MATH 1106, Spring 2017
Week 1, tutorial 1 Solutions
Wednesday 25/01/17
PRELIMINARIES SOLUTIONS
Tutorial 1.1. Exponential functions (exercises from paragraph 2.4)
An expoential function is of the form f (x) = ax for some a > 0.
(a) What happens when a = 1?
(b) Sketch the functions f (x) = 2x and g(x) = 2−x
(c) Solve the following exponential equations:
2
(i) 2x = 32 (ii) (e3 )−2x = e−x+5 (iii) 5x +x = 1
2
(iv) 8x = 2x+4
Solution 1.1. (a) f is the constant function f (x) = 1.
(b) If you’re not sure, try this with a function plotter.
(c) (i) x = 5.
(ii)
e−6x · ex−5 = e−x+5 ex−5
e− 5x − 5 = e0 = 1
−5x − 5 = 0
x = −1
(d) This becomes x2 + x = 0, or equivalently x(x + 1) = 0. Hence
x = 0 or x = −1.
(e) Use the fact that 8 = 23 to reduce to the same base, then solve the
quadratic equation.
Tutorial 1.2. Interest
The simple interest I given principal (initial investment) P and rate of
interest r gained over time t (measured in years) is
I = P rt.
We could also add interest on each accumulated amount, not just on
the initial amount. If an initial deposit of P dollars is invested at a
yearly rate of interest r per year, compounded m times per year for t
years, the compound amount (or total amount on deposit) in dollars is
r
A = P (1 + )tm .
m
We could also compound interest continously. If an initial deposit of
P dollars is invested at a rate of interest r compounded continuously
1
2
PRELIMINARIES SOLUTIONS
for t years, the compound amount in dollars is
A = P ert .
(a) (Chap 2.4 Ex 37) Find the interest earned on $10,000 invested for
5 years at 4% interest compounded as follows:
(i) Annually (ii) Semiannually (iii) Quarterly (iv) Monthly
(v) Continuously
(b) (Chap 2.4 Ex 48) Salmonella bacteria grow rapidly in a warm
place. Suppose the number of bacteria present in a potato salad
at room temperature after t hours is given by
f (t) = 500 · 23t .
(i) If the salad is left unrefrigerated, how many bacteria are present
1 hour later? 1.5 hours later?
(ii) How many were present initially?
(iii) How often do the bacteria double?
(iv) How quickly will the number of bacteria increase to 32,000?
(c) (Chap 2.4 Ex 52) Suppose the quantity in grams of a radioactive
substance present at time t (in months) is
Q(t) = 1000(5−0.3t ).
(i) How much will be present in 6 months?
(ii) How long will it take to reduce the substance to 8g?
(d) * Can you derive the formula for compound interest from the
formula for simple interest?
Solution 1.2. (a) Simply replace P , r, t and m by the given values in
the expressions for compound interest, where m = 1, 2, 4 and 12
in questions (i) -(iv).
(b) (i) Replace t by 1 and 1.5 then compute.
(ii) Initially means t = 0, hence 500.
(iii) The amount of bacteria doubles each time 23t = 2, namely
each time t reaches a multiple of 1/3, or every 20 minutes.
(iv) This comes down to solving f (t) = 32000, namely 23t =
32000/500 = 64, thus in t = 2 hours.
(c) (i) Compute for t = 6. (ii) This comes down to solving the equation 1000 · 5−0.3t = 8 for t, or equivalently 5−0.3t = 0.008. Note that
0.008 = 5−3 . Therefore t = 10.
PRELIMINARIES SOLUTIONS
3
(d) Compute simple interest for the first year, then compute simple
interest for the amount obtained. Iterate this to derive the formula. This is briefly covered in your textbook, but you can ask
me in office hours if you’d like to go over this!
For a > 0, a 6= 1, and x > 0,
Tutorial 1.3. Logarithmic functions
y = loga x means
ay = x.
If a > 0 and a 6= 1, then the logarithmic function of base a is defined
by
f (x) = loga x
for x > 0.
(a) Why do we need to have x > 0?
(b) (Exercises from chap. 2.5) Evaluate (i),(ii) and (iii) without using
a calculator, and solve (iv) and (v)
q for x:
1
(i) log8 64 (ii)log3 81
(iii) log2 3 14 (iv) logx 8 = 34
(v) log2 (x2 − 1) − log2 (x + 1) = 2
(c) (Chap 2.5 Ex 75) Assuming annual compounding, find the time
it would take for the general level of prices in the economy to
double at the following annual inflation rates:
(i) 3% (ii) 6% (iii) 8%
(d) (Chap 2.5 Ex 80) You are offered two jobs starting July 1 2017.
Vegemite Enterprises offers you $45, 000 a year to start, with a
raise of 4% every July 1. At Koala Inc. you start at $30,000 with
an annual increase of 6% every July 1. How long would you have
to work at Koala Inc. before you earnt more than at Vegemite
Enterprises?
Note: you may find the following properties of logarithms useful:
• loga xy = loga x + loga y
• loga xy = loga x − loga y
• loga xr = rloga x
• loga a = 1 and loga 1 = 0
• loga ar = r.
Solution 1.3. (a) The equation ay = x for positive a tells us that for
all y, x must be positive.
(b) (i) Let y = log8 64. Then 8y = 64. Hence y = 2.
(ii) Solve as in (i).
q
(iii) Solve as in (i), noting that
3
1
4
2
= 2− 3 .
3
(iv) This is equivalent to the equation x 4 = 8. Thus, x = 16.
(v) Use the rule loga x − loga y = loga xy .
4
PRELIMINARIES SOLUTIONS
(c) Using the expression for compound interest, the price first douln2
bles when (1 + r)t = 2, or equivalently when t = ln(1+r)
. Substitute in the given values of r and compute.
(d) Translating the text, you want to solve the inequality
30000 · (1.06)t ≥ 45000 · (1.04)t
ln1.5
for t. This becomes t ≥ ln(1.06/1.04)
≈ 21.3. Unless you’re feeling particularly loyal to Koala Inc, maybe consider going to Vegemite Enterprises!