Introduction to Finance and Budgeting
Yves Alvin Carolasan
Laurichelle Bunag
Maureen Cario
Abstract
Proper planning and management of flow of money
can lead to any business success.
This chapter discusses the basics of financing and
budgeting. It introduc
MATH 37 C
Reading Assignment 6
Given tan C wherein there is NO special value for that will satisfy the equation, consider
Case 1:
C is positive.
Arctan C ( P is on the first quadrant) or
Arctan C ( P is on the third quadrant)
Case 2:
C is negative.
Arc
MATH 37 C
Reading Assignment 5
Case 4:
Repeated Quadratic Factors
If Q has repeated linear factors, then for each quadratic factors appearing r
times, ax 2 bx c , write r partial fractions of the form
r
A1 x B1
A2 x B2
Ar x Br
2
r
2
ax bx c ax 2 bx c
MATH 37 C
Reading Assignment 4
Given cot m u csc n u du ,
a. If m is odd, save a factor of csc u cot u and express the remaining factors in terms of
cscu .
b. If n is even, save a factor of csc 2 u and express the remaining factors in terms of cot u .
c.
Graph of a Polar Equation
Unit 4.2
Graphs of Polar Equations
The graph of a polar equation F r , 0,
consists of all points r , whose coordinates
satisfy the equation.
Lines
Form:
C
Graph:
line passing through the pole and
making angle of measure C radian
MATH 37 C
Reading Assignment 3
HYPERBOLIC FUNCTIONS
Definitions of the Hyperbolic Functions
e x e x
2
x
e e x
cosh x
2
sinh x
tanh x
cosh x
1
sinh x
1
sech x
cosh x
cosh x
coth x
sinh x
sinh x
csch x
Properties of the Hyperbolic Functions
1.
f x sin
Vector
Unit 5.2
Vectors
Vector in Plane and Space
1
Example 5.2.1
2
Magnitude
Operations on Vectors
Example 5.2.2
3
Unit Vectors
Theorem
4
Example 5.2.3
Reading Assignment
Required Exercises
End of Unit 5.2
5
Derivation of the Inverse Hyperbolic Trig Functions
y = sinh1 x. By definition of an inverse function, we want a function that satisfies the condition x = = = = e
2y
2ey x =  2xe  1 =
y
sinh y ey  ey by definition of sinh y 2 ey  ey ey 2 ey 2y e 1
Dot Product
Unit 5.3
Dot Product
Example 5.3.1
Theorem
Example 5.3.2
1
Theorem
Theorem
Example 5.3.3
Example 5.3.4
2
Projections
Theorem
Example 5.3.5
Required Exercise
3
End of Unit 5.3
4
MATH 37 C
Reading Assignment 9
Theorem:
The magnitude A B is equal to the area of the parallelogram determined by A
and B .
Example:
Find the area of the parallelogram determined by A 1,0, 1 and B 2,0,0 .
Solution:
i j k
0 1
1 1
1 0
j
k 0i 2 j 0k
A B 1 0
430

CHAPTER 6 APPLICATIONS OF INTEGRATION
we would have obtained the integral
Vy
h
0
L2
L2h
2
2 h y dy
h
3
EXAMPLE 9 A wedge is cut out of a circular cylinder of radius 4 by two planes. One
plane is perpendicular to the axis of the cylinder. The other
The Inverse Hyperbolic Function and Their Derivatives
1. The Inverse Hyperbolic Sine Function a) Definition The inverse hyperbolic sine function is defined as follows: y = sinh 1 x iff sinh y = x with y in (,+) and x in (,+) f ( x ) = sinh 1 x : ( , ) ( ,
436

CHAPTER 6 APPLICATIONS OF INTEGRATION
V EXAMPLE 4 Find the volume of the solid obtained by rotating the region bounded by
y x x 2 and y 0 about the line x 2.
SOLUTION Figure 10 shows the region and a cylindrical shell formed by rotation about the
li
Traces
Unit 5.6
Cylinders and
Quadric Surfaces
Cylinders
1
Example 5.6.1
Quadric Surface
Ellipsoid
Example 5.6.2
2
Elliptic Paraboloid
Example 5.6.3
Elliptic Cone
Example 5.6.4
3
Reading Assignment
Required Exercises
End of Unit 5.6
4
A92

APPENDIX I ANSWERS TO ODDNUMBERED EXERCISES
15. 1615
y
y
x=1
5. 1 1e
x=1
y
x=
(1,1)
y
y=e_
1
0
0
x
x
0
(1,_1)
0
x
1
x
x
7. 16
y
17. 2930
y
y=4(x2)@
y
y=4x+7
7
y
(1,4)
x=
(1,1)
(3,4)
0
x
2
2
x
x
y=
_1
0
x
0
x
9. 212
x=_1
19. 7
21. 10
23. 2
25. 7 1
MATH 37 C
Reading Assignment 10
4.
x2 y2 z 2
1
a 2 b2 c2
Graph:
Elliptic Hyperboloid of One Sheet
Example:
x2 z 2 y 2 1
xy plane:
z 0 x2 y 2 1
Graph: Hyperbola
yz plane:
x 0 z2 y2 1
Graph: Hyperbola
xz plane:
y 0 x2 z 2 1
Graph: Ellipse/Circle
Graph:
1.
APPENDIX I ANSWERS TO ODDNUMBERED EXERCISES

A91
3. 2
Exercises
1. (a) 8
(b) 5.7
y
y
y=
2
y=
2
6
6
0
0
x
2
x
2
5. 162
y
y
y=9
x
4
5. 3
7. f is c, f is b, x0 f t dt is a
9
21
9. 37
11. 10
13. 76
15. 4
17. Does not exist
1
19. 3 sin 1
21. 0
23. 1x 2 ln x
MATH 37 C
Reading Assignment 7
Example: Identify and sketch the graph of the following equations.
1. r 4 3cos
a 4
b 3
Graph: Limaon with a Dent (symmetric with respect to the polar axis)
Since the graph is symmetric with respect to the polar axis, it is s
Plane in Space
Unit 5.5
Equations of Lines
and Planes
Example 5.5.1
1
Example 5.5.2
Theorem
Theorem
Example 5.5.3
2
Line in Space
Example 5.5.4
3
Example 5.5.5
Required Exercises
End of Unit 5.5
4
Unit 4
Polar Coordinate System
Unit 4.1
Cartesian and Polar
Coordinate System
Polar Coordinate System
Polar Coordinates
Plotting Polar Points
1
Example 4.1.1
Cartesian and Polar Coordinates
2
Cartesian and Polar Coordinates
Example 4.1.2
Example 4.1.3
3
E
Hyperbolic Functions
Unit 1.5
Derivatives of and
Integrals Yielding
Hyperbolic Functions
Circular and Hyperbolic Functions
Properties of Hyperbolic Functions
1
Hyperbolic Identities
2
Theorem
Example 1.5.1
Theorem
Example 1.5.2
3
Required Exercises
Readin
Example 2.2.1
Unit 2.2
Integration of Special Powers
of Trigonometric Functions
Pythagorean Identities
Integrals of Powers of Sine and Cosine
1
Example 2.2.2
Integrals of Powers of
Secant and Tangent
2
Example 2.2.3
Reading Assignment
Required Exercises
3
Unit 1.3
Logarithmic
Differentiation
Properties of Logarithm
Theorem
Example 1.3.1
1
Logarithmic Differentiation
Example 1.3.2
Required Exercises
End of Unit 1.3
2
Hyperbolic Sine Function
Unit 1.6
Derivatives of Inverse
Hyperbolic Functions
Inverse Hyperbolic Sine Function
Hyperbolic Cosine Function
1
Inverse Hyperbolic Cosine Function
Other Inverse Hyperbolic Functions
Other Inverse Hyperbolic Functions
Theorem
2
Exponential Function
Unit 1.4
Derivatives of and
Integrals Yielding
Exponential Functions
Theorem
Example 1.4.1
Example 1.4.2
1
Theorem
Example 1.4.3
Example 1.4.4
Required Exercises
End of Unit 1.4
2
Unit 1
Derivatives of and
Integrals Yielding
Transcendental Functions
Unit 1.1
Derivatives of and
Integrals Yielding Inverse
Circular Functions
Recall
Theorem
1
Theorem
Example 1.1.1
Example 1.1.2
Example 1.1.3
Theorem
2
Theorem
Example 1.1.4
3
Required E
First Fundamental Theorem of Calculus
Unit 1.2
Derivatives of and Integrals
Yielding Logarithmic
Functions
Example 1.2.1
The Natural Logarithmic Function
Theorem
1
Example 1.2.2
Theorem
Example 1.2.3
2
Example 1.2.4
Theorem
Example 1.2.5
3
Theorem
Example
LaborManagement Statesmanship
The demands of the public (i.e consumers,
stockholders, all citizens) seem to be forcing decisions
upon management and labor leaders. The public greatly
influence the decisions of the management as well as
the labor leaders.