MATH 190 Final: MATH 190 Louisville Practice Final Solution
MATH 190–03 Practice Final Exam Name:
1. (10 points) Chrome Koran’s latest album, made available for purchase online, would be bought
by 4500 people at a price point of $5. Fan polling suggests that every increase in the price by
$1 would reduce the number of downloads by 500.
(a) (3 points) Find a function describing the demand for the album as a function of price.
Since demand scales directly with the price, demand is a linear function of price. If our
demand function is D(x) = mx +b, then the fact that an increase of 1 in xcorresponds
with a decrease of 500 in D(x) signifies that m=−500, so D(x) = −500x+b. Since
D(5) = 4500, we may solve for bin the equation −500 ·5 + b= 4500 to get b= 7000, so
D(x) = 7000 −500x.
(b) (3 points) Find a function describing the total revenue from album sales as a function
of price.
At a price point x, the band will sell D(x) albums, gaining revenue of xon each sale, for
total revenue of x·D(x). Thus the revenue function is R(x) = xD(x) = 7000x−500x2.
(c) (4 points) Find a sale price for the album which maximizes revenue, and the total
revenue earned at this price. Label which is which.
The function R(x) = 7000x−500x2is a quadratic with negative quadratice coefficient; it
is thus maximized at its vertex, whose x-coordinate is −7000
2·−500 = 7 and whose y-coordinate
is R(x) = 7000 ·7−500·72= 24500, so they can earn a total revenue of $24500 by selling
their album at $7 per copy.
2. (14 points) Answer the following questions about the polynomial function f(x) = 2x3+ 5x2+
x−2.
(a) (3 points) What is the average rate of change of this function between the values x= 0
and x= 2?
The average rate of change is f(2)−f(0)
2−0=36−(−2)
2= 19.
(b) (3 points) What are all the potential rational zeroes of this function?
By the Rational Root Theorem, the possible zeroes are positive or negative fractions
whose numerators are divisors of 2 and whose denominators are also divisors of 2. The
possibilities are thus ±1
2,±1, and ±2.
(c) (4 points) Factor the polynomial into linear terms.
We may perform several synthetic divisions by the results in the previous question; one
which is successful is the division by (x+ 2):
−22 5 1 −2
−4−2 2
2 1 −1 0
so f(x) = (x+ 2)(2x2+x−1). The quadratic can be either factored or solved using the
quadratic formula to find that f(x) = (x+ 2)(2x−1)(x+ 1).
(d) (4 points) What are the x-intercepts, y-intercept, and long term behavior of the func-
tion? Label which is which.
The x-intercepts ca be calculated from the previous question to be x=−2, x=1
2,
and x=−1. The y-intercept is at f(0) = −2, and the long-term behavior, since this
is a polynomial with leading term of odd degree and positive coefficient, is that as x
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MATH 190–03 Practice Final Exam Name:
becomes large in the positive and negative directions, f(x) will likewise become large in
the positive and negative directions respectively.
3. (6 points) Calculate the following trigonometric expressions.
(a) (2 points) arcsin √3
2.
Since sin π
3=√3
2and −π
2≤π
3≤π
2, it follows that arcsin √3
2=π
3.
(b) (2 points) csc 17π
6.
Since 17π
6describes a point with reference number π
6in the second quadrant, its sine is
1
2, so its cosecant is the reciprocal of 1
2, which is 2.
(c) (2 points) tan −5π
3.
Since −5π
3describes a point with reference number π
3in the first quadrant, its tangent is
√3.
4. (10 points) Answer the following questions about logarithms.
(a) (3 points) Find a value of xsuch that 3·22x−1+ 5 = 53.
We can algebraically isolate xstep-by-step:
3·22x−1+ 5 = 53
3·22x−1= 48
22x−1= 16
2x−1 = log216 = 4
2x= 5
x=5
2
(b) (3 points) Calculate the value of the expression log5140 + log52
7−3 log510 exactly.
Using known logarithm laws:
log5140+log5
2
7−3 log510 = log5140+log5
2
7−log5(10)3= log5
140 ·2
7·1000 = log5125 = −2.
(c) (4 points) Calculate the folowing logarithms exactly, giving numerical answers:
•log48.
Since 8 = (√4)3, we know that 8 = 42/3. Thus log48 = 2
3.
•log749.
Since 49 = 72, log749 = 2.
•log31
27 .
1
27 =1
33= 3−3, and so log31
27 =−3.
•log55.
5 = 51, so log55 = 1.
5. (10 points) Answer the following trigonometric questions.
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Document Summary
Name: (10 points) chrome koran"s latest album, made available for purchase online, would be bought by 4500 people at a price point of . Fan polling suggests that every increase in the price by. would reduce the number of downloads by 500. (a) (3 points) find a function describing the demand for the album as a function of price. Since demand scales directly with the price, demand is a linear function of price. If our demand function is d(x) = mx + b, then the fact that an increase of 1 in x corresponds with a decrease of 500 in d(x) signi es that m = 500, so d(x) = 500x + b. D(5) = 4500, we may solve for b in the equation 500 5 + b = 4500 to get b = 7000, so. D(x) = 7000 500x. (b) (3 points) find a function describing the total revenue from album sales as a function of price.
