MTH501 Linear Algebra solved final Term Papers for
Final Term Exam Preparation
For any subspace W of a vector space V, which one is not the axiom for subspace.
0 must be in W.
For all u, v in W and u v must be in W.
For all u, v in W and u.v must be in W.
Applications Of Differentiation
 Related rates
 Increasing functions
 Decreasing functions
 Concavity of functions
Related Rates
Related rates are real life problems.
 These involve finding the rate at which one
quantity changes w.r.t. to another qu
Limit problems of this type can be converted to the form of 0/0 by writing
Indeterminate forms of type 0^0, ^0 and 1^ and 
Limits of the form
Give rise to indeterminate form of type 0^0, ^0 and 1^ .
All these types are treated by first introducing a depe
Practice Exercise For Lecture 13
Q1.
Determine whether
1 cos x
exists or not?
x 0
x
Find the interval on which the given function is continuous:
x3
y 2
x 3x 10
(Ans.
)
(, 2) (2,5) (5, )
lim
Q2.
Q3.
Find the interval on which the given function is continuo
MTH101: Practice Exercise
Lecture No.2: Absolute Value
Q.No.1
Solve for
x7
8.
4 x
x,
Answer:
x
25
39
and x
9
7
Q.No.2
Is the equality
valid for all values of
b b
4
2
reasoning.
Q.No.3
Find the solution for:
Answer:
x 25 x 5
2
x5
Q.No.4
Solve for x:
6 x
Practice Questions for Lecture No. 18
The Chain Rule
Question 1:
Differentiate
with respect to x using the chain rule.
y 5 x 3x x
3
2
Question 2:
Differentiate
y tan x cos x
with respect to x using the chain rule.
Question 3:
Differentiate
y 3sin 2 x5 4co
Operations on Functions
Like numbers, functions can be OPERATED upon
Functions can be added
Functions can be subtracted
Functions can be multiplied
Functions can be divided
Functions can be COMPOSED with each other
Arithmetic Operations on Functions
Limits: A Rigorous Approach
In this section we will talk about
Formal Definition of Limit
 Lefthand and Righthand Limits
So far we have been talking about limits informally
We haven't given a FORMAL mathematical definitions of limit yet
We will giv
The Definite Integral
 Definition of Definite Integral
Definite Integral of continuous functions with nonnegative values
Definite Integral of continuous functions with negative and positive values
Definite Integral of functions with discontinuities
P
Implicit Differentiation
The method of Implicit Differentiation
Derivatives of Rational Powers of x
Differentiability of Implicit functions
Implicit differentiation
Consider this equation xy 1
We want to find its derivative or in other words dy
dx
B
Relative Extrema
Relative Maxima
Relative Minima
Critical Points
First Derivative test
Second Derivative test
Graphs of Polynomials
Graphs of Rational functions
Relative Maxima
Most of the graphs we have seen have ups and downs, much
like Hill and
Moving on. Since we all like theorems so much, here is another one!
Before we prove this theorem, Let's look at a few examples of it to get a
practical feel for what the theorem is saying.
Example
4 4
We used this result earlier without proving it when w
First Fundamental Theorem of Calculus
1st fundamental theorem of calculus.
Relationship between definite and indefinite integrals.
Mean Values theorem for Integrals.
Average Values of a function.
This is the 1st fundamental theorem of calculus.
This
Work and The Definite Integral
In this lecture we will discuss
 Work done by a constant force
 Work done by a variable force
 Fluid Pressure
 Pascals Principle
Work done by a constant force
If an object moves a distance d along a line while a CONSTAN
LIMITS
Calculus was motivated by the problem of finding areas of plane
regions and finding tangent lines to curves
In this section we will see both these ideas
We will see how these give rise to the idea of LIMIT. We will
look at it intuitively, withou
In this process, we see that we have a formula to computed volume of a
solid which is got from revolving a region around the YAXIS
Cylindrical Shells centered on the yaxis
Example 1
Use cylindrical shells to find the volume of the
solid generated by th
COORDINATE PLANES AND GRAPHS
In this lecture we will discuss
So Let's go into details now. We begin with the Coordinate plane.
Just as points on a line can be placed in onetoone correspondence with
the real numbers, so points in the PLANE can be placed
Limits And continuity of Trigonometric functions
Continuity of Sine and Cosine functions
Continuity of other trigonometric functions
Squeeze Theorem
Limits of Sine and Cosine as x goes to + infinity
You will have to recall some trigonometry. Refer to
Continuity
Develop the concept of CONTINUITY by examples
Give a mathematical definition of continuity of functions
Properties of continuous functions
Continuity of polynomials and rational functions
Continuity of compositions of functions
The Interm
The Chain Rule
Derivative of Composition of Functions (Chain Rule)
Generalized Derivative formula
More Generalized Derivative formula
An Alternative approach to using Chain Rule
Derivative of Composition of Functions (Chain Rule)
Suppose we have two
Area as Limits
Definition of Area
Some technical considerations
Numerical approx of area
Look at the figure below
In this figure, there is a region bounded below by the xaxis, on the sides
by the lines x= a and x = b, and above by a curve or the gra
Graphs of Functions
Represent Functions by graphs
Visualize behavior of Functions through graphs
How to use graphs of simple functions to create graphs of
complicated functions
Definition of Graph of a function
We saw earlier the relationship between
Integrations
In this lecture we will look at the beginnings of the OTHER major
Calculus problem.
The Area Problem
Antiderivatives (Integration)
Integration formulas
Indefinite Integral
Properties of Indefinite Integral
The Area Problem
Given a cont
Derivatives of Logarithmic and Exponential Functions and Inverse
functions and their derivatives
Derivative of the logarithmic function f ( x ) log b ( x)
Derivative of the Natural log functions f ( x ) ln( x )
Logarithmic Differentiation
Derivatives
Newtons Method, Rolles Theorem, and the Mean
Value Theorem
Newtons method for approximating solutions to f(x)=0
Some difficulties with Newtons method
Rolles theorem
Mean Value Theorem
Newtons method for approximating solutions to f (x)=0
We have seen
The Derivative
Definition of the Derivative
Notational stuff about the Derivative
Existence of the Derivative
Relationship btw differentiability and continuity
Derivatives at the endpoints of the interval
In the previous lecture we saw that the slop
Sigma Notation
Sigma Notation
Changing Index of Summation
Properties of Sigma Notation
Summation Formulas
Sigma Notation
Sigma notation is used to write lengthy sums in compact form.
Sigma or is an Upper case letter in Greek.
This symbol is called
Integration by substitution
usubstitution
This is like the Chain Rule we saw for differentiation. The idea is to
Integrate functions that are composition of different functions.
Suppose that G is an antiderivative of a function f. So
d
G (u ) f (u )
Volume by Slicing; Disks and Washers
Definite Integrals to find Volumes of three dimensional solids
Cylinders
The method of Slicing
Volumes by cross sections perpendicular to the x axis
Volumes by cross sections perpendicular to the y axis
Volumes o
LHopitals Rule and Indeterminate forms
LHopitals Rule and 0 / 0
Indeterminate form of type
/
Indeterminate form of type 0.
Indeterminate forms of type 0^0, ^0 and 1^ and

If we look at
x2 4
sin x
lim
and lim
x 2 x 2
x 0
x
The top and the bottom bo