Methods TI-nspire CAS Calculater Companion
Methods TI-nspire CAS Calculater Companion
Methods TI-nspire CAS Calculater Companion
MATHS QUEST 12
Mathematical
Methods CAS
RAYMOND ROZEN | BRIAN HODGSON | NICOLAOS KARANIKOLAS
BEVERLY LANGSFORD-WILLING | MARK DUNCAN | TRACY HERFT
LIBBY KEMPTON | JENNIFER NOLAN | GEOFF PHILLIPS
2ND EDITION
Contents
Introduction
iv
CHAPTER 7
Differentiation
CHAPTER 1
CHAPTER 8
Applications of differentiation
CHAPTER 2
19
CHAPTER 9
Integration
CHAPTER 3
27
75
CHAPTER 10
CHAPTER 4
35
45
85
CHAPTER 12
Continuous distributions
CHAPTER 6
83
CHAPTER 11
CHAPTER 5
Inverse functions
63
49
95
69
Introduction
This booklet is designed as a companion to Maths Quest 12 Mathematical Methods CAS Second Edition.
It contains worked examples from the student text that have been re-worked using the TI-Nspire CX
CAS calculator with Operating System v3.
The content of this booklet will be updated online as new operating systems are released by Texas
Instruments.
The companion is designed to assist students and teachers in making decisions about the judicious use of
CAS technology in answering mathematical questions.
The calculator companion booklet is also available as a PDF file on the eBookPLUS under the
preliminary section of Maths Quest 12 Mathematical Methods CAS Second Edition.
iv
Introduction
ChapTer 1
WriTe
Worked Example 8
line as:
p(x) r(x)
Press ENTER .
Write/display
Worked Example 10
Write/display
Worked Example 11
Find the quotient, Q(x), and the remainder, R(x), when x 4 3 x 3 + 2 x 2 8 is divided by the linear
expression x + 2.
Think
1
x+2
Then press ENTER .
Write
Worked Example 12
Write
Worked Example 13
Write
1
x = 2, x = , or x = 3
2
Worked example 16
Sketch the graph of each of the following functions, stating the domain and range of each.
a 4 x 2 y = 8, x [ 3, 3]
b f ( x ) = 1 2 x, x ( , 1)
Think
WriTe/draW/display
When x = 3,
12 2y = 8
When x = 3,
12 2y = 8
2y
2y
= 20
y = 10
= 4
y=2
(3, 2) is the other closed end of the line.
y
2
3
0
4
(3, 10)
8
9
10
11
2 3
4x 2y = 8,
x [3, 3]
When x = 0, y = 4
When y = 0, x = 2
The x-intercept is 2 and the y-intercept is 4.
The domain is [3, 3].
10
(3, 2)
f(x) = 1 2x,
x (, 1)
(2, 5)
(1, 3)
2 1 0
3
Worked Example 22
Sketch the graph of y = 3 + 8 x 2 x 2, showing the turning point and all intercepts, rounding
answers to 2 decimal places where appropriate.
Think
1
Write/draw
y
12
(2, 11)
9
f(x) = 3 + 8x
2x2
3 (0, 3)
(0.35, 0)
(4.35, 0)
x
0
1
4 5
Worked Example 25
Write/draw
y = x3 x2 10x 8
4
Worked Example 27
Write/draw
y
(0, 10)
(1, 0)
( 5, 0)
3 2 1 0
(2, 0)
1
( 5, 0)
x
2 3
Worked Example 28
Sketch the graphs of each of the following equations, showing the coordinates of all intercepts.
Use a CAS calculator to find the coordinates of the turning points, rounding to 2 decimal places as
appropriate.
a y = x2(x 1)(x + 2)b y = (x + 3)2(x 1)2
Think
Write/draw
Then press:
MENU
6: Analyze graph 6
2: Minimum 2
Move the upper and lower bounds
into the correct position to locate the
stationary points. Similarly for the
local maximum at (0, 0).
Sketch the graph of the quartic.
(2, 0)
(0, 0)
0
(1.44, 2.83)
(1, 0)
(0.69, 0.40)
Then press:
MENU
6: Analyze graph 6
2: Minimum 2
Move the upper and lower bounds
into the correct position to locate the
stationary points. Similarly for the
local maximums.
Sketch the graph of the quartic.
y
(3, 0)
(1, 0)
0
(0, 9)
(1, 16)
Worked Example 29
1 0
Think
1
1 2
Write/display
y=
Worked Example 31
ax 7y = 0
2x + (a 9)y = 0
Find the value(s) of a, where a is a real constant. Consider a set of simultaneous equations that
have a unique solution.
Think
1
Press:
MENU b
7: Matrix & Vector 7
3: Determinant 3
to find the determinant in terms of a.
Then complete the entry line as
solve(a2 9a + 14 = 0, a), and press
ENTER .
Write
Worked Example 32
Write
= 0, t
solve det
2 t 5
Press:
MENU b
3: Algebra 3
7: Solve System of Equations 7
2: Solve System of Linear Equations 2
Choose 2 as the number of equations, and x,
and y as the variables.
,{x , y} | t = 3
linsolve
2 x + (t 5) y = 3t
tx 3 y = 6
linsolve
,{x , y} | t = 2
2 x + (t 5) y = 3t
Worked Example 33
Write
linsolve 4 x 3 y + z = 12,{x , y, z}
3 x y z = 14
Worked Example 34
Press:
MENU b
7: Matrix & Vector 7
3: Determinant 3
to find the determinant in terms of k. Then
complete the entry line as
solve(k2 7k + 10 = 0, k),
and press ENTER .
Now press:
MENU b
3: Algebra 3
7: Solve System of Equations 7
2: Solve System of Linear Equations 2
Choose 3 as the number of equations, and
x, y and z as the variables. Complete the
entry lines as:
k y + z = 8
linsolve 3 x + ky + 2 z = 2,{x , y, z}
x 3 y + z = 6
Write
linsolve 3 x + ky + 2 z = 2,{x , y, z} | k = 2
x + 3 y + z = 6
k y + z = 8
3 x + ky + 2 z = 2,{x , y, z}
linsolve
|k =5
x + 3 y + z = 6
k = 2.
Worked Example 35
The cubic function with the general equation y = ax3 + bx2 + cx + 8 passes through the points (1, 2)
(2, 4) and (4, 8). Find the values of a, b and c.
Think
1
Write
On a Calculator page,
Define f(x) = ax3 + bx2 + cx + 8
Now press:
MENU b
3: Algebra 3
7: Solve System of Equations 7
2: Solve System of Linear Equations 2
Choose 3 as the number of equations, and
a, b and c as the variables. Complete the
entry line as:
f (1) = 2
linsolve f ( 2) = 4,{a, b, c}
f (4) = 8
Worked Example 36
Solve these five linear simultaneous equations using matrices and a CAS calculator.
2v + w 3x + 2y z = 12
v + 3w + 4x y + 2z = 13
v 2w + 5x 2y 3z = 32
3v w + 2x y 3z = 18
3v + 3w 4x + 3y 2z = 9
Think
1
a 1b
Write
v = 2, w = 4, x = 1, y = 3 and z = 5.
ChapTer 2
Given the equation y = kx2, determine the effect on the graph y = x2, when k = {2, 3, 4}. Sketch the
graphs.
Think
1
WriTe/display
19
Worked example 6
y
5
3
Think
y = a(x b)3 + c
a = 2
y = 2(x 1)3 + 3
1
2
3
4
20
WriTe/display
Worked example 18
Given f: [0, ) R, where f ( x ) = x and g(x) = af (x) + b, where a and b are positive real
constants, consider the effect on g(x) as a and b increase individually.
Think
1
WriTe
21
Worked example 21
Express f (x) = |5x 4| as a hybrid function, defining the domain of each part and graphing the
function.
Think
1
2
WriTe/draW
5 x 4, where 5 x 4 0
f ( x ) = | 5x 4 | =
(5 x 4), where 5 x 4 < 0
First function: 5x 4
First domain: 5x 4 0
x4
5
4
5 x 4, where x
5
f (x) =
5 x + 4, where x < 4
y
5
4 (0, 4)
3
2
1
2
22
4
5
f(x) = | 5x 4 |
1
( 45 , 0)
Worked example 22
Using matrices, find the location of the point (x, y) under the following transformations of the
point (1, 3):
dilation by a factor of 2 from the y-axis
reflection in the x-axis.
Think
1
WriTe/display
0 1
=
1
3
23
Worked Example 31
Write/display
g(2x) = 20x
2g(x) = 20x
g(2x) = 2g(x)
When g(x) = 10x it satisfies the equation
g(2x) = 2g(x).
Worked example 34
It is believed that, for the data in the table below, the relationship between x and y can be
modelled by y = aaxx 2 + bx + c .
x
y
0
4
1
5.3
2
8.6
3
14.8
5
34.4
Think
4
23
WriTe/display
a = 1.252
b = 0.222
c = 4.096
y = 1.252x2 0.222x + 4.096
Correct to 3 decimal places.
25
Chapter 3
Exponential and
logarithmic equations
Worked example 2
0.4
a 64 3
b 32
c 125 3 .
think
Write/display
32
0.4
2
125 3
Then press ENTER .
Note: If the calculator is set to
Approximate or Auto, the answer
will be displayed as a decimal.
2
64 3 = 16
32
0.4
2
c 125
1
4
1
25
27
Worked example 3
a 2 b4
( a3 b 4 ) 1
a 12 b 1
b 1 2 .
3 b
think
Write/display
a 2b4 (a3b 4) 1
b
1
a 12 b 1
3 1 b 2
Press ENTER after each entry.
a 2 b 4 (a 3b
12 1
a b
b
3 1b2
4 1
1
3
b
3 a
Worked example 4
Simplify
3 n 6 n + 1 12 n 1
.
32n 8n
think
1
Write/display
28
3n 6n + 1 12n 1 3n
=
32 n 8 n
2
b8
a5
Worked Example 9
Write/display
Worked Example 11
Write/display
Worked example 13
Write/display
Worked example 16
Solve the following equations for x, giving your answers both in exact form and correct to
3 decimal places.
a 2x > 5
b 0.5x 1.4
think
1
Write/display
x>
loge (5)
.
loge (2)
7
5
loge (2)
30
Worked Example 18
( ) = 3 .
1
125
Think
Write/display
1: Solve 1
Complete the entry lines as:
solve(logx (4) = 2,x)
solve(log x 1 = 3, x )
125
125
b Solving log x
= 3 for x gives x = 5.
Worked Example 22
Solve for x, showing working. Express your answers in exact form and correct to 3 decimal places.
a ex = 3b ex 3ex = 2
Think
Write/display
1: Solve 1
Complete the entry lines as:
solve(ex = 3,x)
solve(ex 3ex = 2,x)
Press ENTER after each entry.
b Solving ex 3e
3
aSolving
ex
bSolving ex 3e
Worked Example 23
Solve for x, giving your answer both in exact form and correct to 3 decimal places, given that
loge (x) = 3.
Think
Write/display
loge (x) = 3
e3 = x
x = e3
Worked Example 25
Write/display
x = 3ey + 1
Worked example 26
Write
x = 2 log10 (y 1) + 1
10 2
y =
10
y = 10
y = 10
y = 10
1
2
+1
x
10 2
1 + x + 1
2 2
f 1( x)
x 1
2
1
2
+1
+1
+1
x 1
= 10 2
+1
Worked example 27
kx
Solve ekx = 5 + 2e
think
1
solve(e k x = 5 + 2e k x,x)
Note: You must put in a multiplication
sign between the k and the x.
Then press ENTER .
Write/display
x =
33 + 5
1
, k R\{0}.
loge
k
2
33
Worked example 28
1
logg 2 (x
( x ) 5 log 2 ( pp)) = llog 2 (6) where p > 0.
2
think
1
2
Then press ENTER .
Write/display
x = 36p10
Worked example 29
Solve the following equations using a CAS calculator. Give your answers correct to 3 decimal
places.
a ex = x3
b loge (x) = x 2
think
34
Write/display
Chapter 4
Exponential and
logarithmic graphs
Worked example 3
Sketch the graph of f (x) = 2 2x 1, showing intercepts and asymptotes, and stating the domain
and the range
think
Write/draW
0,
35
y
2
Asymptote
y=2
(0, 32 )
1
0
2 1
(2, 0)
x
2
f (x) = 2 2x 1
1
6
36
Worked Example 6
Sketch the graph of f(x) = 2 log10 (3 x) 2, showing intercepts and asymptotes, and stating the
domain, range and transformations. Give exact values or round to 3 decimal places.
Think
Write/draw
To graph y = 2 log10 (3 x) 2 on a
Graphs page, complete the function entry
line as:
f1(x) = 2 log10 (3 x) 2
Press ENTER.
Note: The vertical asymptote at x = 3 is
notdisplayed.
(7, 0)
x
6 4 2 0 2
Asymptote
(0, 2 log10 (3) 2)
x=3
2
f(x) = 2 log10 (3 x) 2
The domain is (, 3) and the range is R.
Reflection in the y-axis, dilation 2 units from
the x-axis, vertical translation 2 units down,
horizontal translation 3 units to the right
Worked example 11
Sketch the graph of f (x) = 2 ex, marking the asymptote and intercepts. State the
transformations, domain and range. Give exact answers. Check using a CAS calculator.
think
1
Write/draW
f (x) = 2 e x
Exponential curve
A reflection in the x-axis and a reflection in the
y-axis. The vertical translation is 2 units up.
If x = 0,
If y = 0,
y=2e 0
=21
=1
or (0, 1) (0, 1) (0, 1)
The y-intercept is 1.
2 ex = 0
ex = 2
1
=2
ex
ex =
1
2
x=
loge 1
2
y
(loge( 12 ), 0)
Asymptote
y=2
(0, 1)
0
1
f (x) = 2 ex
x
38
Worked example 15
Sketch the graph of f (x) = 2 3 loge(1 x), marking the asymptote and intercepts.
State the domain and range.
think
Write/draW
f (x) = 2 3 loge (1 x)
Vertical asymptote is x = 1.
If y = 0, 2 3 loge (1 x) = 0
3 loge (1 x) = 2
loge (1 x) = 23
3
4
If x = 0, y = 2 3 loge (1)
=2
e3 = 1 x
2
x = 1 e3
x 0.95 (to 2 decimal places)
5
y
4
(0, 2)
2
(1 e 3 , 0)
f (x) = 2 3 loge (1 x)
x
2 1 0 1
2 Asymptote
x=1
7
39
Worked example 16
y
4
(0, 2)
(2.44, 0)
4 3 2 1
think
1
0 1 x
Write
[1]
(2.44,
2.44
[2]
For
0 = Ae
0):
+B
a + b = 2
solve
,
a
2.44 + b = 0
a e
Press ENTER .
4
Worked example 23
Sketch the graph of y = x2ex using a CAS calculator. Show all axis intercepts and any asymptotes.
think
40
Write/diSplaY
y = 02e0
y=0
(0, 0)
y = 0 is an asymptote.
Worked example 24
Write/draW
f1(x) = 2 lnn ( x + 2 ) 3
Then press ENTER . Press t and
select the absolute value template.
2
1
7 6 5 4 3 2 110
1 2 3 4 5 x
2
3
Asymptote
x = 2
2 loge ( x + 2) 3
y=
2)) 3
2 loge ( ( x + 2))
2 loge (x + 2) 3 = 0
3
logge ( x + 2) =
2
x + 2 = e2
for x > 2
for x < 2
for
x = e2 2
x = 2.481 69
4
+ 2) = e 2
3
x = e 2 2
x = 6.481 69
5
2 2, 0 and e 2 2, 0
e
41
y = 2loge 0 + 2 3
y = 2loge 0 + 2 3
y = 2loge 2 3
y = 2loge 2 3
y = 1.613 71
42
x R\{2}
yR
Worked example 26
Write/draW
2
3
Find W when t = 0.
Write the answer in a sentence.
Find W when t = 2.
W = 100e0.03t
When t = 0, W = 100
The initial size of the population is 100 wombats.
When t = 2, W = 100e0.03 2
= 100 1.0618
106 (nearest whole number)
After 2 years there are 106 wombats.
When t = 10, W = 100e0.03 10
= 100 1.3499
135 (nearest whole number)
After 10 years there are 135 wombats.
t = 2020 1998
= 22 years
When t = 22, W = 100e0.03 22
= 193.479
= 193 (nearest whole number)
In the year 2020, there are approximately
193 wombats.
43
44
Let W = 250.
Divide both sides by 100.
250 = 100e0.03 t
2.5 = e0.03 t
loge (2.5) = loge (e0.03 t )
0.03t = loge (2.5)
1
t=
loge ( 2.5)
0.03
t = 30.543
t = 31 (nearest year)
There will be 250 wombats in the year 2029.
CHAPTER 5
Inverse functions
WORKED EXAMPLE 6
If f(x) = ln(x + 1) + 1,
a find f 1(x)
b draw the graph of f(x) and its inverse f 1(x).
THINK
WRITE
Let y = f (x).
Interchange x and y.
y = ln(x + 1) + 1
x = ln(y + 1) + 1
(x),
open a Graphs page.
Complete the function entry line as:
f 1(x) = ln(x + 1) + 1
Then press ENTER .
Complete the function entry line as:
f 2(x) = ex 1 1
Then press ENTER .
f 1(x) = ex 1 1
b
45
WORKED EXAMPLE 12
a Sketch the graph of f(x) = x2 3x + 3, showing the turning point and relevant
intercept(s).
b Find the rule of the inverse by an algebraic method and sketch this graph
find any points of intersection between the original curve and its inverse.
y
4
3
2
1
0
of f 1(x).
46
(x) exists.
WRITE/DRAW
f(x) = x2 3x + 3
(1.5, 0.75)
1 2 3 4
x = y2 3y + 3
f 1( x) =
4x 3 + 3
2
y
4
3
2
1
f(x) = x2 3x + 3
y=x
1 2 3 4 5 6 7
f(x) = x2 3x + 3
y
4
3
2
1
y=x
(1.5, 0.75)
1 2 3 4
f : ( , 32 ) R
R,, f ( x)
x ) = x 2 3x + 3
a=
3
2
47
ChapTer 6
Circular (trigonometric)
functions
Worked example 1
Convert the following to degrees, giving the answer correct to 2 decimal places.
9
a 2c
b 6.3c
c
10
Think
WriTe
9 c
= 162.
10
49
Worked Example 5
If sin ( ) =
12
13
and
Think
1
Write
sin = cos 1
13
tan = cos 1
13
Given sin ( ) = 12 ,
13
cos( ) =
tan( ) =
5
13
12
5
< < ,
2
Worked Example 7
3
6
Think
1
WRITE
a sin
5
6
Press ENTER after each entry.
b tan
4
3
=
sin
2
3
5
3
=
tan
3
6
Worked Example 9
2
2
2
, x ) | 0 x 2
2
Then press ENTER .
solve (cos( x ) =
x=
3 5
,
4 4
Worked Example 10
Find all solutions to the equation sin () = 0.7 in the domain [0, 4]. Give your answers correct to
4 decimal places.
Think
1
Write
Worked Example 14
solve(2sin (2 x ) = 3, x )
Then press ENTER .
solve(2sin (2 x ) = 3, x ) | 0 x 2
Then press ENTER .
Write
2
and x =
n = 0: x =
6
3
5
5
n = 1: x =
and x =
6
3
11
8
and x =
n = 2: x =
6
3
2 5 5 11
, , ,
For 0 x 2, x =
3 3 6 6
Worked Example 15
Find the general solution of the equation sin (3x) = cos (3x) and hence find all solutions for x in the
domain 0 x 2.
Think
1
Write
x=
5 3 13 17 7
, , ,
,
,
12 12 4 12 12 4
Worked Example 18
Sketch the graph of y = 12 sin (3 ) for one complete cycle stating the amplitude, period and range.
Think
1
Write/draw
Amplitude =
Period =
1
2
2
3
1 1
Range = 2 , 2
y
1
2
0
1
2
Worked example 25
While out in his trawler John North, a fisherman, notes that the height of the tide in the harbour
can be found by using the equation:
h = 5 + 2 cos t ,
6
where h metres is the height of the tide and t is the number of hours after midnight.
a What is the height of the high tide and when does it occur in the first 24 hours?
b What is the difference in height between high and low tides?
c Sketch the graph of h for 0 t 24.
d John North knows that his trawler needs a depth of at least 6 metres to enter the harbour. Between
what hours is he able to bring his boat back into the harbour?
Think
f 1( x ) = 5 + 2 cos x
6
f 2( x ) = 6
Press ENTER after each entry.
Select an appropriate window.
WriTe
t = 0, 12, 24, . . .
A high tide of height 7 m occurs at midnight, noon
the next day, and midnight the next night.
b
For minimum h,
cos t = 1
6
So h = 5 + 2 1 = 3
Alternatively, min. value =
median amplitude so h = 5 2 = 3.
2
h
6
4
2
0 2 4 6 8 1012 14 16 18 202224 t
55
Press:
MENU b
6: Analyze Graph 6
4: Intersection 4
Move the cursor to the left of the
first point of intersection, press
ENTER , then move the cursor
to the right of the first point of
intersection, press ENTER ,
the coordinates of the first point of
intersection are displayed. Repeat for
the other points of intersection.
Worked example 26
Using addition of ordinates, sketch the graph of y = sin (x) + cos (x) for the domain [0, 2].
Think
1
draW
y
2
1
0
1
2 x
2
3
y
2
1
0
1
2
56
Worked Example 27
Sketch the graph of y = |3 cos (2x)| over the domain [0, 2].
Think
1
draw
y
3
2
1
0
1
2 x
2
3
Worked Example 28
Find the domain and sketch the graph of the product function y = x sin (x). Use a CAS calculator
for assistance.
Think
1
Write/draw
Worked Example 30
Write/draw
f ( g( x )) = cos( x )
Worked Example 32
Consider a remote island where global warming has caused the temperature to increase by
0.1degree each month. The mean daily temperature is modelled by the function
T ( m) = 16 + 0.1 m + 6 cos m ,
6
where T is the temperature in degrees Celsius and m is the number of months after January 2008.
a Sketch a graph of the function for a five year period from January 2008, using a CAS calculator
for assistance.
Write
f 1( x ) = 16 + 0.1x + 6 cos x
6
Press:
MENU b
6: Analyze Graph 6
4: Intersection 4
Move the cursor to the left of the first
point of intersection, press
ENTER , then move the cursor
to the right of the first point of
intersection, press ENTER ,
the coordinates of the first point of
intersection are displayed.
m = 11.5631
Hence, the first time the temperature reaches
23 degrees Celsius will be during the 12th month
after January 2008. That is, during January 2009.
CHAPTER 7
Differentiation
WORKED EXAMPLE 12
If f ( x ) = x 3 2 x 2 +
a f (x)
8
, use a CAS calculator to find:
x
b f (2).
THINK
1
WRITE
a f (x) = 3 x 2 4 x
b f (2) = 2
8
x2
CHAPTER 7 Differentiation
63
WORKED EXAMPLE 14
If f ( x ) =
1
2 x2
3x
THINK
WRITE
f (x) =
Express y as a function of u.
Express u as a function of x.
du
= 4x 3
dx
3
dy
= f ( x ) = 12 u 2 (4 x 3)
dx
3x
y = (2 x 2 3 x )
Let y = u
dy
=
du
1
2
1
2
where u = 2x2 3x
3
1
2
u
2
u = 2x2 3x
=
=
=
2x 2
(4 x 3) (2 x 2 3 x )
2
(4 x
3
2
3)
3
2(22 x 2 3 x ) 2
3 4x
2 (2
(2 x 2 3 x)
x )3
d
1
dx 2 x 2 3 x
Press ENTER .
10
64
f ( x ) =
(4 x
3)
2 x (2 x 3) x (2 x 3)
WORKED EXAMPLE 24
WRITE
y = sin (5x)
du
Express u as a function of x and find
.
dx
Let u = 5x so
du
=5
dx
dy
.
du
y = sin (u) so
dy
= cos((u)
du
Find
dy
using the chain rule.
dx
dy
= 5cos(u)
dx
= 5 cos (5x)
dy
= 5cos(55 x )
dx
CHAPTER 7 Differentiation
65
WORKED EXAMPLE 33
WRITE
As f (x) = | x2 4x | is a composite
function, apply the chain rule to find
the derivative,
f (x) where g(x) = x2 4x and
h(x) = | x |.
66
f (x) = h(g(x))
f (x) = g (x) h(g(x))
2
1 iff x 4 x > 0
f (x) = 2x 4
2
1 iff x 4 x < 0
2
2 x 4 if x 4 x > 0.
f (x) =
2
2 x + 4 if x 4 x < 0
y
4
3
2
1
0
1
1
2
3
4
1 2 3 4 5x
)
|x|.sign
f'(x) = 2x 4, x > 4
10
8
6
4
2
f(x) = | x 4x | 2
x
3 2 1 0 1 2 3 4 5 6 7
2
4
f '(x) = 2x + 4, 0 < x < 4
6
8
f '(x) = 2x 4, x < 0
CHAPTER 7 Differentiation
67
CHAPTER 8
Applications of differentiation
WORKED EXAMPLE 2
WRITE
y = 3 loge (2x)
Evaluate y when x = 1.
At x = 1, y = 3 loge (2)
Find
dy
.
dx
dy
when x = 1 to obtain the
dx
gradient of the tangent at x = 1.
Evaluate
dy 3(2)
=
dx
2x
3
=
x
dy 3
=
dx 1
=3
So gradient of tangent is 3.
a At x = 1,
Equation of tangent is
y 3 loge (2) = 3(x 1)
= 3x 3
y = 3x 3 + 3 loge (2)
b Gradient of normal is
1
.
3
Equation of normal is
y 3 loge (2) = 13 (x 1)
3y 9 loge (2) = 1(x 1)
= x + 1
x + 3y = 1 + 9 loge (2)
69
70
WORKED EXAMPLE 3
a
b
c
d
Find the stationary points and their nature for the function f (x) = x3 + 5x2 8x 12.
Show that the curve passes through (1, 0).
Find the coordinates of all other intercepts.
Hence, sketch the graph of f (x).
THINK
WRITE/DRAW
f (x) = x3 + 5x2 8x 12
f (x) = 3x2 + 10x 8
d
( f ( x))
dx
solve(3x2 + 10x 8 = 0, x)
{ }
4,
Gradient table:
2 400
, 27
3
2
3
f (x)
Slope
).
71
Solve f (x) = 0.
d
(4, 36)
f(x)
(6, 0)
(1, 0)
(2, 0) x
(0, 12)
)
( 23 , 14 22
27
72
WORKED EXAMPLE 6
WRITE
f (x) = (x a)(3x a + 6)
b
a 6 4(a + 3)3
and (a, 0) are the coordinates
,
3
27
of the stationary points.
a6
f
, a .
3
Press ENTER .
Write the stationary points in
coordinate form.
73
5 256
and (1, 0) are the stationary points.
,
3 27
f (x)
Slope
+
/
74
+
/
Press ENTER .
Select the appropriate value of a
given that it is positive.
3 27
4 ( a + 3)3 5(a 6)
solve
=
+ 15, a
27
3
1 6 4(1 + 3)3
and (1, 0)
3 ,
27
a = 3( 5 2)
2
ChapTer 9
Integration
Worked example 5
5
Antidifferentiate 2 x + 3 .
Think
1
WriTe
5
5
dx = loge 2 x + 3 + c
2x + 3
2
ChapTer 9 Integration
75
Worked Example 12
Find [2 e4 x 5sin(2 x ) + 4 x] dx .
Think
1
Write
(2e4 x 5sin(2 x ) + 4 x ) dx
Press ENTER .
Note: When integrating trigonometric
functions, ensure the CAS is set to radian
mode.
(2e 4 x 5sin(2 x ) + 4 x ) dx
=
e 4 x 5cos(2 x )
+
+ 2x 2 + c
2
2
Worked Example 18
f(x)
Write
Worked Example 21
0 (3 x 2 + 4 x 1) dx
1 (2 x + 1)3 dx
Think
1
Write
0 (3x 2 + 4 x 1) dx
2
1 (2 x + 1)3 dx
Press ENTER after each entry.
2
0 (3x 2 + 4 x 1) dx = 42
1 (2 x + 1)3 dx = 225
16
Worked Example 23
If
0 8 x dx = 36, find k.
Think
1
Write
Press ENTER .
Chapter 9 Integration 77
Worked Example 27
Area =
2 ( x 3 4 x ) dx 0 ( x 3 4 x )
2 ( x 3 4 x ) dx 0 ( x 3 4 x ) dx
Press ENTER .
Alternatively, take the absolute value
for the area below the x-axis as
shown.
Write
y = x3 4x
dx
Worked Example 34
4
and g(x) = x intersect.
x
b
Sketch the graphs on the same axes. Shade the region between the two curves and x = 1 and
x = 3.
cFind the exact area between f(x) and g(x) from x = 1 to x = 3.
aFind the values of x where the graph of the functions f(x) =
Think
Write/draw
f(x) =
4
, g(x) = x
x
f(x) = x4
g(x) = x
01 2 3
1
2
4
x
Area =
1 ( x x ) dx + 2 ( x x )
2
dx
( 4x x ) dx + ( x 4x ) dx
3
Press ENTER .
4
The area is 4 loge + 1 square units.
3
Chapter 9 Integration 79
Worked Example 36
Find the average value of f(x) = loge (2x) for the interval [2, 4]. Give your answer in exact form.
Think
1
Write
yav =
4
1
loge (2 x ) dx
2
42
1 4
( ln (2 x )) dx
2 2
Press ENTER .
Worked example 39
The rate of change of position, velocity, of a particle travelling in a straight line is given by
dx
WriTe
ii
dx
.
dt
dx
= 40 10e 0.4 t
dt
= 40 10e 0.4 0
0
= 40 10e
= 40 10
= 30
b 1 Substitute
dx
= 35.
dt
dx
= 40 10e 0.4 10
dt
= 40 10e 4
= 39.82
dx
= 40 10e 0.4 t
dt
35 = 40 10e
Solve for t.
5 = 10e
0.4 t
0.4 t
e 0.4 t = 0.5
0.4t = ln(0.5)
ln(0.5)
0.4
= 1.73 s
t=
ChapTer 9 Integration
81
c
To graph dx = 40 10e 0.4 t , on a
dt
Graphs page, complete the function
entry line as:
Antidifferentiate.
dx
= 40 10e 0.4 t
dt
x=
10
(40 10e
10
82
0.4 t
) dt
0.4 0
ChapTer 10
Find the expected value and variance of the following probability distribution table.
x
Pr(X = x)
0.15
0.12
0.24
0.37
0.12
Think
1
WriTe
E(X) = 3.19
Var(X) = 1.5339
83
Worked Example 28
Pr(X = x)
1
4
3
8
1
8
1
4
Calculate the expected value, the variance and the standard deviation.
Think
1
Write
E(X) = 1.375
Var(X) 1.2344
SD(x) 1.1110
ChapTer 11
WriTe
x
Pr(X = 0)
Pr(X = 1)
Pr(X = 2)
Pr(X = 3)
Pr(X = 4)
Pr(X = 5)
Pr(X = 6)
Pr(X = x)
0.046 656
0.186 624
0.311 04
0.276 48
0.138 24
0.036 864
0.004 096
85
Worked Example 4
A new drug for hay fever is known to be successful in 40% of cases. Ten hay fever sufferers take
part in the testing of the drug. Find the probability, correct to 4 decimal places, that:
a four people are cured b no people are cured c all 10 are cured.
Think
Write
Evaluate.
0.2508
n = 10
p = 0.4
X ~ Bi(10, 0.4)
X = the number of people cured; therefore x = 4.
Press ENTER .
10
X ~ Bi(10, 0.4)
As X = the number of people cured,
therefore x = 0.
Pr(X = 0) = binomPdf(10, 0.4, 0)
= 0.0060466
The probability that no people are cured is 0.0060.
X ~ Bi(10, 0.4)
As X = the number of people cured,
therefore x = 10.
Pr(X = 10) = binomPdf(10, 0.4, 10)
= 0.00010486
The probability that no people are cured is 0.0001.
Worked Example 5
Grant is a keen darts player and knows that his chance of scoring a bullseye on any one throw
is 0.3.
a If Grant takes 6 shots at the target, find the probability, correct to 4 decimal places, that he:
i misses the bullseye each time
ii
hits the bullseye at least once.
b
Find the number of throws Grant would need to ensure a probability of more than 0.8 of scoring
at least one bullseye.
Think
Write
2
3
4
5
variables.
Note: Pr(X 1) would involve
adding probabilities from
Pr(X = 1) to Pr(X = 6). Using the
fact that Pr(X 1) = 1 Pr(X = 0)
allows us to solve the problem
using fewer terms.
Substitute the values into the
rule.
Pr(X = x) = nCxpxqn x
n=6
Let X = the number of bullseyes, therefore
x = 0, 1, 2, 3, 4, 5, 6
p = 0.3
q = 0.7
Pr(X = x) = nCxpxqn x
Pr(X = 0) = 6C0(0.3)0(0.7)6
= 1 1 0.117 649
= 0.117 649
0.1176
The probability that Grant misses the bullseye
each time is 0.1176.
ii n = 6
Worked Example 8
So Jung has a bag containing 4 red and 3 blue marbles. She selects a marble at random and then
replaces it. She does this 7 times. Find the probability, correct to 4 decimal places, that:
a at least 5 marbles are red
b greater than 3 are red
c no more than 2 are red.
Think
Write
Press ENTER .
Pr(X 5) = binomCdf(7, 4 , 5, 7)
7
= 0.359345
The probability that at least 5 red marbles are
chosen is 0.3593.
b
Pr(X 2) = binomCdf(7, 4 , 0, 2)
7
= 0.126584
The probability that no more than 2 marbles are
chosen is 0.1266.
Worked Example 13
Using the above data for attending the gym or aerobics class, find:
a the proportion of people attending the gym and aerobics class on the 5th day
b the number of people attending the gym or aerobics class in the long term.
Think
3
4
Write
0.8 0.7
T=
0.2 0.3
150
S0 =
50
0.8 0.7
0.2 0.3 t
150
50 s 0
S4 = T4 S0
150
0.8 0.7
=
50
0.2
0.3
155.555
=
44.445
7
S50 = T50 S0
50
150
0.8 0.7
=
50
0.2
0.3
155.555
=
44.445
3
Chapter 12
Continuous distributions
Worked example 4
logg e ((0.5 x )
, 2 x 2e
f
(
x
)
=
2
A random variable, X, has its frequency curve defined as
.
0,
elsewhere
Calculate the probability, correct to 4 decimal places, that X is:
a less than 4
b between 2.5 and 3.5.
think
Write
2
2
ln(0.5 x )
dx
2
Press ENTER .
Write the solution, rounding to
4 decimal places.
3.5 log
ge ((0.5 x )
dx
= 0.2004
95
Worked example 5
1 1 x
2
Write
y
1
(0, 2)
0
b
96
k 0
A = lim
1
e
2
1
= lim e
k 2
= klim
e
ddx
k
2 0
0
k
1
lim
= k 1 x
e 2 0
1
1
lim k
= k
0
e 2 e
=0+
1
1
=1
Since f (x) 0 for all x and the total area under the
curve is 1, f (x) is a pdf.
c i Pr(X < 3) =
31
0 2
1
x
2
ddx
1
= x
e 0
1
2
1
1
= 3 0
e2 e
= 0.7769
ii
=1
2.5 1
ddx
2.5
1
=1 1
e 2 x 0
1 1
= 1 2.5 0
2 e
e
= 1 0.7135
= 0.2865
2.5 2 e 2
ddxx
Press ENTER
iii
1
x
2
1e
2.5 2
= 0.2865
ddx
0.06337
0.7769
= 0.0816
=
97
Worked example 8
Find the variance and standard deviation for the following probability density function.
1
x ,1 x2
f (x) =
.
2
0,
elsewhere
think
1
xf ( x ) dx
1
x 2 f ( x ) dx
1
98
Write
Worked example 9
Write
= E( X ) = x loge ( x ) dx
0
x dx
1 ( x ln( x))
Then press ENTER .
= E( X ) = x loge ( x ) ddx
As 1 m e, so m must be the
bigger of the two possible solutions.
Answer the question and round to
3 decimal places.
= 2.097
b
logge ( x ) ddx =
1
2
1
2
for m implies
m = 0.1866 or m = 2.1555.
The median is 2.156.
99
1 ( x 2 ln( x )) dx (2.09726)2
e
100
SD(X) = Var( X )
= 0.176047371282
= 0.420
2 X + 2
= 1.2581 X 2.9364
= 1.2581 X e, since 2.9364 > e (the upper
value).
Calculate Pr( 2 X + 2)
using the CAS calculator.
Pr( 2 X + 2) =
2 = 2.097 2 0.420
= 1.2581
+ 2 = 2.097 + 2 0.420
= 2.9364
1.2581
logge ( x ) ddx
= 0.969
Note that in this example, 96.9% of the data lies
within 2 standard deviations of the mean, which is
close to the estimated value of 95%.
Worked example 15
Write
0
2
Press ENTER .
0.728 0
101
2.02
102
0 1.59
Worked example 16
Write
50 55
2
Press ENTER .
28
2
50
65
103
Pr (X < 40)
Pr (X < 70)
Region required
40 50 70
2
nor
normCdf
( ,,40,50,8)
nor
normCdf
( ,,70,50,8)
0.105650
=
0.993790
Pr(( X < 40)
Pr(( X < 40 | X < 70) =
Pr(( X < 70)
nor
normCdf
( ,,40,50,8)
=
nor
normCdf
( ,,70,50,8)
=
= 0.1063
104
Worked example 20
Write
a
57%
0 c
2
Press ENTER .
c = invNorm(0.57,0,1)
= 0.176
b
91%
c
2
105
106
c = invNorm(0.09, 0, 1)
= 1.341
Worked Example 23
X is normally distributed with a mean of 10 and a standard deviation of 2. Calculate x1, correct to
3 decimal places, if:
a Pr(X x1) = 0.65
b Pr(X > x1) = 0.85.
Think
Write
0.65
10 x1
2
Press ENTER .
x1 = invNorm(0.65,10,2)
= 10.771
b
0.85
x1
10
x1 = invNorm(0.15,10,2)
= 7.927