(t) = √3−t−√2 + t.
We have two square roots appearing in this function, so both of them must have non-
negative arguments in order for the function to be evaluatable. Thus, it is necessary that
3−t≥0 and 2 + t≥0; from these we respectively derive the conditions t≤3 and t≥ −2,
so −2≤t≤3, or, in interval form, tmust lie in [−2,3].
(b) (3 points) f(x) = 2x3−5
x2+x−6.
This function contains a fraction, so the denominator must be nonzero; we thus have the
condition x2+x−66= 0. Using the quadratic formula, it thus follows that xis not equal
to either value of −1±√1+24
2=−3,2. Thus x6=−3 and x6= 2, or, in interval form, xis in
(−∞,−3) ∪(−3,2) ∪(2,∞).
4. (8 points) There are currently 2000 inhabitants of the luckless town of Dunwich, MA, and its
population decreases by 7% each year due to emigration, death, and mysterious disappearances.
(a) (4 points) Construct a function f(t)to describe the population of Dunwich in tyears.
Since the population each year becomes 93% of its former value, after one year the popu-
lation would be 2000 ·0.93, after two years it would be 2000·0.93 ·0.93, and so forth. Thus
after tyears it has gone through this depletion procedure ttimes, resulting in a population
of f(t) = 2000(0.93)t.
(b) (4 points) The remaining denizens of the town will abandon it when there are only 100
people left. When will this occur?
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Document Summary
Any answers which require logarithms to be expressed should be put in terms of natural or common logarithms. = (2a2 + 4ah + 2h2 3a 3h) (2a2 3a) = 4ah + 2h2 3h: (3 points) find the equation of the line through the points ( 2, 8) and (2, 2). The slope of this line is given by m = 2 8 y = 3. Plugging in either of the known points on the line lets us solve for b: 5 = b so the resulting equation is y = 3. 2 x + 5: (6 points) identify the domains of the following functions: (a) (3 points) g(t) = 3 t 2 + t. We have two square roots appearing in this function, so both of them must have non- negative arguments in order for the function to be evaluatable.
