6
Applying calculus
to further modelling
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Describing curve properties
Curve properties from calculus
Investigating new models
Identifying models from graphs
Modelling problems
Surge and terminal velocity models
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B
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D
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Contents:
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208
APPLYING CALCULUS TO FURTHER MODELLING
(Chapter 6)
In this chapter we will apply the calculus studied in Chapter 5 to various modelling questions. We
will extend the modelling work done in previous chapters, and consider models of the following
types:
9
² linear
=
already examined in previous work
² exponential
;
² power
¾
² quadratic
polynomial models
² cubic
² logarithmic
y = b + a ln t
² logistic
y=
² terminal velocity
y = A(1 ¡ e¡bt ), A > 0, b > 0
² surge
y = Ate¡bt , A > 0, b > 0
A
C
, A > 0, b > 0, C > 0
1 + Ae¡bt
DESCRIBING CURVE PROPERTIES
When we describe the graphical features of models, it is important to use language that we all
understand.
LOCAL AND GLOBAL MAXIMA AND MINIMA
A local maximum is a maximum turning point.
local maximum
A local minimum is a minimum turning point.
local minimum
The global maximum is the point with the maximum value of y on the domain of the function.
The global minimum is the point with the minimum value of y on the domain of the function.
global
maximum
y
local maximum
global maximum
(and local maximum)
local minimum
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global minimum
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global
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APPLYING CALCULUS TO FURTHER MODELLING
(Chapter 6)
209
ASYMPTOTES
y
This curve has a horizontal asymptote y = 1
and a vertical asymptote x = 3.
As y gets very large and positive, and as x gets
very large and negative, the graph gets closer
and closer to the asymptotes.
y=1
x
x=3
INCREASING AND DECREASING FUNCTIONS
Graphs like
are increasing for all x.
A graph is increasing on some interval if, as the x value increases, the y value increases also.
As we move from
left to right we are
going up the hill.
increase in y
increase in x
Graphs like
are decreasing for all x.
A graph is decreasing on some interval if, as the x value increases, the y value decreases.
decrease in y
As we move from
left to right we are
going down the hill.
increase in x
Some graphs contain intervals which are increasing, as well as intervals which are decreasing.
For example:
g
incr
easi
d
3
si n
rea
ec
incr
eas
ing
ng
y
y
(4,-1)
x
(3, 2)
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This function is decreasing for
0 6 x 6 4 and increasing for
x 6 0 and for x > 4.
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This function is decreasing for x 6 3
and increasing for x > 3.
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210
APPLYING CALCULUS TO FURTHER MODELLING
(Chapter 6)
Example 1
Find intervals where the following graphs are increasing or decreasing:
a
y
b
y
(4, 3)
x
x
(2, -1)
(8, -2)
b Decreasing for x 6 2.
Increasing for x > 2.
a Increasing for 0 6 x 6 4.
Decreasing for
x 6 0 and for 4 6 x 6 8.
EXERCISE 6A.1
1 For each of these graphs:
i give the position and nature of any turning points
ii state intervals where the graph is increasing or decreasing
iii state the equation and nature of any asymptotes.
y
a
y
b
y
c
(2, 3)
y
d
x
x
x
y
e
y
(Qw , 6_Qr_)
f
(0, 2)
x
x
x
(1, -1)
g
(-1, 2)
y
y
h
x
x
(2, -2)
(-4, -3)
y
i
y = 10
2
x
y = -2
2 Find the global minimum and global maximum for the following graphs:
a
b
y
y
c
(3, 17)
(4, 5)
(-5, 6)
(-4, 10)
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(2, 8)
x
(-1, 1)
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(-3, -4)
(4, -5)
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APPLYING CALCULUS TO FURTHER MODELLING
3
(Chapter 6)
211
Consider the illustrated line. For fixed x-steps, there are
fixed y-steps, so the slope of the line is constant.
Comment on the slopes of the following curves as the
x-value increases.
a
b
d
e
c
f
B
A
4 Use question 3 to answer the following:
a What is the slope of a curve:
i at a maximum turning point or minimum turning point
ii on an interval where the function is increasing
iii on an interval where the function is decreasing?
b What is happening to the slopes of the curves at:
i point A in 3d
ii point B in 3f ?
INFLECTIONS AND SHAPE
My cave looks like:
When a curve, or part of a curve, has shape:
we say that the shape is concave
we say that the shape is convex.
Consider this concave curve:
As x increases, the slope of the tangent
decreases.
m=0
m=1
m = -1
So, the rate of change is decreasing.
m=2
m = -2
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y = -x 2
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APPLYING CALCULUS TO FURTHER MODELLING
(Chapter 6)
Now consider this convex curve:
As x increases, the slope of the tangent
increases.
y = x2
m = -2
So, the rate of change is increasing.
m=2
m = -1
m=1
m=0
POINTS OF INFLECTION
A point of inflection is a point on a curve at which there is a change
of curvature or shape. At this point, the curve changes from concave
to convex, or vice versa.
DEMO
or
point of
inflection
point of
inflection
The tangent at the point of inflection, also called the
inflecting tangent, crosses the curve at that point.
EXERCISE 6A.2
1 For the following graphs, identify which of the points A, B, C or D is a point of inflection:
a
b
y
c
y
y
B
C
A
B
C
B
D
A
A
x
D
x
C
x
D
2 For the following functions, find:
i any points of inflection
ii intervals where the rate of change is increasing
iii intervals where the rate of change is decreasing.
a
b
c
y
y
y
y = 10
(-2, 11)
(1, 7)
(6, 5)
(4, 3)
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APPLYING CALCULUS TO FURTHER MODELLING
d
e
y
y
f
(0, 4)
(-1, 3)
213
(Chapter 6)
y
(8, 5)
(1, 3)
(5, 2)
x
x
x
(2, -1)
(2, -1)
3 What happens to the slope of the curve at a point of inflection?
Hint: Consider Exercise 6A.1 question 4 b again.
B
CURVE PROPERTIES FROM CALCULUS
So far we have described features of a curve by visual inspection.
However, we are often given the equation of a function, and we want to identify features of its
graph such as turning points, and intervals where the graph is increasing or decreasing.
Even if we draw the graph of the function, it is difficult to precisely locate its features.
In situations like this, we can use the calculus we learnt in the previous chapter to determine key
features of the graph.
dy
or f 0 (x), we can find the slope of the tangent to the curve at any
dx
value of x by substituting that value of x into it.
Given the slope function
For example:
If y = x3 ¡ 5x ¡ 2 then
Now, when x = 1,
y = x3 - 5x -2
y
dy
= 3x2 ¡ 5.
dx
dy
= 3(1)2 ¡ 5
dx
= ¡2
1
x
So, at the point where x = 1, the tangent has slope ¡2.
slope -2
TURNING POINTS
The tangents at the turning points are
horizontal and therefore have slope 0:
local maximum
A
At turning points,
B
local minimum
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When a function
is increasing, the
slope of the tangent
is positive.
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dy
= 0.
dx
When a function
is decreasing, the
slope of the tangent
is negative.
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APPLYING CALCULUS TO FURTHER MODELLING
(Chapter 6)
A function is increasing on an interval if
dy
> 0 for all x on that interval.
dx
A function is decreasing on an interval if
dy
6 0 for all x on that interval.
dx
POINTS OF INFLECTION
y = f(x)
Consider the graph of y = f(x) alongside, where
A is a point of inflection. The graph of its derivative
y = f 0 (x) is given below it.
A
Notice that:
² The turning points of f (x) correspond to the
x-intercepts of f 0 (x). This is because f 0 (x) = 0
at the turning points.
x
² The inflection point of y = f (x) corresponds to
the turning point of f 0 (x).
y = f'(x)
The slope of y = f(x)
is maximised at A, the
point of inflection.
So, to find the inflection points of y = f(x), we graph the derivative function y = f 0 (x), and
find the x-coordinates of any turning points. We can determine the y-coordinate of an inflection
point by substituting the x-coordinate into y = f (x).
Example 2
Consider the function y = x3 ¡ 3x2 + 2.
dy
dy
a Find
, and solve
= 0.
dx
dx
b Use technology to obtain a graph of the function.
c Find any turning points.
d Find any points of inflection.
e Find intervals where the function is:
i increasing
ii decreasing
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y = x3 ¡ 3x2 + 2
dy
)
= 3x2 ¡ 6x
dx
dy
So,
= 0 when
dx
3x2 ¡ 6x = 0
) 3x(x ¡ 2) = 0
) x = 0 or 2
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APPLYING CALCULUS TO FURTHER MODELLING
(Chapter 6)
215
c When x = 0, y = 03 ¡ 3(0)2 + 2 = 2
When x = 2, y = 23 ¡ 3(2)2 + 2 = ¡2
So, there is a local maximum at (0, 2) and a local minimum at (2, ¡2).
d
dy
= 3x2 ¡ 6x has the graph given alongside:
dx
dy
has a turning point at x = 1, so
dx
the function has an inflection point at
x = 1.
When x = 1, y = 13 ¡ 3(1)2 + 2 = 0,
so there is an inflection point at (1, 0).
We can use technology to
find turning points.
Consult the graphics
calculator instructions at
the front of the book.
e
i
ii
iii
iv
The
The
The
The
function is
function is
function is
function is
increasing for x 6 0 and x > 2.
decreasing for 0 6 x 6 2.
concave for x 6 1.
convex for x > 1.
EXERCISE 6B
1 For each of the following functions:
dy
dy
and solve
=0
dx
dx
ii use technology to obtain a graph of the function
iii find the turning points and points of inflection of the function
iv find intervals where the function is increasing, decreasing, concave, and convex.
i find
a y = x3 ¡ 3x + 1
b y = ¡2x2 + 2x ¡ 1
c y = 2x3 ¡ 12x2 + 3
d y = x3 ¡ 9x2 + 24x ¡ 15
p
g y= x
e y = 5e2x
f y = 2e¡x
h y = 8(1 ¡ e¡2x )
i y = 2 + ln x
2 Consider the function y = 2te¡4t for t > 0.
dy
= 2(1 ¡ 4t)e¡4t .
a Show that
dt
dy
b Explain why
= 0 only when t = 14 .
dt
c Use technology to graph the function.
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d Find any turning points and points of inflection.
e Find intervals where the function is:
i increasing
ii decreasing
iii concave
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APPLYING CALCULUS TO FURTHER MODELLING
(Chapter 6)
20
for t > 0.
1 + 3e¡2t
120
dy
.
= 2t
a Show that
dt
e (1 + 3e¡2t )2
dy
> 0 for all values of t > 0.
b Explain why
dt
c Use technology to graph the function. What is the significance of b to the graph?
3 Consider the function y =
C
INVESTIGATING NEW MODELS
Since linear, exponential and power models have been examined in an earlier chapter, and quadratic
and cubic models have been examined in Stage 1, we will now investigate the remaining four
models. They are the logarithmic, surge, terminal velocity, and logistic models.
INVESTIGATION 1
THE LOGARITHMIC MODEL
The logarithmic model has the form y = b + a ln t where a
and b are constants. In this investigation we examine this family
of curves as the values of a and b change.
GRAPHING
PACKAGE
The use of the graphing package is recommended. Click on the icon to open this package.
A graphics calculator is also appropriate. It is desirable for students to experience both
technologies.
What to do:
1 Let a = 2, so y = b + 2 ln t.
Graph the following functions:
a y = 5 + 2 ln t
b y = 3 + 2 ln t
c y = 1 + 2 ln t
d y = 0:5 + 2 ln t
Write down your observations, including any turning points, asymptotes, intervals where
the function is increasing or decreasing, and intervals where the function is convex or
concave.
2 Let b = 2, so y = 2 + a ln t.
Graph the following functions:
a y = 2 + 5 ln t
b y = 2 + 3 ln t
c y = 2 + ln t
d y = 2 ¡ ln t
e y = 2 ¡ 3 ln t
f y = 2 ¡ 5 ln t
Write down your observations.
INVESTIGATION 2
THE SURGE MODEL
Ate¡bt
where A and b are
A surge model has the form y =
positive constants. The independent variable t is usually time,
t > 0.
GRAPHING
PACKAGE
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This model is used extensively in the study of medicinal doses
where there is an initial rapid increase to a maximum, and then
a slower decay to zero.
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APPLYING CALCULUS TO FURTHER MODELLING
217
(Chapter 6)
What to do:
1 Let b = 4, so y = Ate¡4t . Graph the following functions:
a y = 10te¡4t
b y = 20te¡4t
c y = 40te¡4t
Write down your observations.
2 Let A = 20, so y = 20te¡bt . Graph the following functions:
a y = 20te¡t
b y = 20te¡2t
c y = 20te¡3t
d y = 20te¡5t
Write down your observations.
INVESTIGATION 3
THE TERMINAL VELOCITY MODEL
A terminal velocity model has the form y = A(1¡e¡bt ) where
A and b are positive constants. The independent variable t is
usually time, t > 0.
GRAPHING
PACKAGE
This model is often used when an object is falling under the
influence of gravity and approaches a limiting velocity.
What to do:
1 Let b = 2, so y = A(1 ¡ e¡2t ). Graph the following functions:
a y = 100(1 ¡ e¡2t )
b y = 60(1 ¡ e¡2t )
c y = 30(1 ¡ e¡2t )
d y = 10(1 ¡ e¡2t )
Write down your observations.
2 Let A = 100, so y = 100(1 ¡ e¡bt ). Graph the following functions:
a y = 100(1 ¡ e¡5t )
b y = 100(1 ¡ e¡2t )
c y = 100(1 ¡ e¡t )
d y = 100(1 ¡ e¡0:5t )
Write down your observations.
INVESTIGATION 4
THE LOGISTIC MODEL
A logistic model has the form y =
C
where A, b and
1 + Ae¡bt
C are positive constants.
GRAPHING
PACKAGE
The independent variable t is usually time, t > 0.
The logistic model is useful in limited growth problems, where growth cannot
go beyond a particular value due to predators or lack of resources.
What to do:
C
.
1 + 2e¡2t
Graph this function for:
a C = 30
Write down your observations.
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1 Let A = b = 2, so y =
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APPLYING CALCULUS TO FURTHER MODELLING
(Chapter 6)
30
.
1 + Ae¡2t
Graph this function for:
a A=1
b A=3
Write down your observations.
30
.
3 Let A = 2, C = 30, so y =
1 + 2e¡bt
Graph this function for:
a b=1
b b=2
Write down your observations.
2 Let b = 2, C = 30, so y =
c A=5
d A = 10
c b=3
d b = 10
From these investigations, we can make the following observations:
Logarithmic Model y = b + a ln t, a and b constants, t > 0.
y
y
t
t
a>0
a<0
² The y-axis is a vertical asymptote.
² There are no turning points or inflection points.
² The function is either always increasing (a > 0) or always decreasing (a < 0).
Surge Model y = Ate¡bt , A and b positive constants, t > 0.
² The t-axis is a horizontal asymptote.
² The function passes through the origin.
² There is a maximum turning point when
1
t= .
b
2
² There is an inflection point when t = .
b
y
Qx
Wx
t
Terminal Velocity Model y = A(1 ¡ e¡bt ), A and b positive constants, t > 0.
² y = A is a horizontal asymptote.
² The function passes through the origin.
² There are no turning points or inflection
points.
² The function is always increasing, and
the rate of change is always decreasing.
Logistic Model y =
y
A
y=A
t
C
, A, b, C positive constants, t > 0.
1 + Ae¡bt
y
² y = C is a horizontal asymptote.
² The function is always increasing.
² There is an inflection point when
C
y= .
2
y=C
C
_w
C
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APPLYING CALCULUS TO FURTHER MODELLING
219
(Chapter 6)
Example 3
A swine flu patient is prescribed a course of an antivirus drug. The amount of drug
in the bloodstream t hours after the first dose is taken, is given by the surge model
A(t) = 100t £ e¡2t mg.
a Sketch the graph of the function for 0 6 t 6 4.
b Show that A0 (t) = 100e¡2t (1 ¡ 2t).
c Find the value of t for which A0 (t) = 0, and determine the maximum amount of drug
in the system.
d Explain what is happening to the amount of drug in the bloodstream over time.
e Draw a graph of A0 (t) = 100e¡2t (1 ¡ 2t).
f Find A0 (0:25) and interpret your answer.
g Find the t-coordinate of the point of inflection, and interpret your answer.
a
20
b
A
A(t) = 100te
_ -2t
15
A(t) = 100te¡2t is the product of
u(t) = 100t and v(t) = e¡2t
) u0 (t) = 100 and v 0 (t) = ¡2e¡2t
) A0 (t) = u0 (t) v(t) + u(t) v 0 (t)
10
fproduct ruleg
= 100e¡2t + 100t(¡2e¡2t )
5
1
3
2
t
4
c A0 (t) = 0 when 1 ¡ 2t = 0
= 100e¡2t (1 ¡ 2t)
fe¡2t 6= 0g
) t = 12
¡
¢
1
Now A( 12 ) = 100 12 £ e¡2( 2 ) ¼ 18:4
So, the maximum amount of drug in the system is 18:4 mg, after half an hour.
d The amount of drug in the system
e
A'(t)
rapidly increases for the first half
100
hour until it reaches a maximum of
80
18:4 mg. It then decreases quickly
at first, then slowly as it approaches
60
A'(t) = 100e
_ -2t(1 - 2t)
zero.
40
20
1
2
3
4
t
f A0 (0:25) = 100 £ e¡2(0:25) (1 ¡ 2(0:25))
-20
¼ 30:3
After a quarter of an hour, the amount of drug in the system is increasing at a rate of
30:3 mg per hour.
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g From e, A0 (t) has a turning point when t = 1.
) A(t) has a point of inflection at t = 1.
After 1 hour, the rate of decrease of the drug from the body is maximised.
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APPLYING CALCULUS TO FURTHER MODELLING
(Chapter 6)
EXERCISE 6C
1 When a new pain killing injection is administered,
the effect is modelled by E = 750te¡1:5t units,
where t > 0 is the time in hours after the injection
of the drug.
a Sketch the graph of E against t.
b What type of model is this?
c What is the effect of the drug after:
i 30 minutes
ii 2 hours?
dE
= 375e¡1:5t (2 ¡ 3t), and hence determine when the drug is most
d Show that
dt
effective.
e In the operating period, the effective level of the drug must be at least 100 units.
i When can the operation commence?
ii How long has the surgeon to complete the operation if no further injection is
possible?
f Find t at the point of inflection of the graph. What is the significance of this point?
2 The number of ants in a colony after t months is modelled by A(t) =
25 000
.
1 + 4e¡t
a Draw a sketch of the function.
b Identify the model type.
c What is the initial ant population?
d What is the ant population after 3 months?
e Is there a limit to the population size? If so, what is it?
f At what time does the population size reach 24 500?
g
100 000e¡t
.
(1 + 4e¡t )2
ii Find the time at which the growth rate of the ant population is maximised.
iii Find the ant population at this time.
i Show that A0 (t) =
3 Alexa has a headache and takes 1000 mg of paracetamol. The concentration of this drug in
her system after t hours is modelled by the function P (t) = 150t £ 0:3t parts per million.
a Use technology to copy and complete the table of values below:
0
0
t
P (t)
0:5
41:1
1
1:5
2
2:5
3
3:5
4
4:5
5
b Sketch the graph of y = P (t) for 0 6 t 6 5.
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c Describe how the concentration of paracetamol changes over the 5 hours.
d For the drug to be effective, it is necessary for the concentration of paracetamol to be
above 15 parts per million. Find the interval of time for which the drug is effective.
e Find the time at which the concentration of paracetamol is at a maximum, and determine
the concentration at this time.
f Find the rate of change of the paracetamol concentration after 1 hour.
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APPLYING CALCULUS TO FURTHER MODELLING
(Chapter 6)
221
4 The speed of descent of a skydiver t seconds after jumping from a plane, is given by
v(t) = 180(1 ¡ e¡0:2t ) m s¡1 .
a Sketch the graph of y = v(t).
b What type of model is this?
c Describe the velocity of the skydiver over time.
d Find the velocity after:
i 1 second
ii 3 seconds.
e Find the average rate of change of the velocity over the time interval t = 1 to t = 3.
f Find v 0 (t).
g Find v 0 (3) and interpret your answer.
h Find the limiting velocity of the skydiver.
5 During a storm, the volume of water collected in a tank after t hours is modelled by
V (t) = 12 + 10 ln t litres, t > 1.
a Draw a graph showing the volume collected over 20 hours.
b Find V 0 (t).
c Add the graph of the derivative to your graph in a.
d Find V (6) and V 0 (6), and interpret your answers.
e Describe what is happening to both the volume and rate of change of volume over time.
6 The number of customers in a restaurant t hours after noon is modelled by
N (t) = 2t3 ¡ 15t2 + 24t + 25, 0 6 t 6 6.
a Find N 0 (t).
b Find t when N 0 (t) = 0, and interpret your answer.
c What was the least number of customers in the restaurant?
d During what time interval was the number of customers decreasing?
e Sketch the graph of N 0 (t), and explain the significance of the point where N 0 (t) is
a minimum.
7 Suppose f (t) = Ate¡bt , where A and b are positive constants.
a Show that f 0 (t) = Ae¡bt (1 ¡ bt).
1
b Show that f (t) has a turning point when t = .
b
D
IDENTIFYING MODELS FROM GRAPHS
The speed of a motorbike is measured at one second
intervals as it accelerates from a stop sign.
speed (kmh
_ -1)
25
The following data is obtained, and displayed on the
scatter plot:
20
15
Time (s)
1
2
3
4
5
6
Speed (km h¡1 )
10
14
17
20
22
24
10
5
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0
0
2
4
6
time (s)
100
50
75
25
0
5
95
100
50
75
25
0
5
95
100
50
75
25
0
5
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100
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0
5
Based on the shape formed by the scatter plot, a
logarithmic or power model appear most appropriate
for this data.
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APPLYING CALCULUS TO FURTHER MODELLING
(Chapter 6)
The following summary may prove useful in identifying appropriate models:
² linear
y
m<0
c
c
m>0
t
² power
y = mt + c
y
t
m<0
m>1
0<m<1
t
² exponential
y = atm
y
y
y
t
t
y = aebt
y
y
b>0
a
b<0
a
t
t
² quadratic
y
y
a>0
c
a<0
c
t
vertex t
² cubic
y
t
a>0
d
turning points exist
y
d
t
t
no turning points exist
y
d
t
a<0
a<0
turning points exist
² logarithmic
y = at3 + bt2 + ct + d
y
a>0
d
y = at2 + bt + c
vertex
no turning points exist
y
y = b + a ln t
y
a<0
a>0
t
t
² terminal velocity
y = A(1 ¡ e¡bt )
y
y=A
A > 0, b > 0
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t
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APPLYING CALCULUS TO FURTHER MODELLING
² surge
y = Ate¡bt
y
t
A > 0, b > 0
² logistic
223
(Chapter 6)
y
y=
y=C
C
1 + Ae¡bt
C > 0, A > 0, b > 0
t
EXERCISE 6D
²
²
²
²
²
1 From the list given alongside, determine
the possible models for data with the
following scatter plots:
a
b
Mass
c
Power
t
d
e
h
Kilometres
i
Tonnage
k
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100
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t
75
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0
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Q
t
l
S
x
5
t
t
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Force
x
x
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t
f
Demand
t
j
quadratic
exponential
logarithmic
terminal velocity
x
Weight
g
²
²
²
²
linear
cubic
power
logistic
surge
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N
t
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APPLYING CALCULUS TO FURTHER MODELLING
(Chapter 6)
2 For the following scatter plots, explain why the suggested model is not acceptable:
a
b
y
P
t
t
A logarithmic model of the form
P = b + a ln t, a > 0
A power model of the form
y = atm , a > 0, m > 1
d
c
T
D
t
t
A cubic model of the form
D = at3 + bt2 + ct + d
A logistic model of the form
C
T =
, A, b and C > 0
1 + Ae¡bt
e
f
V
N
t
t
A terminal velocity model of the form
V = A(1 ¡ e¡bt ), A and b > 0
An exponential model of the form
N = aebt , a > 0, b < 0
3 For the following data sets:
3
1:7
4
2:8
6
6:7
8
14:1
10
23:9
12
32:1
14
36:7
16
38:7
3
17:7
5
12:6
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60
5
95
40
22
20
39:8
80
1
100
30
45
50
20
81
17
39:1
15
22:5
75
15
100
13
15:7
25
10
110
10
10:1
0
7
105
95
50
75
3
67
8
9:2
5
0
0
8
19:6
95
t
y
7
19:4
100
d
6
19:0
50
0
30
5
18:4
75
t
y
4
17:3
25
c
3
15:5
0
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0:7
2
12:6
5
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y
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100
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i draw a scatter plot of the data
ii suggest the most appropriate model for the data.
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APPLYING CALCULUS TO FURTHER MODELLING
E
(Chapter 6)
225
MODELLING PROBLEMS
In this section we will attempt to fit an appropriate model to a set of data.
We have previously fitted linear, exponential, and power models to data. In this section we will
examine data which may be:
² exponential
² cubic
² power
² logarithmic
² quadratic
² logistic
In Chapter 4 we used natural logarithms to transform data into linear data, then used linear
regression to find the equation connecting the variables. Now we will use technology to find
the model of best fit, in the same way we find linear models. Consult the graphics calculator
instructions at the front of the book if you require assistance.
Example 4
Bacteria are cultivated on agar plates. The weight W of bacteria present is recorded at two
hour intervals.
2
0:09
Time (t hours)
Weight (W grams)
4
0:36
6
1:02
8
1:82
10
2:27
12
2:45
14
2:46
a Draw a scatter plot of the data.
b Explain why the logistic model is most appropriate for the data.
c Find the logistic model of best fit.
d Estimate the weight of the bacteria after 7:5 hours.
e At what rate is the bacteria weight increasing after 6 hours?
a
W
2
or
1
t
2 4 6 8 10 12 14
b The data points increase slowly at first, then faster, then more slowly again as they
approach a limiting value. This behaviour is consistent with a logistic model.
c
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Using technology, the logistic model of best fit is W ¼
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2:484
.
1 + 94:69e¡0:6960t
SA_12MET-2
226
APPLYING CALCULUS TO FURTHER MODELLING
(Chapter 6)
2:484
¼ 1:64
1 + 94:69e¡0:6960(7:5)
d When t = 7:5, W ¼
After 7:5 hours, the weight of the bacteria is 1:64 g.
dW
¼ 0:417
dt
After 6 hours, the weight of the bacteria is increasing
at 0:417 grams per hour.
e Using technology, when t = 6,
If there is more than one possible model which appears suitable for the data, we can use the
r2 values associated with each model (except for the logistic model) or else the context of the
problem, to determine the most appropriate model.
Example 5
A bottle of water has been left out in the sun, and is now placed inside a refrigerator. The
water temperature is measured every 15 minutes. The temperatures are:
Time (t mins)
o
Temperature (T C)
15
30
45
60
75
90
105
120
31
19
12
7
4
3
2
1
a Draw a scatter plot of the data.
b The data appears to follow either a logarithmic or an exponential curve.
i Find the model of best fit for each of these model types. Include the r2 values.
ii Which model is most appropriate? Explain your answer.
c Use the model to estimate the water temperature after 20 minutes in the refrigerator.
d Find the rate at which the water is cooling when t = 40.
a
40
T
30
or
20
10
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150
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50
25
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95
100
50
75
25
0
The logarithmic model of best fit is
T ¼ 69:1 ¡ 14:7 ln t
r2 ¼ 0:977
5
95
100
50
75
25
0
i Logarithmic
95
30
100
0
5
95
100
50
75
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0
5
b
t
75
0
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APPLYING CALCULUS TO FURTHER MODELLING
227
(Chapter 6)
The exponential model of best fit is
T ¼ 49:0e¡0:0318t
r2 ¼ 0:995
Exponential
ii The exponential model has a higher value of r2 . Also, the logarithmic model will
head towards ¡1 over time, whereas the exponential model will approach zero
from above, which is more consistent with how the temperature will behave. So,
the exponential model is most appropriate.
c Using the exponential model,
when t = 20, T ¼ 49:0e¡0:0318(20) ¼ 26:0.
So, after 20 minutes, the temperature of the
water is 26:0o C.
d
T ¼ 49:0e¡0:0318t
dT
¼ 49:0(¡0:0318)e¡0:0318t ¼ ¡1:56e¡0:0318t
)
dt
dT
¼ ¡1:56e¡0:0318(40) ¼ ¡0:437
When t = 40,
dt
So, when t = 40, the water is cooling at 0:437o C per minute.
NOTE ON EXPONENTIAL MODELS
Exponential models in this chapter are written in the form y = aebt , as opposed to y = abt in
Chapter 4. This allows us to differentiate the model, and makes the model more consistent with
other models discussed in this chapter which also involve e.
When using technology to find an exponential model, the Casio gives the model in the form
y = aebt , as shown in the above example.
However, the Texas Instruments calculators give the
model in the form y = abt , as shown alongside.
We can use the rule b = eln b to convert this to the
form y = aebt :
T ¼ 49:0(0:969)t
) T ¼ 49:0(eln 0:969 )t
) T ¼ 49:0(e¡0:0318 )t
) T ¼ 49:0e¡0:0318t
EXERCISE 6E
1 A hemispherical bowl is filled with water. The
volume of water required to reach different water
levels is given in the table below.
40 cm
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APPLYING CALCULUS TO FURTHER MODELLING
0
0
h (cm)
V (mL)
3
550
6
2050
9
4350
(Chapter 6)
12
7250
15
10 600
18
14 250
a Obtain a scatter plot of the data.
b A cubic model is the best fit for the data. Find the cubic model for this data.
c Use the model to find the volume when the bowl is full.
d The volume of a sphere is given by V = 43 ¼r3 . Use this formula to check your answer
to c.
e Use the model to estimate the volume of water when the height is 10 cm.
10 cm
2 Water is kept at a constant temperature as it flows through a thick
walled pipe. The diameter of the pipe is 20 cm and the walls of the
pipe are 10 cm thick. The temperature of the pipe wall varies from
inside to outside as the heat escapes through the walls. The temperature
measurements are:
11:2
81
Distance (x cm)
Temperature (T o C)
13:8
66
15:7
55
16:9
51
17:4
48
10 cm
x cm
19:1
42
cross-section
of the pipe
a Draw a scatter plot of the data.
b Find the logarithmic model of best fit.
c Find the temperature at the point half-way through the wall.
d Estimate the temperature of:
i the inner wall
ii the outer wall.
dT
e Find
, and hence determine the rate at which the temperature is falling at the point
dx
half-way through the wall.
3 Air is pumped into a spherical balloon at a constant rate, and the radius of the balloon is
measured at various times:
18:2
16
Volume (v litres)
Radius (R cm)
27:8
19
39:6
21
58:2
24
73:1
26
96:8
28
a Draw a scatter plot of the data. Explain why a power model appears suitable.
b Find the power model of best fit.
c What is the radius when there is 80 litres of air in the balloon?
d At what rate is the radius increasing at the instant when v = 80? Give your answer
in cm L¡1 .
4 The mass of a radioactive substance is measured each year for six years:
0
5:7
Time (t years)
Mass (M grams)
1
5:3
2
4:9
3
4:6
4
4:3
5
4:0
6
3:7
a Draw a scatter plot of the data.
b For the scatter plot, it appears that either an exponential or a linear model is most
appropriate.
i Find the model of best fit for each of these model types. Include the r2 values.
ii Which model is the most appropriate? Explain your answer.
c Use your model to estimate the remaining mass of the substance after 10 years.
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d The ‘half-life’ of a radioactive substance is the time taken for it to reduce to half its
original mass. What is the half-life of this substance?
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APPLYING CALCULUS TO FURTHER MODELLING
229
(Chapter 6)
5 In a small community, everyone eventually hears a rumour. A small group of people
inadvertently start a rumour, and the proportion of people who have heard the rumour is
recorded on an hourly basis during the day.
0
Hours (h)
1
2
3
4
5
6
7
8
9
10
Proportion (P ) 0:02 0:04 0:09 0:18 0:33 0:54 0:72 0:86 0:91 0:96 0:98
a Draw a scatter plot of the data.
b What is the most likely model type for the data? Find the model of best fit.
c Estimate the proportion of people who have heard the rumour after 5:5 hours.
d Find the rate at which the rumour is spreading after: i 2 hours ii 5 hours.
6 A contractor digs wells to a maximum depth of 100 m. He wants a formula to find the cost
of digging wells of various depths. Using data from some of his previous jobs, he constructs
a table of costs:
Depth (x m)
15
30
45
58
68
84
Cost ($C)
8000
14 000
21 000
32 000
46 000
84 000
a Draw a scatter plot of the data.
b The contractor suspects that either a quadratic or a cubic model is most appropriate.
i Find the model of best fit for each of these model types. Include the r2 values.
ii Which model is the most appropriate?
c Find the cost of digging a well which is:
i 50 m deep
ii of maximum depth.
dC
when the depth is 50 m. What is the meaning of this result?
d Find
dx
dC
start increasing?
e At what depth does the rate of change of cost
dx
7 Consider again the data for the motorbike on page 221:
Time (t seconds)
1
2
3
4
5
6
km h¡1 )
10
14
17
20
22
24
Speed (S
a The
i
ii
iii
data appears to follow either a logarithmic or a power curve.
Find the logarithmic model of best fit.
Find the power model of best fit.
Which model do you think is most appropriate? Explain your answer.
b Using the model of best fit, estimate:
i the speed of the motorbike after 4:5 seconds
ii the rate at which the motorbike’s speed is increasing after 3 seconds.
8 The profit when making and selling x speed boats per month is given in the following table:
Number of boats (x)
5
7
10
14
19
24
Profit ($P ’000)
50
136
233
300
290
172
a Obtain a scatter plot of the data.
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b Which model type is most likely to fit the data?
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APPLYING CALCULUS TO FURTHER MODELLING
(Chapter 6)
c Find the model of best fit.
d Use the model to estimate the profit for the month if:
i 0 boats are sold
ii 8 boats are sold.
e Explain your answer to d i.
f How many boats should be sold to maximise the profit?
g Check your answer to f by differentiating the model.
h Give a possible reason why the curve decreases for larger production.
i For what production level does a loss occur?
9 Diamonds are categorised according to their quality. The values of 7 high quality grade-F
diamonds of different weights are given in the table below:
Weight (c carats)
0:32
0:41
0:67
0:81
1:03
1:27
2:11
Value ($D)
430
690
1750
2500
4000
5900
15 500
a Draw a scatter plot of the data.
b Find the:
i power model
ii exponential model of best fit.
c Which model best fits the data? Explain your answer.
d Use the model from c to find the value of a 1:5 carat grade-F diamond.
dD
e What is the value of
when c = 1:5? Interpret this result.
dc
dD
f Why is
positive for all values of c?
dc
10 A wine glass has a parabolic cross-section. Wine is added to the
empty glass in 20 mL lots, and each time the height of the wine
above the vertex is measured.
v (mL)
20
40
60
80
100
H (cm)
3:6
5:0
6:2
7:2
8:0
9 cm
H cm
a Obtain a scatter plot of the data.
b Find the model which best fits the data.
c Estimate the capacity of the glass.
vertex
d Find the height of the wine when the glass
is at half capacity.
11 The times taken for a population of rodents to reach certain levels are shown in the table
below:
Population (p)
48 150 290 560 2150
0
Time (w weeks)
5
8
11
17
a From a scatter plot of the data, which model type seems to be a possible fit for the data?
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b Find the model of best fit.
c Use the model to estimate the number of weeks needed for the population to reach:
i 400
ii 3000:
d Suggest reasons why this model may be inappropriate for larger population sizes.
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231
(Chapter 6)
12 Damien throws a cricket ball vertically upwards. The distance above the ground at various
times is determined from digital video evidence. The following table of values gives the
distance above the ground t seconds after the ball was released.
Time (t seconds)
1:2
1:8
2:4
2:9
3:6
4:2
4:7
5:2
Distance (D m)
30:1
39:2
45:0
47:1
45:8
40:7
33:9
24:9
a Using scatter plot evidence, what model seems to fit the data?
b
c
d
e
Find the model of best fit.
Use the model to estimate the height of the ball 2 seconds after release.
How far above the ground was the ball when released?
Find the greatest height, and the time after release when this height was reached.
f Find the total time the ball was in the air.
dD
of the ball:
g Find the velocity
dt
i when it was released
ii after 3 seconds
iii when it hit the ground.
h Show that the acceleration of the ball was constant throughout its flight.
Hint: Acceleration is the rate of change of velocity.
13 Several years ago the Willtinga Golf Course greens were ‘burnt off’ due to watering with
highly saline water from its dams. This occurred late in summer when the dams were very
low in water. As the water level lowered due to use and evaporation, the salt concentration
increased. During the following summer, the head greenkeeper kept records of the volume
of water in the dam and the salt concentration. The results were:
Volume (V ML)
45
32
24
14
8
4
Salt concentration (C%)
0:012
0:023
0:036
0:068
0:125
0:258
a
b
c
d
From a scatter plot, suggest the most likely model type for the data.
Find the model of best fit.
What was the salt concentration when the volume of water in the dam was 20 ML?
At what rate was the salt concentration increasing when the volume of water in the dam
was 20 ML?
e Water from the dam should not be used when the concentration exceeds 0:2%. What is
the volume of water in the dam when this occurs?
14 A small island off Cape York has an abundance of rodents which are the major food source
of a threatened species of brown snake. A few pairs of brown snakes were introduced to the
island which was previously snake free. Gulls and other birds on the island keep the snake
population in check. The population of snakes over many years has been recorded.
Time (t years)
1
2
3
4
5
6
7
8
Snake population
47
108
215
384
560
675
753
780
a Draw a scatter plot of the data and suggest a possible model.
c Estimate the population after 4:5 years.
b Find the model of best fit.
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d Is there a limit to the population size? If so, what is it?
e At what rate was the population increasing after:
i 2 years
ii 4 years
iii 6 years?
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APPLYING CALCULUS TO FURTHER MODELLING
(Chapter 6)
15 The times taken for a radioactive substance to reach certain masses are given in the following
table:
Mass (m grams) 3:0 2:5 2:0
1:5
1
0:5
4:0
Time (t hours)
7:9
12:6
18:7
27:4
42:0
a Obtain a scatter plot of the data and indicate which model types best fit the data.
b Find the model of best fit.
c Use the model to find the initial mass of the substance.
d Use the model to estimate the time required for the substance to decay to:
i 1:8 grams
ii 0:2 grams.
e Which of the two estimates in d would be more reliable? Explain your answer.
f Kaye produced a 5:37 gram sample of the substance and observed its decay. A table of
values was found. What would be the model for m(t) in this case?
F
SURGE AND TERMINAL VELOCITY MODELS
If you examine your graphics calculator menu for possible models, you will notice that it does
not directly handle surge or terminal velocity situations.
However, if we perform simple algebraic transformations on data which we suspect is fitted by
one of these models, we can then use a graphics calculator to complete the task.
SURGE MODELS
y
= Ae¡bt , which is an exponential model.
t
Notice that if y = Ate¡bt then
So, if we suspect the graph of y against t is fitted by a surge model, we can plot the graph of
y
against t, and check whether an exponential model is a good fit for these points.
t
Example 6
The effect of a pain-killing injection after t hours is shown in the following table:
Time (t hours)
0:00
0:10
0:25
0:50
0:75
1:00
1:25
1:50
1:75
2:00
Effect (E units)
0
56
84
84
58
42
22
10
5
3
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0
5
95
100
50
75
25
0
5
95
100
50
75
25
0
5
95
100
50
75
25
0
5
a Obtain a scatter plot of the data and suggest a suitable model for it.
E
against t using technology, and fit an exponential model to this data.
b Plot values of
t
c Rewrite the model in the form E = Ate¡bt .
d When is the chemical most effective?
e An operation can only take place when the effect is at least 60 units.
i At what time can the operation commence?
ii How long has the surgeon to complete the operation?
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233
(Chapter 6)
a
A surge model appears most suitable.
b
t
E
against t
t
E E
t
We remove the point
(0, 0) as Pp is
undefined.
E
¼ 804:5e¡3:147t
t
c The surge model is E ¼ 804:5te¡3:147t
dE
= 804:5e¡3:147t + 804:5t(¡3:147e¡3:147t )
d
fproduct ruleg
dt
¡3:147t
(1 ¡ 3:147t)
= 804:5e
dE
Now, the chemical is most effective when
=0
dt
1
¼ 0:3178 hours ¼ 19 minutes
) t=
3:147
e
i When E = 60 for the first time,
t ¼ 0:103 hours ¼ 6 min 11 s.
60
So, the operation can start 6 minutes and
11 seconds after the injection is given.
ii When E = 60 for the second time,
t ¼ 0:721 hours ¼ 43 min 16 s.
So, the surgeon has 43 ¡ 6 = 37 minutes
to complete the operation.
The exponential model is
TERMINAL VELOCITY MODELS
If y = A(1 ¡ e¡bt ) = A ¡ Ae¡bt ,
then A ¡ y = Ae¡bt , which is an exponential model.
So, if we suspect y and t follow a terminal velocity model, we can plot A ¡ y against t (where
A is the limiting value of the data), and check whether an exponential model is a good fit.
Example 7
A metal sphere of mass 1 kg is dropped into a deep reservoir. Its speed is measured at
various times as follows:
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2:9890
50
2:9551
75
2:8178
25
2:6807
0
2:4409
5
2:0212
95
1:2864
100
0
50
2
75
1:5
25
1
0
0:8
5
0:6
95
0:4
100
0:2
50
0
(m s¡1 )
75
25
0
5
95
100
50
75
25
0
5
S
t (s)
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APPLYING CALCULUS TO FURTHER MODELLING
(Chapter 6)
a Draw a scatter plot of the data and suggest a suitable model for it.
b Is there a value which S is approaching for larger values of t? Let this value be A.
c Draw a scatter plot of_ A ¡ S _against t and fit an exponential model to the data.
d Rewrite the model in the form S = A(1 ¡ e¡bt ).
e Use the model to find the speed of the sphere after 0:5 seconds.
f The rate of change of speed with respect to time is acceleration. What is the acceleration
after 1 second?
b Yes, S is approaching 3, so A = 3.
a
A terminal velocity model appears
suitable.
c
A - S against t
t
S A-S
The exponential model is A ¡ S ¼ 3e¡2:804t
d Since A = 3, the model is 3 ¡ S ¼ 3e¡2:804t
) S ¼ 3 ¡ 3e¡2:804t
) S ¼ 3(1 ¡ e¡2:804t )
e When t = 0:5, S ¼ 3(1 ¡ e¡2:804(0:5) ) ¼ 2:26.
So, after 0:5 seconds, the speed of the sphere is 2:26 m s¡1 .
dS
f Using technology, when t = 1,
¼ 0:510.
dt
So, after 1 second, the acceleration is 0:510 m s¡2 .
EXERCISE 6F
1 The following table gives the average pain relief effectiveness (APRE) of a 500 mg aspirin
tablet over a 12-hour period.
Time (t hours)
0
0:5
1
2
3
4
6
9
12
APRE (P units)
0
2:9
3:6
5:0
5:5
5:2
3:9
2:5
1:1
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100
50
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0
5
a Obtain a scatter plot of the data and state the model type which best fits the data.
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APPLYING CALCULUS TO FURTHER MODELLING
(Chapter 6)
235
P
against t, and find the exponential model for this data.
t
c Rewrite the model in the form P = Ate¡bt .
d What is the APRE at time
i t=5
ii t = 10?
e At what time is there maximum pain relief?
b Plot
f At what time does the rate of change of APRE start to increase?
g If the tablets produce acceptable pain relief when the APRE level is 2 or more units,
between what times does the tablet produce acceptable relief?
2 A pharmaceutical company develops a new pain-killing chemical which is supposed to be an
improvement on the one illustrated in Example 6. The effect of the new chemical over time
is given in the table below:
Time (t hours)
0:00
0:10
0:25
0:50
0:75
1:00
1:25
1:50
1:75
2:00
Effect (E units)
0
59
90
93
64
46
22
8
4
1
a Obtain a scatter plot of the data and suggest a model which fits it.
E
against t using technology and fit an exponential model to this data.
b Plot
t
c Rewrite the model in the form E = Ate¡bt .
dE
, and hence determine when the chemical is most effective.
d Find
dt
e The operation can only take place when the effect is at least 60 units.
i At what time can the operation commence?
ii How long has the surgeon to carry out the operation?
f Comment on the differences between this drug and that of Example 6.
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Time (t weeks)
0
1
2
3
4
5
6
7
8
Interest (I people)
0
42
67
83
58
32
15
7
4
Time (t weeks)
1
2
3
4
5
6
7
8
Cumulative sales (s items)
8
34
67
92
101
108
111
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the cumulative sales data:
draw a scatter plot, and suggest the most likely model type
find the model of best fit
determine the week in which sales were made at the greatest rate.
100
b For
i
ii
iii
50
the interest data:
draw a scatter plot, and suggest a model which closely fits the data
find the model of best fit
use the model to estimate the day of highest interest
estimate the interest in week 9:
75
a For
i
ii
iii
iv
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3 The owner of a computer software shop advertises a new information game called D’FACTO.
Interest in the new product from telephone calls and shop enquiries is recorded on a weekly
basis, along with the cumulative sales of the product. The data on interest and cumulative
sales was tabulated and is shown below.
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APPLYING CALCULUS TO FURTHER MODELLING
4 A stomach virus spreads throughout a high school.
The number of students N affected after t days
is shown on the scatter plot alongside.
µ ¶
N
is plotted against t, the data
When ln
t
is linear, and the equation of the line of best fit
µ ¶
N
= ¡0:351t + 3:92.
is ln
t
(Chapter 6)
60
N
50
40
30
20
10
0
t
0
5
10
15
20
N
N
and t, in the form
= Ae¡bt .
t
t
b Hence, find the surge model connecting N and t.
c How many students were affected after one week?
dN
d Find
, and hence determine the rate at which N is increasing when t = 2.
dt
a Find the equation connecting
5 A skydiver jumps from a small aeroplane and her speed is recorded at certain times.
Time (t seconds)
0
Speed (S km h¡1 )
0
2
4
7
10
15
20
30
40
50
42:7 76:2 113:7 139:8 166:9 181:9 194:5 198:4 199:5
a Draw a scatter plot of the data and suggest a model which best fits it.
b Is there a value which S is approaching as t gets larger? Call it A.
c Draw a scatter plot of A ¡ S against t, and fit an exponential model to the data.
d Rewrite the model in the form S = A(1 ¡ e¡bt ).
e Use the model to find the speed of the skydiver 6 seconds after she jumped from the
aeroplane.
f What is her acceleration after 6 seconds of free-fall?
6 The current in a circuit is shown in the following table:
Time (t milliseconds)
0
Current (I amps)
0
40
120
360
740
1240
2500
3250
3800
6:15 17:07 41:06 61:79 73:30 79:46 79:88 79:96
a Draw a scatter plot of the data and suggest a suitable model.
b Is there a value which I is approaching as t gets larger? Call it A.
c Draw a scatter plot of A ¡ I against t, and fit an exponential model to the data.
d Rewrite the model in the form I = A(1 ¡ e¡bt ).
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100
50
75
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0
5
e Use the model to find the current in the circuit when:
i t = 500 milliseconds
ii t = 8000 milliseconds.
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(Chapter 6)
237
REVIEW SET 6A
1 For each of the following graphs, find:
i
ii
iii
iv
v
the
the
the
the
the
position and nature of any turning points
intervals where the graph is increasing or decreasing
position of any points of inflection
intervals where the graph is concave or convex
equations of any asymptotes.
a
y
b
(1, 4_Qw_)
(-1, 1)
c
y
x
(-2, 3)
(-Qw , 3_Qw_)
y
4
(-4, 1)
x
(2, 0)
(-4, 2)
1Qw
y = -1
x = -3
(-2, -Qw_)
4 x
x=2
2 The stopping distance of a car is the distance travelled in the time between the driver
applying the brake, and the car coming to a complete halt. A test was carried out to find
the stopping distances of a car at various speeds. The results were:
Velocity (v km h¡1 )
40
60
70
80
90
110
120
Stopping distance (s metres)
29
55
67
96
112
168
223
a Draw a scatter plot for the data.
b Find the equation of the cubic model of best fit.
c Use the model to estimate the stopping distance at: i 100 km h¡1 ii 65 km h¡1 .
ds
for v > 0. Hence find the intervals where the curve is:
d Find
dv
i increasing or decreasing
ii concave or convex.
3 Diamonds can be synthetically produced in specially equipped laboratories.
Unfortunately, with current technology, it is a slow and expensive process. The costs of
producing diamonds of various thicknesses are given below.
Thickness (T atoms)
100
200
300
400
500
1000
2000
Cost (C dollars)
27:9
84:5
162
256
366
1109
3363
a Draw a scatter plot for this data and suggest a possible model.
b Find the model which best fits the data.
c Use this model to predict the cost of making a diamond:
i 250 atoms thick
ii 1500 atoms thick
iii 5000 atoms thick.
dC
, and determine the rate at which the cost is increasing for thicknesses of:
d Find
dT
i 200 atoms
ii 2200 atoms.
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5
4 The cartoon channel of a Pay-TV network regularly conducts surveys of its subscribers
to find out which cartoons are popular and which are not. It then uses the results to
decide which cartoons to show next season. The popularity of the cartoon ‘Poker Man’,
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APPLYING CALCULUS TO FURTHER MODELLING
(Chapter 6)
a cartoon about pocket sized monsters who could be caught and trained to play poker,
are as follows:
Date of survey
Jan 2008
June 2008
Jan 2009
April 2009
June 2009
Viewers (V )
0
17 524
21 930
19 464
17 976
Date of survey
Sept 2009
Jan 2010
June 2010
Dec 2010
Viewers (V )
15 026
11 890
7921
4929
a By letting t be the number of months after January 2008 (t = 0 for January 2008,
t = 5 for June 2008, and so on) construct a scatter plot of V against t.
b Suggest a possible model type for this data. Find the model of best fit.
c Use the model to predict the number of viewers:
i when t = 8
ii in September 2010.
dV
d Find
, and hence find the month in which ‘Poker Man’ was most popular.
dt
dV
when t = 8, and interpret your answer.
e Find
dt
f If management decides to scrap all shows whose popularity drops below 1000
viewers, when will ‘Poker Man’ be cancelled?
5 Vivian runs a small gardening store, specialising in garden gnomes. She purchases all
of her gnomes from Muhlack and Dangerfield Gnomes of Distinction, who sell to her at
a discount rate which varies according to the size of her order. The cost of an order of
g gnomes is given by C = 609 ln g ¡ 672 dollars.
a Find the cost of ordering 25 gnomes.
b Find the number of gnomes Vivian must order for the total cost to be $1500.
dC
, and show that the cost is always increasing. What can be said about the
c Find
dg
rate of increase?
REVIEW SET 6B
1 For each of the following graphs, find:
i
ii
iii
iv
v
the
the
the
the
the
position and nature of any turning points
intervals where the graph is increasing or decreasing
position of any points of inflection
intervals where the graph is concave or convex
equations of any asymptotes.
y
a
y
b
(2, 4)
y=2
-1
(3, 3)
(Qw , 2)
(-2, Qw)
x
1
y
c
y=5
x
(-1, -Qw)
2
x
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239
(Chapter 6)
2 A group of conservationists, primarily concerned with the welfare of an endangered breed
of lemming, learned of an island off the South Australian coast with an abundant food
supply for lemmings and very few cliffs. They introduced a few pairs of lemmings to the
island and then recorded the population at yearly intervals. The population of lemmings
t years after the program began is modelled by
2570
P (t) =
.
1 + 84e¡0:9t
a Estimate how many pairs of lemmings were originally introduced to the island.
b Show that the rate of change in the lemming population is given by
dP
194 292e¡0:9t
.
=
dt
(1 + 84e¡0:9t )2
c Find the lemming population and the rate at which it was changing after 30 months.
d Will the lemming population increase indefinitely, or will it approach some limiting
value?
e Find the value of t at which the rate of growth of the population is at a maximum,
and find this maximum rate of growth.
3 Technology has many applications in mathematics. One of these applications is the
simulation of projectile motion. For example, an irate author threw her malfunctioning
computer off the top of a tall building and then recorded its velocity at various times on
its way to the ground.
The velocity of the computer t seconds after it was thrown is given by
V = 50(1 ¡ e¡0:35t ) m s¡1 .
a Find the velocity of the computer:
i as it passed a 10th storey window, 6:6 seconds after being thrown
ii just before it hit the ground, after 8:3 seconds.
dV
dV
.
c Evaluate
when t = 4. Interpret your answer.
dt
dt
d Does the velocity approach a maximum value? If so, what is it?
b Find
4 Kathy and Karina run the computer store where the author from 3 bought her replacement
computer. Kathy is the salesperson and Karina constructs and delivers the computers.
Between them, they are able to sell and dispatch up to 150 computers per week. Their
profit figures for the last eight weeks are:
Computers sold (c)
43
115
0
101
82
19
144
62
132
2790 3250 ¡700 2820 2710 1260 5180 3090 4230
Profit ($P )
a Draw a scatter plot for the data.
b Which model type will best fit the data? Find the model of best fit.
c Use the model to predict the profit made by selling 90 computers.
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d How many computers need to be sold to make a profit of $5000 per week?
dP
.
e Find
dc
f Suggest how many computers Kathy and Karina should sell per week to maximise
their profit. What is this maximum weekly profit?
Hint: Remember that they can only produce 150 computers per week.
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(Chapter 6)
5 The population of zebras in a herd after t years is given by Z(t) = 80 + 35 ln(2t + 1),
t > 0.
a Sketch the graph of Z(t) against t.
b Find:
c
ii the population after 5 years.
i the initial population
Z 0 (t).
i Find
ii Hence, determine the rate at which the zebra population is increasing after
2 years.
6 A sample of radioactive material was studied over a 20 year period. The mass M of the
radioactive material remaining was recorded over that time:
Time (t years)
0:5
1
1:5
2
5
10
15
20
Mass (M grams)
48:0
46:1
44:1
42:4
32:7
21:2
14:7
9:7
a Construct a scatter plot for this data.
b The data fits either a logarithmic, a power, or an exponential model.
i Find the model of best fit for each of these model types. Include the r2 values.
ii Which model is most appropriate? Explain your answer.
c Use your model to calculate the mass of radioactive material:
i when the sample was taken
ii after 7 years.
d The ‘half-life’ of a radioactive substance is the time taken for the mass of the
substance to halve. Find the half-life for this material.
e Find the rate of decay after:
i 1 year
ii 10 years.
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f Is the rate of decay increasing or decreasing?
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