Calculus
Writing Assignment: Area Between Two Curves
Each problem is worth 5 points.
Total Points: 50
Answer each of the problems. Make sure to show all your work.
1) Find the area bounded b the curve
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 a
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 b
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
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=5.18
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
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 studen
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
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
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
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 t
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 integ
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
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 equat
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 doub
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.
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
Normal differential comparisons (ODEs) emerge in numerous settings of arithmetic and science, (social
and additionally regular). Scientific depictions of progress use differentials and subordinates. D
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 intric
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
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 comp
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 repr
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 edg
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
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.
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)
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
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
= limsi
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.