Name: Student ID: Date:
Class Test # 2 – 201-015-RE – Winter 2020
Unless otherwise specified, you should ALWAYS justify your answers.
PART ONE: You only need to answer THREE of the following FOUR questions:
(2.5 points) 1. Recall that all exponential functions have the form
f(x) = acb(x−h) + k while all logarithmic functions have the form
g(x) = a logc
(b(x − h)) + k, and graph the following functions. For each, also
determine the equation of its asymptote and its x and y-intercepts, whenever
applicable:
(a) p(x) = −2(2x−4
) + 1 (b) q(x) = ( 1
4
)
3x+3 − 5
(c) u(x) = log10 (x + 3) + 9 (d) u(x) = − ln (−x)
(2.5 points) 2. (a) Find either the equation or the graph of the inverse of y = 102x−3 + 4 :
(b) Find either the equation or the graph of the inverse of y = ln (x − 1) + 2 :
(c) If F(x) = e
x and G(x) = ln (x), determine (F ◦ G)(x) and (G ◦ F)(x) :
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(2.5 points) 3. Suppose you invest $2000 at a yearly interest rate of 3% over the
course of 10 years:
(a) Approximately how much money will you earn if interest is compounded yearly?
(b) How much will you earn if interest is compounded every six months?
(c) How much will you earn if interest is compounded continuously?
(d) If interest is compounded continuously, determine after how many years you will have
earned exactly $3000:
(2.5 points) 4. For each of the following properties, determine which ones are
FALSE and show why they are false (you may use counterexamples):
(a) log (a + b) = log (a) + log (b)
(b) logc
(x) = log (x)
log (c)
(c) log (0) = 0
(d) log (a) log (b) = log (ab)
(e) log (x
2
) = (log (x))2
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PART TWO: You only need to answer FOUR of the following FIVE questions:
(2 points) 5. At the start of 1999, Zimbabwe’s yearly inflation rate was 48% –
meaning that the price of selected goods would increase by 48% after a year
(for example: if an item cost $1 then, it would cost $1.48 one year later)
(a) At this inflation rate, if an item costs P dollars, find an equation for its price after t
years:
(b) Suppose a loaf of bread cost 3 Zimbabwean dollars (ZWD) in 1999. Assuming the rate
of inflation did not get worse (though in reality it did) or better, approximately what
would be the cost of that same loaf of bread in 2020?
(c) At this rate, after how many years would the cost of a loaf of bread be exactly 3 million
Zimbabwean dollars?
(d) How long would it take for a $10 item whose price is increasing at an inflation rate of
12% to reach the exact same price as a $3 item increasing at a rate of 48%?
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(2 points) 6. Suppose you take a 60mg dosage of medicine. After 24 hours, only
half of the medicine remains in your body; after another 24 hours, only half of
what remained (so half of half) stays in your body, and so on, meaning it has a
“half-life” of 24 hours:
(a) Find an equation for the amount of medicine (in milligrams) remaining in your body
after t days:
(b) How many milligrams of the drug are remaining in your system after two and a half
days?
(c) How much of the drug will remain in your system in the far future (for example, after
many, many years)?
(d) Most drugs lose their effectiveness when less than 10% of the initial dosage remains in
your system. Under this assumption, after exactly how many days will this medicine stop
working? (Answers can include fractional or decimal amounts of days)
(2 points) 7. Solve the following inequalities:
(a) 4x
2 + 16x + 7 < 0
4
(b) 16x
2 ≥ 25
(c) (x+2)(x−8)
(x−4)(x+6) > 0
(2 points) 8. Solve the following systems of equations:
(a) x + y = 13
x
2 + y
2 = 16
(b) 3x
2 − y
2 = −1
7x
2 = 4y
2
(c) Determine all points at which the line y = 6x + 9 intersects the parabola y = −x
2
:
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(2 points) 9. Suppose that the amount of digits in any number – even excessively
large numbers where simply counting would be impractical – can be found
using some method involving logarithms:
(a) Come up with a theory, procedure, etc. that would allow us to find the amount of
digits in any number. Feel free to demonstrate using simple examples:
(b) Suppose 282589922 − 1 is the largest known prime number (though there are infinitely
many prime numbers). Determine how many digits it has:
Equations of Some Functions/Other Useful Formulas:
A = P(1 + r
n
)
nt
A = P ert
h(x) = a logc
(b(x − h)) + k
j(x) = acb(x−h) + k
Quadratic Formula: −b±
√
b
2−4ac
2a
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