The University of Sydney
MATH1111 Introduction to Calculus
Semester 1
Week 6 Exercises (Thurs/Fri)
2017
Important Ideas and Useful Facts:
(i) Exponential functions: The simplest form of an exponential function f is
y = f (x) = ax
where a is some fixed positive real number. When a > 1 the graph of f is sloping upwards.
When a < 1 the graph of f is sloping downwards. When a = 1, the graph is the line
y = 1. In all cases the y-intercept is y = 1.
y
y
y = ax for 0 < a < 1
y = ax for a > 1
1
1
x
x
When a 6= 1, the domain of f is R, the range is (0, ∞), the graph satisfies the vertical
line test so f −1 exists (described next in (ii)), and the x-axis is a horizontal asymptote.
(ii) Logarithmic functions: Suppose a is positive and a 6= 1. The inverse of the function y = ax
is called the logarithmic function to the base a and denoted by
y = loga (x) .
The graph of the logarithm function is obtained by reflecting the graph of y = ax in the
line y = x. Its domain is (0, ∞), its range is R and the y-axis is a vertical asymptote.
y
y
y = loga x for 0 < a < 1
y = loga x for a > 1
1
x
x
1
In the special case that a = e, where e is Euler’s number, we write
y = loge x = ln x ,
and ln x is called the natural logarithm of x.
(iii) Logs and exponentials undo each other: For all x, all positive A and all positive a 6= 1,
loga (ax ) = x ,
ln ex = x ,
1
aloga A = A ,
eln A = A .
(iv) Exponential growth and decay: A useful general form of an exponential function f is
y = f (x) = Aeλx
where A and λ are nonzero constants. In applications, A is typically positive and x
represents time, in which case f models exponential growth when λ > 0, and exponential
decay when λ < 0.
(v) Exponential laws: For all x and y, and for all positive a and b,
ax ay = ax+y ,
ax /ay = ax−y ,
(ax )z = axz ,
(ab)x = ax bx .
(vi) Logarithmic laws: For all positive x and y, for all positive a 6= 1, and for all real k,
(a)
(b)
(c)
loga (xy) = loga x + loga y , loga (x/y) = loga x − loga y , loga (xk ) = k loga x .
ln(xy) = ln x + ln y , ln(x/y) = ln x − ln y , ln(xk ) = k ln x .
ln x
loga x =
.
ln a
Tutorial Exercises:
1.
Use your calculator to find the following, correct to 3 decimal places:
(i) 22.5 , 32.5 , 21.5 and 62.5 .
(ii) 22.5 × 21.5 ,
22.5
21.5
,
22.5 ×32.5
62.5
and
22.5√
×32.5
6
.
Can you give simple explanations why the answers to (ii) should become whole numbers?
2.
You are asked to draw the graph of y = 2x for 0 ≤ x ≤ 40 with a scale of 1 unit per
millimetre. How long does your piece of paper need to be? Compare this length to the
distance from the earth to the moon (which is about 400, 000 kilometres).
3.
Which is larger, log4 17 or log5 24? Explain, without using a calculator. Now use your
calculator to find each of these to 3 decimal places.
4.
Solve the following for x, giving expressions for exact answers and also final answers
correct to 3 significant figures:
(i) 3x = 17
(ii) 20 = 50(1.04)x
∗
(iii) 2x = ex+1
∗
(iv) 3e2x = 5e4x
∗
5.
An animal skull still has 20% of the carbon-14 that was present when the animal died.
The half-life of carbon-14 is 5730 years. Estimate the age of the skull to the nearest
thousand years.
∗
6.
You are offered a very well-paid job lasting one month (31 days), and can accept payment
in one of two ways:
(i) You receive ten million dollars at the end of the month.
(ii) After the first day you receive one cent. After the second day you receive two cents.
After the third day you receive four cents. After the fourth day you receive eight
cents. Each day, until the 31st day you receive double the amount of cents you
received the previous day.
Which of (i) or (ii) is the most lucrative option?
2
Further Exercises:
7.
Use logarithmic laws to derive the following equations:
(i) ln(1/2) = − ln 2
8.
(ii) ln(2e) = 1 + ln 2
(iii) ln 100 = 2(ln 5 + ln 2)
Use your calculator and the formula
S = 0.007184W 0.425H 0.725 ,
where S is the body surface area in m2 , W is the body mass in kg, H is the body height
in cm, to find the surface area (skin area) of a person 150 cm tall, mass 76 kg, quoting
your answer to 2 significant figures.
∗
9.
In 2010 the Australian population was approximately 21, 374, 000 and the growth rate
estimated at about 1.69% per annum.
(i) Assuming exponential growth, find a formula linking the Australian population P
to the number t of years since 2010.
(ii) Use your formula to make a forward estimate of P in 2050 to the nearest million.
(iii) Use this formula to make a backward estimate of P in 1900 to the nearest million.
∗
10.
Caffeine is eliminated from the body at a continuous rate of approximately 17% per hour.
A standard cup of coffee contains 150 mg of caffeine.
(i) Write a formula for the amount of caffeine C in mg as a function of the number t of
hours after drinking a cup of coffee.
(ii) Find the number of hours (to the nearest hour) it takes for half of the caffeine from
a cup of coffee to be eliminated from the body.
(iii) How many hours (to the nearest hour) does it take for only 1.5 mg of the caffeine
from a cup of coffee to remain in the body.
3
Short Answers to Selected Exercises:
1.
(i) 5.657, 15.588, 2.828, 88.182. (ii) 16.000, 2.000, 1.000, 36.000.
2.
Over two and a half times the distance to the moon.
3.
2.044, 1.975
4.
(i) 2.58 (ii) −23.4 (iii) −3.26 (iv) −0.255
5.
13, 000 years
6.
(ii) is more lucrative.
8.
1.7 m2
9.
(ii) 42 million (iii) 3 million
10.
(ii) 4 hours (iii) 25 hours
4