37
The
table
alongside
shows the probability
distribution for X.
E(X)
If
38
=
x
= x)
P(X
0
1
2
3
0.3
0.2
m
n
The derivative
find the value of m.
1.55,
DIFFERENTIATION
l' (X)
function
li
f(x+h)-f(x)
h~
h
provides:
An investigation into the weight of packed vegetables found the
following:
Number of tomatoes
in a 1.5 kg bag
Median weight of
tomatoes in bag (g)
15
90
f
with respect to x
•
the rate of change of
•
the gradient of the tangent to
of x.
y
= f (x)
for any value
y
y = f(x)
h)
f(x+
11
125
14
110
12
125
f--h---j
14
136
x
17
82
12
115
10
150
f(x)
q
,
x+h
x
When we use the limit definition to find a derivative, we call
this the method of first principles.
a Find the correlation coefficient r.
C
d
Write down the equation of the least squares regression
line.
Hence estimate the median weight of the tomatoes if there
are:
i 13
ii 20 in the bag.
X ~ N(13,
40
Suppose X is normally distributed with
0"2)
=
and P(X:::;; 15)
Given
0.613,
find
P(X:::;; 24)
=
0
nxn-1
cu(x)
cu'(x)
u'(x)v(x)
u'(x)v(x)
u(x)
vex)
0".
0.035
ef(x)
addition rule
product rule
- u(x)v'(x)
[vex )]2
ef(x)
quotient rule
1'(x)
exponentials
1'(x)
f(x)
In f(x)
CALCULUS
Name of rule
+ v'(x)
+ u(x)v'(x)
u'(x)
u(x)v(x)
and P(X ;;:::33) = 0.262.
Find the mean and standard
deviation of X correct to 3 significant figures.
TOPIC 6:
c
xn
u(x)+v(x)
e Comment on the reliability of your estimates in d.
39
l' (x)
f(x)
b Describe the relationship between the variables.
logarithms
sinx
cosx
cosx
-smx
tanx
1
cos2 x
trigonometric
functions
LIMITS
If f (x) can be made as close as we like to some real number A
by making x sufficiently close to a, we say that f (x) approaches
a limit as x approaches a, and we write
f (x) =
lim
A.
Chain rule
If y
= g(u)
where
u
= f(x)
then
dy
dy du
dx
du dx'
f(x)
is the derivative
x"'" a
We say that as x approaches a, f(x)
We can use the idea of limits as x
f(x) -4 ±oo to find asymptotes.
converges to A.
-4
±oo
and as
A graph will never cross its vertical asymptotes, but may cross
its horizontal asymptotes.
Higher derivatives
The second derivative of the function
of 1'(x).
f(n)(x)
The instantaneous rate of change of a variable at a particular
instant is given by the gradient of the tangent to the graph at
that point.
dy
IT
gives the rate of change in y with respect to x.
If dy
is positive, then as x increases, y also increases.
If dy
is negative, then as x increases, y decreases.
=~
f (x)
dxn
is
f(x)
PROPERTIES OF CURVES
Tangents and normals
For the curve y = f(x),
dx
dx
The nth derivative of tile function
RATES OF CHANGE
dx
= ~(f'(x))
f"(x)
is mT
the gradient of the tangent at x = a
= 1'(a).
The equation of the tangent to the curve at the point
dx
Mathematics
SL - Exam Preparation
& Practice Guide (3rd ed;tj~n)
is
Y- b
x-a
= l' (a).
A( a, b)
="
ient of the normal
at
mN
KINEMATIC PROBLEMS
y=f(x)
1
= - flea)"
An object moves along a straight line. Its position from the
origin at time Us given by a displacement function set).
tangent
.
I . . .
ds
IS given by v = dt'
Its instantaneous ve ocity
ildue
normal
.Ift::lSi°n,g and decreasing
increasing on S
{o}
on S
{o}
- decreasing
a < b.
I'(x)
j(a) (f(b)
{o}
for all
Signs of set):
decreasing on S
I'(x)
{o}
(0
for all x in S.
. ns which have the same behaviour for an x E]R; are
monotone increasing or monotone decreasing.
- nary points
Signs of v(~):
tionary point ofiifunction
is a point such that
I' (x) = O.
should be able to identify and explain the significance
and global maxima and minima, and inflections both
~:DIary and non-stationary.
-
tionary
point
maximum
minimum
Sign diagram of
near x=a
4
+
i' (x)
I
4
!1\
\J
• X
+
I
a
• X
I
stationary
inflection
~x
a
is a point of inflection at z
r
ive
a
=a
x~a
if
f"
(a)
If Set)
=0
Interpretation
=0
P is instantaneously
>0
P is moving to the right
<0
P is moving to the left
a(t)
at rest
Interpretation
>0
velocity is increasing
<0
velocity is decreasing
=0
velocity may be a max. or min.
represents the speed then S
= Ivl.
are the same then the speed of
If the signs of vet) and aCt)
the object is decreasing.
are different then the speed of
Optimisation
problem
solving method
Draw a large, clear diagram of the situation.
.
•
.'
Construct an equation with the variable to be optimised
as the subject. It should be written in terms of one other
variable such as x,
Write down any restrictions on the value of z.
3 Find the first derivative of the formula, and the values
of x which make it zero.
4
(0
for all x E S, the curve is
ncave downwards on the interval S.
f"(x)
vet)
P is located to the right of 0
It is important to remember that a local minimum or maximum
does not always give the minimum or maximum value of the
function in a particular domain. You must check for other turning
points in the domain and whether the end values of the domain
give higher or lower values.
;:- f"(x)
ncave
<0
P is located to the left of 0
If the signs of vet) and aCt)
the object is .increasing.
2
a, b)
=0
>0
OPTIMISATION PROBLEMS
and the
-- the tangent at a point of inflection is
.::mizontal, we say we have a stationary
ection point.
x=a
Interpretation
P is at 0
x~a
_ of
(x) changes on either side of x = a. It corresponds
a change in shape of the curve.
;:-the tangent at a point of inflection ~..
. not horizontal, we say we have a
n-stationary inflection point.
aCt):
set)
The speed at any instant is the magnitude ofthe object's velocity.
do,~
or
.L.
Signs of
Shape of curve
near x=a
-
a
dv
or a = v ds'
for all a, b E S
~ f(b)
jmt
c.,
2
ds
= dt2
You should also understand the physical meaning of'the-different
combinations of signs of velocity and' acce,l~ration.
~ 0 for all x in S.
f(a)
d iIts instantaneous
.
I
. b
acce eranon y a
You should understand the difference between instantaneous
velocity or acceleration, and average velocity or acceleration
over a time period.
functions
- increasing on an interval S
such that a < b.
x
an
~ 0 for all x E S, the curve is
on the interval S.
El
Show, using a sign diagram or second derivative test,
that you have a maximum or minimum stationary point.
Test the stationary points and end points of the domain
to find the optimal solution.
Mathematics
SL - Exam Preparation & Practice Guide (3rd edition)
Consider the graph:
I kf(x)
I [j(x)
D
y
B
=
dx
I f(x)
k
+ g(x)]
=
dx
dx,
k a constant
I f(x)
+ I g(x)
dx
x
Integral
Function
C
kx +c
k
A
xn+l
--+c
n+1
xn
A is a global minimnm as it is the minimum value of y
t' (x) =
downwards at that point.
C is a local minimum as f'(x)
0 and the curve is concave
-1x
= 0 and the curve is concave
y or f(x)
antidifferentiation /
or integration
F/(x)
is a function where
=
f(x),
then
F(x)
is
antiderivative
Area under a curve
F(x),
y
y= f(x)
If f(x) is a continuous positive function
on the ipterval a (; x (; b, then the
area under the curve between x = a
I:
f(x) dx.
I feu)
=
~: dx
I feu)
du
· I: f(x) dx =
· I: !(x) dx = - I: f(x)
0
• I: f(x)
· I:
· I:
=
set)
=
c (; x (; d where
f(x)
< 0,
± g(x)] dx =
J vet)
'--
I: f(x)
dx
I:
!(x) dx
±
J:
g(x) dx
dt.
Ib !(x)
dx
dx.
/'
'"
vet)
=
differentiate
'"
ds
dt
aCt)
velocity
integrate /
For the time interval
The total shaded area
c
=
differentiate
displacement
For example:
I: f(x)
[f(x)
dx
!(x) dx
set)
y= !(x)
f(x) dx.
!(x) dx -
+ I: f(x)
cf(x) dx = cl:
/'
f(x) < O.
I:
dx
dx
Kinematics
The displacement function is determined by the integral
To find the total area enclosed by y = f (x) and the z-axis
between x
a and x = b, we need to be careful about where
i-
__1 cos(ax + b) + c, a i- 0
a
Definite integrals
= F(b) - F(a).
the area is -
sin(ax + b)
Integration by substitution
Fundamental theorem of calculus
For a continuous function f(x) with
On an interval
~ sineax + b) + c, a i- 0
a
of f(x).
the antiderivative
and x = b is
cos(ax + b)
~~ or f'(x)
'<.
= I:
1
-In
lax + b\, a i- 0
a
ax+b
/ ""?">,
f(x) dx
+ c, a i- 0
a
(ax + b)n+1
(
) + c, n i- -1
an+1
1
INTEGRATION
I:
1ax+b
_e
(ax + b)n
anywhere on the domain.
ni--1
'
+ c
In \x\ + c
eax+b
upwards at that point.
D is a global maximum as it is the maximum value of y
If F(x)
eX
eX
anywhere on the domain.
B is a local maximum as
dx
=
'--
=
dv
dt
2
=
ds
dtZ
acceleration
integrate ..'
h (; t (; t2 :
s(tz) - s(h) = Jt,
rt2 vet) dt
•
displacement
•
total distance travelled = Jt,
rt2 \v(t)\ dt.
The area between two functions is given by
To find the total distance travelled given
A = I:(yu - YL) dx
where yu ~ YL on the domain a (; x ~ b.
a (; t (; b, we:
• draw a sign diagram for vet)
vet)
so we can determine when
any changes in direction occur
Indefinite integrals
When performing an indefmite integral, we use the rules for
differentiation in reverse. Do not forget to include the constant
of integration.
If F'(x)
I f(x)
AAnthpmnfir:s
= f(x)
then
I f(x)
dx
= F(x) + c.
dx is the indefinite integral of f(x) with respect to z.
SL _ Exam Preparation
rd
& Pradice Guide (3 edition)
• determine set) by integration
• find sea), s(b), and set)
changes
• draw a motion diagram
• determine the total distance
diagram.
at each time the directio
travelled
from the moti
SOLIDS OF REVOLUTION
9 Find
When the region enclosed by y = f(x),
the z-axis, and the
vertical lines x = a, x = b is rotated about the x-axis to
generate a solid, the volume of the solid is given by
dy
dx
for:
10 Find
d2y
dX2
11
f(x)
Let
1-
a y=
for:
2x
Tx
b y = 2x(1
3
X2
a y=-
+ 2x)~
b y = X2 sin3x
= x-4.
x+2
a Find the equations of the asymptotes of the function.
--""l:t---r-:----ti-:-----+-x
f'(x).
b Find
e Find the equation of the tangent to y
where
=
x
LL BUILDER QUESTIONS (NO CALCULATORS)
12
1 The derivative of f(x)
x -
f (x
+ h)
-
h
h~
b Find the equation of the tangent to
i
=
x
f (x )
(x + ~)
=
y
4.
b F· d dy
dx'
m
d2y
e Given that
>
dX2
for all x =I- 0, discuss the shape
0
of the curve.
1.
Consider the curve
at the point
a Find the asymptote of the function.
.
f (x) = 7 x - X2.
y = f (x) at the point
a Use this definition to differentiate
where
Consider the curve
is defined by
i' ( ) - r
= f(x)
3.
=
y
_2X2
d
+ 3.
x
Find the point for
minimum value.
>
0 at which the function has its
a Differentiate the function from first principles.
b Find the gradient of the tangent to the curve at the point
where x = -l.
e
Find the equation of the normal to the curve at the point
where x = -l.
13 For the first 6 seconds of its motion, a particle moving in a
straight line has velocity given by v = t3 - 9t2 + 24t m S-I,
where t is the time in seconds.
a Find the acceleration function for the particle.
b Find the greatest
6 seconds.
d At what point does the normal in e meet the curve again?
3 Find
:~
e y
Let
=
3x+
..;x
1
b y=-X4
x2v"f=X2
g(x)
=
d (2x
1
+9
Show that
(x
+ l?(x
s(t) = 12t - 3t3
f (x)
x3
~
2X2
+ 1 cm
where
a Find the velocity
16 Let
and acceleration
a Find:
(2+x)~.
i g'(x)
ii gl/(x).
y
= g(x)
i increasing
17 Let
v'3x
+1
and
y
=
y
=
x(x2
.
a Find
e Find the coordinates of the local maximum.
=
b
v'5x - X2.
dy
dx
-
12x
+ 45).
2
d y
and
dx?'
Find the turning points of the function.
e Find the point of inflection of the function.
at this
release is given by h(t)
=
100
+ 32t
e
For what range of values of a does the equation
18
12x2
+ 45x
- a
=
0
- 12x
+ 45).
Sketch the graph of
-
y
= x(x2
d
x3
c Find the equation of the common tangent.
A man standing on a cliff above the ocean throws a ball high in
the air. The height of the ball above the water t seconds after
IS:
ii concave up.
a Find the point at which these curves meet.
Show that the curves have the same gradient
intersection point.
for the
g(x)=x2e-(x+2).
a State the domain of the function.
y
functions
ii velocity decreasing?
b Find intervals where
dy
F·ill d dx'
function
t? 0 is in seconds.
i speed decreasing
reasons for your answers.
b
3x - 2.
b When is the particle's:
b Find the greatest and least values of the function, giving
Consider the curves
at
particle's movement.
on the domain
a Find f'(x).
b
-
+ 2x + 1
15 A particle moves in a straight line with displacement
Consider the function
-l~x~l.
=
- 2) = x3
in the first
c At which point does this tangent meet the curve again?
-xcosx.
b Find the gradient of the tangent to the graph y = g( x) at
the point where x = ~.
Y
y = x3
a Find the equation of the tangent to
the point where x = -l.
b
+ 3)5
a Find g'(x).
Let
of the particle
for:
14
a y=--
velocity
have three distinct real roots?
Find the equation of the normal to the curve
point where
x
=
y
2.
=
8
x2
at the
- 4t2 m.
a Find h'(t).
b How high above the water will the ball reach?
19
a Find the derivatives with respect to x of:
x
••
eX
11
Mathematics
2x
+1
e2x _ 1
e
SL - Exam Preparation
& Practice
Guide
(3rd edition)
---
b
20
---
a Show that p(x)
c Find the stationary points of
has a stationary point at x
p(x)
b Suppose
Find:
b Find the zeros of f(x).
dx.
= x3 + ax2 + b.
p(x)
Consider the polynomial
+ e2x+1)
J(x2
Without simplifying, find
d
= O.
also has a stationary point at (-2,
27
i-
h
O.
Show that
given by A = xy
perimeter is 48 m.
a
=
-~h
iii the coordinates
inflection.
and
28
x.
a
!,(x)
=
f(x)
non-stationary
4x3
3x4,
-
point
of
showing
the
4x3
3x4
-
=
k
= 1-
4x
+4
-2--'
y
=
f(x).
Clearly show the
information found above.
d y=ln(:2~~)'
=
In
29 Find !,(x)
x>4
(~2-:~),
then
2X2 - 2x - 4
(1 _ 2x)(x2 + 2)'
b On what intervals is f(x)
24
f( x ')
c Sketch the graph of
1
+ l' xi-- 2
=
the
b Find the position and nature of the stationary points of the
graph of f(x).
x>-l
Show that if
of
a Find the axes intercepts.
e2x
y= -2x
C
.
Consider
X
.
a y=xln(x+1),
find:
has exactly two distinct positive solutions.
for:
dx
3x4
-
c Find values of k for which the equation
Its
b Hence
find the maximum
possible area of the figure.
dy
clearly labelling the
the intercepts on the axes
shown is
+ (0.6)X2.
a Express y in terms of
22 Find
= 4x3
y
b Sketch the graph of y
information found in a.
21 The area of the figure
and their nature.
ii the coordinates and nature of the stationary points
+ ~ha
b=k
Sketch the graph of y = f(x),
information in a, b, and c above.
a For the graph of
Suppose instead that the second stationary point of p( x)
is at (h, k) with
= f(x)
6).
i the values of a and b
ii the nature of both stationary points.
C
y
for these functions:
a f(x)=3sin(x-4)
b f(x)
=
12x-2cos(J)
c f(x) =
d
f(x)
=
vsin(2X
sin2x
1 + 2x
e f(x) = e2sinx
decreasing?
30
The diagram below shows an open trough for which the
cross-section is an isosceles right-angled triangle. The total
outside surface area is fixed at 27 square metres.
Let
f(x)
+ 1)
f(x) = tan(3x - 4)
= x+2
rx=-r
.
a State the values of x for which f(x)
is defmed.
b Find the equation of the normal to the curve y
at x = 10.
= f(x)
y
31
f(X,")
a By considering the total outside surface area of the trough,
show that
b
x2
P(0,2)
+ 2xy = 27.
.---------~~--------_x
Find an expression for the volume V of the trough in terms
of x only.
c Hence show that the dimensions of the trough which yield
the maximum volume are x = y = 3 metres.
25
A comet travels in an orbit which can be
equation
On the graph of y = f(x)
shown, A and B are stationary
points, and C is a point of
inflection.
Copy
and
complete
the
following using +, 0, or - :
!,(x)
observer at the point
sex)
b
!,,(x)
B
a
Mathematics
-
+ 4.
Find the least and the greatest distance between the comet
and the observer for -2::;; x::;; 2.
C = 48 (3: + x)
4
= x + 5 + -.
x
State the asymptote of y
SL - Exam Preparation
";X4
P(O, 2) is given by
a By letting x be the length in metres of the side fenced with
corrugated iron, show that the cost of fencing is
D
f(x)
=
3x2
32 /Ierry wants to fence off a rectangular garden plot of area 48 m2.
Three sides will be fenced with strong wire mesh costing $1
per metre, and the remaining side will be fenced with corrugated
iron costing $30 per metre.
C
Consider
y = X2 as shown in the diagram.
a Show that the distance of the comet at C(x, y)
A
26
2
-2
&
b
= f (x).
Practice Guide (3rd edition)
El
dollars.
Find the dimensions ofthe garden plot which will minimise
the cost of fencing.
~ ._~_
ture.
~ ..~
g the
oil rig is at point R, 8 km from a straight shore.
-- on the shore directly opposite the rig. A refinery
;tloce at S which is 11 km from P.
44
R
45
-- to be constructed
R to reach the
point Q. From Q a
ID be taken overland
cost of the pipeline is
per km under the sea
- 53 million per km overland.
Find
the
=
y
If
area
1 - X2
=
Y
of the
xv4
- x,
~-=-='-J::=from
ints
int
of
w the
47
j~
f(x)
dx
5-
f(x)
dx
J2 f(x)
dx
8
C
= f(x)
=
and simplify your answer.
y
y = f(x)
+ 4.
dx.
m s"
1.
at time t seconds,
i
=
=
a If its position at time
at time t.
t
0 is s
> O.
0, find its position
b What is the acceleration at time t?
=
=
y
6
f(x)
Find:
Jo
1r
a
sinx dx
b
I:
cos2x
dx
4
x
area
/
a
4,
=
49
By considering
50
If
d
--(x21nx),
dx
4
2
Y = x e-x,
J
quarter circle
Hence find
Y= f(x)
2
find Jxlnx
dx.
dy
2
dx = 2x(1 - x)e
show that
X2
x(l - x2)e-
_x2
.
dx.
a.
'---~--~~a------~x
51
a Find the points of intersection of the graphs of
y
the integrals:
J(2x2
+X
-
3) dx
52
d
1
o
y
--dx
2x + 1
g'(x) = 2cos3x
53
pose
g(-3:) = 4.
and that
1'(x)
= (x
+ 2)2
and that
f(l)
=
!so
1'(x)
=
v'4x
+5
and that
l' (x)
f(O)
= - ~.
54
defined?
and
y
=-
2x - 4.
A particle moves on a straight line with acceleration
given
at rest?
find
k(l - x)
(3 - 2x)~'
f' (x),
J
Hence, find
2
3 Use the identity
dx
b
_x_-_1"""""7"1 dx.
(3 - 2X)2
1
X2
28
to find
2x
1
sin28
= ~- ~ cos
+1
55
dx
J sin 3x
2
=
t = 0
1.
b Find the total distance moved by the particle in the interval
from t = 0 to t = 1.
and write your answer
5 3
2X2 - X - 3
x
At time t seconds, the velocity of a particle moving in a straight
line is given by v = 2t - 3t2 m S-l.
to t
f(x)=x(3-2x)~,
J
4 - x2
a Find the change in displacement of the particle from
Find:
a
=
c Find the displacement function of the particle.
a For what values of x is
in the form
- 1.
b At what other time is the particle momentarily
b Find f(x).
If
=x
a Find the velocity function of the particle.
2
Find f(x).
uppose
y
by 2 - 3t m S-2. Initially when t = 0 s, the particle has
displacement 3 m. When t = 1 s, the particle is momentarily
at rest.
?":nd g(x).
-
and
b Hence find the area bounded by the graphs.
4_1dx
o v'x + 4
pose
- 3
a Find the points of intersection of the graphs of
1
1
-
= X2 + 2x
b Hence find the area bounded by the graphs.
1
minimise
J~2 f(x)
y5t+4
area
y
dx
parabola
8~
dx.
o y4 - x
X2 - 4x
= ~ 10
V
48
that
_ f(x)
-
the
A particle undergoing straight line motion has velocity
when the shaded area is ~ A.
Y
~~
below
-3.
b Does the value
obtained in a represent
the shaded area?
Explain your answer.
ornamental pond of area A is to be
. t with straight sides and semi-circular
as shown. The cost of tiling per unit
gth is 25% more along the rounded
~
than along the straight walls. Show
the total cost of tiling the walls is
the function shown,
x3
a Find
b What is the minimum cost of the pipeline?
of the
=
f(x)
that the cost to construct
the pipeline from R to S is
C(x) = 5v'x2 + 64 + 33 - 3x million dollars.
4=k
find
=
y
Consider the graph of
a If Q is x km from P, show
- g the
lying
12
Hence, evaluate
46
region
and above the line
dx.
m
The line
y
=
x - 2
is a tangent to the curve
(x - 1)2(x
+ 2) = x
3
= x3
-
2x.
+ 2.
a
Show that
b
Explain why the point of contact T occurs when x = 1.
Find the coordinates of T, and the other point of
intersection P of the line and the curve.
c Find the axes intercepts of
Mathematics
y
=
3x
y
-
x3
-
2x.
SL - Exam Preparation & Pradice Guide (3rd edition)
d
c to sketch
Use the results of band
y =
x3 -
2x
y = x - 2
and
64 Find the volume of revolution
when the shaded region
about the x-axis.
0
revolved through 360
on the same set of axes.
a
e Find the area of the region bounded by the curve and the
b
y
segment [PT] of the given straight line.
IS
y = 2x
+3
The shaded area is
~ units''
56
Find a given a
>
x
1
O.
65
A parabola passes through the points (-7r,
0), (7r, 0), and
(0, o ). The area between the parabola and the x-axis is 4 units".
Calculate the possible values of
57
58
Let
f((}) = 2 ~i~~S(}'
'((})
f
b
Find the minimum value of f((}).
=
f((})=cos(}sin2(},
bFind
f(x)
27r about the x-axis.
1- 2 cos ()
sirr' ()
Sh ow th at
Let
66 Find the volume of the solid formed when the region enclosed
by the graph of y = X2 - 2x and the x-axis is rotated through
e « %.
a
67
a Write down the derivative with respect to x of:
i f(x)=lnx
ii F(x)=xlnx-x.
b
O<(}::::;%.
What is the relationship between
the maximum value of f((}).
is said to be odd if f(-x)=-f(x)
fl f(x)
Evaluate
c Show that
J~l
Show that the area bounded by [OP], the curve C,
and the z-axis, is t - ~t In t - 1 units''.
f(x).
+ ~sin
(e-2X +x3 cos2x) dx.
68
2t cm at any time t ~ 0 s.
a Determine:
61
Hence describe the motion of the point during the time
interval 0::::; t ::::;
27r.
a Sketch the graph of
-27r ::::;x ::::;
27r.
b
a On the same set of axes, sketch the two functions
f(x) = sinx
and g(x) = sin2x
for 0::::; x::::;tt .
b
the position and velocity of the point at t = 0 sand
t = 27r S
ii the times when the point comes to rest for
o ::::;t ::::;27r, and the acceleration at each of these
instants.
b
+ i)
f(x) = sin(x
Find the area that lies above the line y
the graph of y = f(x)
for
= ~
Show the zeros and turning points of both
63
a If
y = In (tan x),
for which
b
dy
69 A piece of wire 40 cm long is bent
to form the boundary OPQ of a
sector of a circle with centre 0
and radius x cm.
The angle of the sector is
() radians and the length of the arc
is s cm.
b
for -27r::::; x ::::;
27r.
0
<
x
Given that
<
%'
find the constant k
3'
SL -
EXllm
•
=
s
(}x
and the area of the sector is given by
show that
A = 20x -
undergoing
-
X2
straight
line
ern".
motion
3et metres per second at time
has
t seconds,
velocity
t ~ O.
Show that the particle is stationary when t = In 3 seconds.
c If the initial position of the particle is 1 metre to the right
of the origin, show that its position at t = In 5 seconds
is also 1 metre to the right of the origin.
x
7r
IS
Prepl'mtion
~(}X2,
A particle
b
1
y=-sin2x
"6
~ In 3 units".
Mnthemlltics
Q
a Find the initial velocity.
y
i
=
70
v(t) = e2t
Show that the shaded
area enclosed by the
curve, the x-axis, and
the lines x =
and
X
scm
c Find the value of x for which the area A is a maximum.
d Hence find () when the area is a maximum.
Alongside is a graph of
1
y = sin Sz '
7r
P
a Show that s = 40 - 2x.
and below
k
sin2x
dx
and g.
graphs.
1~(Sin3x+5cOSX)dx.
Find
f
c Find the area of each of the two regions bounded by the
A=
62
For what
value of t will this area be maximised?
ii -Write down the equation of the tangent to the curve
C at the point P(t, In t). Find the condition on t for
this tangent to pass through the origin.
is an odd function.
The position of a point moving on a straight line is given by
x = sin t
IS
d Let 0 be the ongm, and let P be a point on C with
x-coordinate t, 1 < t ::::;
e.
for all x.
for any odd function
f (x) = x3 cos 2x
d Hence evaluate
60
dx
and F(x)?
y = lnx.
a Sketch the graph of one odd function.
b
f(x)
c Sketch the graph of the curve C whose equation
!'((})=sin(}(3cos2(}-1).
a Show that
59
<
0
Cl!.
7r
"4
7r
3
71
Suppose
f(x)
a Find f"(x).
& Practice Guide
(3'" edition)
=
3~,
e
0::::; x::::; 4.
b Show that the graph y = f(x)
d
has:
Let
g(8)='iTsin28+2cos28.
f'(8)
Show that
a maximum turning point at
ii a non-stationary
c Sketch the graph
(1,~)
e The diagram below shows the graph of g(8).
(2, :2)'
inflection point at
f (x),
y =
y
3
showing the information
y=2
.• ·····2·
found above.
F(x) = - 3(x + 1)
d Show that
= g(8) - 2.
is the antiderivative
eX
7r
of
2"
-,
f(x).
e Show that the area bounded by y = f(x),
~
0.2 0.4 0.6 0.8 1.0 1.2 1.4
-1
the x-axis,
-2
and the line x = 4, is 3 _ 15 units''.
e4
Read the solution of
la sed
ough
o
f' (8)
= 0 from this graph for
0< 8 <-§-.
f(x) = -(x - 2)2 + 4. The point P(a, f(a)),
< a < 4, lies on the graph of y = f (x) such that the
Let
ii Show that this value of 8 maximises
f (8).
normal L to the graph at P passes through the origin.
a Sketch
y =
f (x),
showing the vertex and axes intercepts.
-:E.
b
?
Show that the gradient of the normal at point P is
Let
c Find the equation of the line L in terms of a.
on is
d
C with
a = 3±
Show that
~.
f(x)
=
_X2
+ 4x.
a Find
lim f(x + h) - f(x)
h
value represents.
and
explain
what
this
h~O
e For the case a = 3 + ~,
explain why the area of the
region enclosed by Land
e C,
13+ J, {
For what
curve
on t for
SKILL BUILDER QUESTIONS (CALCULATORS)
1
2a- 4'
The diagram
y = v'3x
circle
ctions
~ 'iT.
a
X2
shows
y
a line
by the
+ y2 = r2,
c Suppose this tangent has positive gradient and passes
through (4, 9). Find the value of k.
y
y = v'3x
>
r
O.
Show that et = ~.
A
_--~--L----ffi~--~-~~-~~x
_-flt--....J...---_x
=
Rectangle ABCD is inscribed under one arch of y = cosx.
Suppose the point C has x-coordinate x.
~;;;---;;
1
V r? -
show that
a Write an expression for A, the area of rectangle ABCD, in
terms of.z.
v'3r2
X2
dx - --.
o
b Find the coordinates of C such that ABCD has maximum
area.
8
~
1
c Using parts a and b above, show that
rx=":
o
V r2
d Hence show that
-
X2
1
10
dx
'iTr2
=-
12
v'3r2
+ --.
3
length of [AB] is 2y,
-DB =
The function
f
is defined by
f (x)
= x -
1
Vx'
8
a For what values of x is the function defined?
v36 - 9x2 dx = 'iT+ 3f.
b Find f' (x) and explain why the function is increasing
over its whole domain.
the diagram alongside, 0 is the
eentre of two concentric circles.
larger circle has radius 1, and
smaller circle has radius x.
:_.\B] and (CD] are parallel
rds of the larger circle and
gents to the smaller circle.
-
at the point
and an arc of the
b By considering A as the
area between the two curves
X2 + y2 = r2 and y = v'3x,
cm
b Find the equation of the tangent to y = f(x)
where x = k.
dx.
+1
Hence state the arc
length I and area A of
the sector shown.
g.
is given by
2 v'2) x)
+ (4 -
_X2
f (x)
y =
c Find any axes intercepts.
d
Write the equations of any asymptotes.
e Sketch the graph of
y =
f(x).
Find the area enclosed by the function, the x-axis, and the
line x = 4.
4 The weight of a radioactive substance is given by the function
and
t
W(t) = 100 x e-2(j grams where t is the time in days, t;;:, O.
28.
Let S be the area of the shaded region.
Show that
S =
'iTy2
a Find the initial amount of radioactive substance present.
= 'iTsin2 8.
b Find the time necessary for half of the mass to decay.
Find the area of the shaded region that is below [AB].
e Hence show that the shaded area that is between [AB] and
[CD] is given by
f( 8) = 'iTsin2 8 + sin 28 - 28.
.
c Find
d
El
dW
--
dt
i
.. sign.
an d mterpret
Its
Discuss W as t increases.
Mathematics
SL - Exam Preparation
& Practice
Guide
(3rc/edition)
e
5 Let
e By using your graph, or otherwise, explain why
Sketch the graph of W against t showing the information
obtained above.
J(x)
= xe-2x2
In x ~::
0 ~ x ~ 2.
on the interval
11 Let
a Find the stationary point of J(x) on this interval.
y = J (x) and the x-axis, from x = 0
2, is revolved through 360 about the x-axis.
c The area between
=
- 2x).
c At what point(s) on the graph of y
0
Write an integral which gives the volume
resulting solid.
= In(xv1
J(x)
> o.
for x
a State the domain of the function.
1- 3x
b Show that l' (x) = (
).
x 1- 2x
b Hence determine the maximum and minimum values of
J (x) on this interval.
to x
e
normal have gradient
of the
J(x)
12 Consider the function
ii Evaluate the integral to find the volume.
- ~?
=
(2 -~)
e-x,
x>
O.
a Find the zero of J(x).
6 An open cylindrical bin is to be
made from PVC plastic and is
to have a capacity of 500 litres.
Suppose the bin has radius r m.
open
I
b Discuss the behaviour
x --> 00.
c Find the position and nature of the stationary point.
d
a Show that the surface area
of the bin
A(r)
1
is given
+ 1fr2
= r
of
by
Sketch the graph of
information.
= J (x),
y
e Find the area enclosed by
y=x-l.
m2.
showing all the above
= J (x)
y
and the line
b Find the dimensions of the bin which minimise the plastic
required.
= J(x),
3x - l.
7 Consider the graph of y
J (x)
J(x)
= 1
a Find
l' (x),
13 Let
= 3x3 + 3x2
-
where
1
_ 1·
and show that the graph of
J (x)
has no turning
points.
and g(x) = eX on the same
set of axes.
Hence determine the number of real roots of the equation
b Draw the graphs of J(x)
a Find the y-intercept.
b Find the stationary points and their nature.
c
+ 2x
Find the location and nature of the inflection point.
(2x - l)eX - 2x
d Sketch the graph of y = J(x), -2 ~ x ~ 1, clearly
labelling the information in a, b, and c above.
e Use technology to find the x-intercepts.
=
O.
c Solve the equation in b.
14 A particle moves from rest along a straight line. Its velocity is
given by v
seconds.
8 A psychologist proposes that the ability of a child to memorise
during the first two years can be modelled by the continuous
function
J(x) = xlnx + 1, where x is the age in years,
0< x ~ 2.
= 20 -
t ms-I,
where
0 is the time in
t)
a Find the acceleration function of the particle.
b Show that the direction of motion changes at
a During which month is the ability at a minimum in the
t
first two years?
=4
seconds.
c Find the total distance travelled in the first nine seconds
of motion.
b When is it a maximum during this period?
d If the particle starts at the origin, find its displacement
9 The population of insects in a colony at time t may be described
N
where N is a positive
by the function
J(t) =
_.1.
1 + 2e 2
constant.
a Find
l' (t)
1" (t)
and show that
l' (t) >
J(t)
for
15
a Determine algebraically, the area of the region bounded by
the curve y
0 for all values of t.
b Find
and show that the maximum rate of growth
of the population occurs when t = In 4. Find the size of
the population at this time.
c Sketch the graph of
function.
t )
0,
showing its
x =
b
=
Jx,
the x-axis, and the lines x
9.
0
16 Let J be the function defined by
J(x)
= ~ __
a Find the value ofx for which J(x)
g(x) = lnx
x
for
0
<x ~
Find g' (x)
and gl/ (x),
and hence show that the graph of
c Describe the behaviour of g(x) for x close to
Mathematics
graph
of 9 (x),
SL - Exam Preparation & Pradice
Guide
showing
(3rd edition)
all
o.
the
above
=0
c Show that if 1"(x)
o.
then
(x - 2)3
= 4x3.
Hence find the coordinates of the point on the graph for
which JI/(x)
decimal places.
Find the coordinates of these points, and find the gradient
of the graph at the point of inflection.
Sketch the
information.
=
of f.
g(x) has one stationary point and one point of inflection.
d
4_.
x - 2
x
b Find the coordinates and the nature of the stationary points
5.
a Find the point where the graph of g(x) cuts the z-axis,
b
1 and
If the area in a is rotated 360 about the x-axis, what is
the volume of the resulting solid?
asymptotes and axes intercepts.
10 Let
=
=
d
Draw the graph of
e
Find algebraically,
O.
Give your answer accurate to two
J, showing
all the above information.
the area enclosed by the graph, the
x-axis, and the lines x
=~
and x
=
1 ~.
a The tangent to
y
=
3x
X2 -
at the point where x
=
has gradient l.
=
X2
-
3x
and the line
= Vi
y
at
apart on a straight road. Each plant emits pollutants into the
atmosphere.
At a point x km from Cinder, on the road towards Puff-Out,
the total concentration of pollutants is given by
8
1
G(x) = x2+ (2-x)2
units, D<x<2.
= x.
y
dx = 2.
25 Two industrial plants, Cinder and Puff-Out, are situated 2 km
b Find the exact area of the region enclosed by the graphs
y
a
b Find the equation of the normal to the curve
the point where x = a.
i Find the value of a.
ii Find the equation of the normal to the curve at this
point.
of
J: Vi
a Find a given that
24
a,
y
y = sinx
'~'~\~.
b
The shaded region has area 0.42 units", Find k.
9
a
Sketch the curve
y
=
3x2
b Find the area between
x ~ 3.
+1
for
the curve
traight line has a velocity of 27 cm S-l as it starts from O.
a Find the velocity function for the particle.
b Find the displacement function for the particle.
c Find the total distance that the particle has travelled when
it comes momentarily to rest for the second time.
Find exactly the solution of
_ (7 -t ) e t-6
dN
--
Find the coordinates of the:
dt
.
ii point of inflection
iii t- intercept.
c Use these coordinates to state:
the time when all the bacteria are dead
ii the maximum number of bacteria reached
sample
in the
iii the time at which the rate of increase of the bacteria
is a maximum.
metres per second.
i metres
t
i turning point
+ 2)2 dx.
+ 2 = O.
d
a Find the value of t for which the acceleration of the particle
is -
of
N
b
A particle moves along the z-axis with velocity
= e-2t
8.
0 ~ t ~ 8.
Show that
c Hence find the volume of the solid obtained when the curve
y = x3+ 2, from its z-intercept to the y-axis, is revolved
through 27r about the x-axis.
vet)
=
occurred, is modelled by the function
millions,
a
3
x3
(8 - t)et-6
The graph of this function
is shown alongside, and
takes into account the
effect
of
the
body's
immune
system,
which
eventually'
kills
the
bacteria.
Its acceleration t seconds later is (6t - 30) cms-2.
b
=
N
A particle moving on the
J (x
3
26 The number of bacteria found in a sample of human tissue
for
t hours after infection
a Find the indefinite integral
(2 ~ x)
0 ~ x ~ 3.
and the z-axis
c Find a constant k such that the area between the curve and
the x-axis for 0 ~ x ~ k is 10 units".
is a point on a straight line.
0 when
c Find the value of x for which the total concentration
pollutants is minimised.
o~
o
=
Show that G'(x)
per second per second. Give your answer to
Copy the graph and indicate clearly which points on the
graph represent your answers to c.
3 significant figures.
b At time t = 0, the particle is 2 metres to the right of the
origin. Find:
i the displacement
27
Y
3
function for the particle
ii the position of the particle at time t = 1 second.
~----~------~ffi---------~--~~x
-7r
The velocity of a boat travelling in a straight line is given by
vet) = 30 - 20e-O.2t m s-1, t? O.
a What is the boat's initial v.elocity?
b
What is the velocity after 2 seconds?
correct to two decimal places.
a The function
Give your answer
-7r ~
b
e
Calculate
positive.
Graph
d.
vet)
Vi
(t)
as t
->
f (x)
= a cos bx,
=
f(x)
at the
ii Find the values of c such that the normal passes
through the origin.
oo?
and show that the acceleration is always
c Sketch y = f (x)
same set of axes.
against t, showing the information from a to
9 Find the formula for the boat's displacement
set)
initial position is 10 m in the positive direction.
has the form
Find the equation of the normal to y
point where x = c.
20 m S-l? Give your answer correct to two decimal places.
What happens to vet)
f
zr.
State the values of a and b.
c How long does it take for the boat's velocity to reach
d
X ~
28
if its
and the normals found in b ii on the
Find the volume of revolution when the region between
y
= sin
x, the x-axis,
and the lines x
=i
and x
=
5611',
is
rotated through 27r about the z-axis,
El
Mathematics
SL - Exam Preparation
& Practice
Guide
(3rd edition)
29
= x + sin
f(x)
Consider
a Sketch y = f(x)
x,
x)
O.
b
for 0 ~ x ~ 27r.
b Find the area enclosed by the curve, the x-axis,
X=
C
and
5
~.
s: I
(3,7)
Y
+ sin x
and y =
2
f(t)
= sin t
- sint
where
= f(t)
a Sketch the graph of y
f'(x)
6 Let
=
..;x
2x - ~
Ix~
=4
=
34
b Write E(X)
seconds.
Find f(x).
= 3.
f'(x)=
8 Points
0 ~ x ~ 2.
< a < 27r
0.15
n
2m
+ n.
in terms of m only.
=
E(X)
2.7.
(6 marks)
A(2, -1,.3)
lie on line L1.
and B(I, 2, -4)
A second line L2 is parallel to
(2x-l)cosx.
(2:)
and is perpendicular
to L1.
Find the minimum value of f(x)
Find a given that 0
m
Section B
+ 2xsinx,
- sinx
c Find m if
0 its velocity is 2 m s"".
b Find the total distance travelled in the interval t = 0 to
t = 5 seconds. Give your answer to two decimal places.
= 2cosx
= x)
a Find the value of 3m
k
At t
S-2.
a Find the velocity at t
b
fU)
and
7 Random variable X has the following probability distribution:
4 cos t m
a Show that
Find the coordinates of A.
(7 marks)
At time t seconds a particle is moving in a straight line with
f(x)
b Find a.
(4 marks)
-k
33 Let
+ k.
over the given domain.
The shaded area is
24 units". Find k.
acceleration
a x - h)2
0 ~ t ~ 2.
P(X
32
and y-intercept
can be written in the
X
a Find h and k.
x
X2.
b Find the area enclosed by the curve, the t-axis, and t = 2.
31
:
form
C
30 Consider
for
3,)
2..5.
and g(x) = X2 meet at
Find a correct to 3 decimal places.
d Find the area enclosed by y = x
-..ana
a quadratic function with
.-
~
= x + sinx
f(x)
= a, a > O.
-=
(6 marks)
~.
Suppose
x
Hence, find
-7r ~ (J ~-.
--->
for 0 ~ x ~ 2.
Jaa+2
and
sin x dx
=
a Find BA.
b Find a vector equation for L1 in the form
0.3.
r = p
+ tq.
c Find m.
C(2, 3, k),
find a vector
e If L1 and L2 intersect, find the value of k.
(16 marks)
d
TRIAL EXAMINATION
1
Paper 1 • Mo calculators
(1 hour 30 minutes)
If L2 passes through the point
equation for L2.
Section A
y
The first three terms of an infinite geometric
24, x, and 6, where x > O.
a Write down x.
C
Find
d
U5.
sequence
B
are
b Find the common ratio r,
6
Find the sum to infinity of the sequence.
x
A
(7 marks)
2 Let
h(x)=e-xcosx.
o~
b Find h'(~).
c Find the equation of the tangent to the curve at the point
where
x
= ~.
(8 marks)
d
<
<:
b
A
;~n~' y~h:nd :;obability
that Bevan
target.
c Find
hits
the
1
-3
A'
Z
~
Z
P(A' I E).
a If
sin 28 = tan 8,
cos8 =
Mathematics
show that either
±~.
SL - Exam Preparation & Pradice
(.;f (x
f' (x) =
By considering
and B.
tt
(.;f (x
sin
f' (x) =
- 1)).
0, find the x-coordinates
sin 8 = 0
f(x)
=
2x - 1 and
b
g
0
f
function
d
h(x).
=
3x2
-
1.
through
Find
to produce the
h(x).
in the form
a(x - h)2
+ k.
State the coordinates of the vertex of:
i g(x)
e
g(x)
is translated
c Write h(x)
or
tt .
(gof)(x).
a Find
E'
of A
(15 marks)
10
BE'
- 1)) + 2,
b Find the x-intercept.
e Find the area of the shaded region in terms of
(7 marks)
4
-4 cos
for the events
R
down the values
=
A and B are turning points of the function.
c Show that
Let A be the event of Adrian hitting the target,
and E be the event of Bevan hitting the target.
The tree diagram below shows the probabilities
x ~ 6.
a Find the y-intercept.
3 Adrian and Bevan are competing in an archery competition.
A .'"~:'~
f (x)
The graph shows the function
a Find h'(x).
Find c given that
ii
h(x).
y = 2x
+c
is a tangent to h(x).
(14 marks)
Guide (3rd edition)