Faculty of Actuaries
Institute of Actuaries
EXAMINATION
26 April 2010 (am)
Subject CT5 Contingencies
Core Technical
Time allowed: Three hours
INSTRUCTIONS TO THE CANDIDATE
1.
Enter all the candidate and examination details as requested on the front of you
EXPONENTIAL DIFFERENTIATION
Recall:
c
b
ab
b
-c
b -c
1. ab a c = a b + c
3. ( a b ) = a bc = ( a c )
2. a c = a a = a
4. Equations involving ln x and e x
If y = ln x = loge x , y = log e x e y = x
5. eln x = x
Proof:
Let y = eln x ;
ln y = ln eln x ; ln y
SMA 104: CALCULUS I
BACKGROUND INFORMATION
Functions
Example 1:
Suppose we have 2 functions, f and g, both having as a domain and suppose one of them
squares each member of a domain and the other doubles each member of a domain.
We wrote f(x) to represent
LOGARITHMIC FUNCTIONS
Recall:
1. log1 = 0
b
2. log b = 1
b
. logb ac = logb a + logb c
4. log b
a
= log b a - log b c
c
log b a r = r log b a
1
2
1
log b a
2
7. If log c a = y c y = a
Taking logs on both sides
e.g.
log b a =
6. log b
1
= log b c -1 = -1lo
IMPLICIT FUNCTIONS
Explicit functions if x e.g y = x 2 - 5 x + 4
Here, y is given as an expression in x .If however y is given implicity by an equal such as
x = y4 - y + 1
He cannot express y in terms of x.
Consider
1
x = y2 , y = x 2
1
dy 1 - 2
1
\ = x =
1.
5.184
Solution
3
givena linear scale factor :l . s . f = ,
5
3
3
3
27
(
)
volume scale factor : v . s . f = l . f . s =
=
5
125
()
thus , volume after dilation :V ' =v . s . f V
V '=
21.
27
24=5.184 c m 3
125
Solution
making P ( centre of dilation ) th
The median value of a home in a particular market is decreasing exponentially. if the value of a home
was initially $240,000. then its value two years later is $235,000. Answer the following.
6) Write a differential equation that models this situation. Le
Calculus
Writing Assignment: Growth and Decay Models
Each portion of each problem is worth 5 points.
Total Points: 50
Answer each of the problems. Make sure to show all your work.
The number of students at a school increases at a rate proportional to its
In Problems 11-15, let R be the region bounded by the lines y = —2x — 4. y = 6, and x = —2. Each problem
will describe a solid generated by rotating R about an axis, Find the volume of that solid (you :19 evaluate
these problems).
11) The solid is generat
In Problems 6-10, let R be the region bounded by y = x2 + 1 and y = x + 1.Each problem will describe a
solid generated by rotating R about an axis. Write an integral expression that can be used to ﬁnd the
volume of the solid (do not evaluate).
6) The soli
Calculus
Writing Assignment: Volume of Solids of Revolution
Each problem is worth 5 points.
Total Points: 75
Answer each of the problems. Make sure to show all your work.
In Problems 1—5, let R be the region bounded by y = J)? , y = 1, and x = 4. Each pro
In Problems 11-15, let R be the region bounded by the lines y = 4 — x, x = —2, and the x-axis. Each
problem will describe the cross sections of a solid. Find the volume of that solid (you Q evaluate these
problems).
11) The cross sections are squares perp
in Problems 6-10. let R be the region bounded by the semicircle x = J4— y2 and the y-axis. Each
problem will describe the cross sections of a solid that are perpendicular to the y-axis. Write an integral
expression that can be used to find the volume of t
Limits
The concept of limits of a function is one of the fundamental ideas that distinguishes Calculus
from other areas of mathematics e.g. Algebra or Geometry.
If f(x) becomes arbitrarily close to a single number L as x approaches a from either side, the
PARAMETRIC EQUATIONS
Consider x = f ( t ) and y = g ( t ) ,then x and y are both functions of (t).These equations are called
parametric equations for x and y and the variable t is called a parameter.
Example of parametric equation is x = 2t , y = t 2 - 1
Normal differential comparisons (ODEs) emerge in numerous settings of arithmetic and science, (social
and additionally regular). Scientific depictions of progress use differentials and subordinates. Different
differentials, subsidiaries, and capacities ge
In science the Laplace change is a vital change named after its pioneer Pierre-Simon Laplace (/lpls/).
It takes an element of a positive genuine variable t (regularly time) to a component of an intricate
variable s (recurrence).
The Laplace change is fund
Section 7.4 Matrix Representations of Linear Operators
Definition.
B : V Rn defined as
B (c1v1+c2v2+ cnvn) = [c1 c2 cn]T.
Property:
[u + v]B = [u]B + [v]B and [cu]B = c[u]B for all u, v V and scalar c.
1
Section 7.4 Matrix Representations of Linear Opera
Notes 15 Linear Mappings and Matrices
In this lecture, we turn attention to linear mappings that may be neither surjective nor injective. We show that once bases have been chosen, a linear map is completely determined
by a matrix. This approach provides t
Denition 3. The dimensions of the kernel and
image of a transformation T are called the transformations rank and nullity, and theyre denoted
rank(T ) and nullity(T ), respectively. Since a matrix represents a transformation, a matrix also has
a rank and n
APPLICATIONS OF MAXIMUM AND MINIMUM VALUES
Examples:
1.The figure below represents a rectangular sheet of metal measuring 8cm by 5cm.Equal squares
of side xcm are removed from each corner, and the edges are then turned up to make an open box
of volume vcm
KINEMATICS
VELOCITY AND ACCELERATION
The velocity (v) is instantaneous rate of change of position. The velocity of a moving particle can
be positive or
a negative, depending on whether the particle is moving in the
positive or negative direction along a l
RATES OF CHANGE
dy dy dt
The identity
=
is useful in solving certain rate of change problems.
dx dt dx
dy
dx
=rate of change of y w.r.t. time t;
=rate of change of x w.r.t time t.
dt
dt
Examples:
1. A spherical balloon is blown up so that its volume incr
APPLICATIONS OF DIFFERENTIATION
EQUATIONS OF TANGENTS AND NORMALS
Definition: A normal to a curve at a point is the straight line through the point at right angles to
the tangent at the point.
y=f(x)
normal
tangent
Finding the equations of tangents and no
THE DERIVATIVE OF FUNCTIONS
Definition:The d.erivative of the function f is the function f ' defined by
f ( x + h) - f ( h )
f ' ( x ) = lim
for all x for which this limit exists.
h0
h
The function f is differentiable at x = a if lim f ( x ) = f ( a ) exi
DERIVATIVES OF TRIGONOMETRIC FUNCTIONS
Recall
1. limsinh = 0
lim cosh = 1;
h0
h 0
sinh
cosh
sinh
= lim
= 1 (by L' Hospital Rule); \ lim
=1
h0 h
h 0
h0 h
1
lim
Also,
lim
h 0
1 - cosh
sinh
= lim
= limsinh = 0
h 0 1
h0
h
Recall also the factor formulae
P+Q
P
Calculus
Writing Assignment: Volume of Solids Using Cross Sections
Each problem is worth 5 points.
Total Points: 75
Answer each of the problems. Make sure to show all your work.
In Problems 1-5, let R be the region bounded by y = x2 + 3, y = 2x, x = 0, an
The velocity of a particle varies directly as the product of its position and time squared. The particle has
known positions s(0) = 3 and 3(2) = 5. Answer the following.
11) Write a differential equation that models this situation. Let 8 represent the pos
Suppose the cost of an object appreciates at a rate inversely proportional to the sum of its squared cost
and 300. The object cost $240 when first purchased, but is worth $45 more after one year. Answer the
following.
6) Write a differential equation that
Renaissance (Renaissance)
Renaissance (Renaissance) (Renaissance), the era of intellectual and artistic flowering that
began in Italy in the 14th century and reached a peak in the 16th century and had a significant
influence on European culture. The term
Solution of systems of linear algebraic equations
The system of two linear equations with two unknowns is as follows:
a 11 x 1 + a 12 x 2 = b 1
a 21 x 1 + a 22 x 2 = b 2
where x 1 and x 2 - unknown; a 11, a 12, a 21, a 22 - coefficients of the system; b 1
Solution of linear equations with examples
Equation with one unknown, which, after opening brackets and similar terms, takes
the form
AX + b = 0, where a and b are arbitrary numbers, is called a linear equation with one
unknown.
For example, all of the eq
Sum (Difference) matrices.
The sum effect of simple matrices.
NOT all the matrices can be added. To make adding (subtracting) matrices, it is
necessary that they are identical in size.
For example, if a given matrix "two by two", it can add "two by two" o