NOTES The Mathematical Method via Onevariable Calculus Ma1a Fall 2010
DINAKAR RAMAKRISHNAN TausskyToddLonergan Professor of Mathematics Oce: 278 Sloan Extension: 4348
1
0
Proofs in Mathematics
It s not dicult to write mathematical proofs of statements,
7
Improper Integrals, Exp, Log, Arcsin, and
the Integral Test for Series
We have now attained a good level of understanding of integration of nice
functions f over closed intervals [a, b]. In practice one often wants to extend
the domain of integration an
6
Fundamental Theorems, Substitution, Integration by Parts, and Polar Coordinates
So far we have separately learnt the basics of integration and dierentiation.
But they are not unrelated. In fact, they are inverse operations. This is
what we will try to e
5
Integration
We will rst discuss the question of integrability of bounded functions on
closed intervals, followed by the integrability of continuous functions (which
are nicer), and then move on to bounded functions with negligible discontinuities.
The m
4
Dierential Calculus
4.1
Basic Notions
Denition 4.1 The function f dened on a neighborhood of a R is called
dierentiable at a if the limit
f (a + h) f (a)
h0
h
lim
exists. It is called the derivative of f at a and is denoted as f (a) (or as
df
(a).)
dx
R
3
Limits of functions, Continuity
After introducing the basic notions on functions, limits and continuity, we
will go on to Bolzanos theorem, and the Intermediate Value Theorem (IVT)
which follows from it, as well as the Extremal Value Theorem (EVT).
3.1
2
Sequences and series
We will rst deal with sequences, and then study innite series in terms of
the associated sequence of partial sums.
2.1
Sequences
By a sequence, we will mean a collection of numbers
a1 , a2 , a3 , . . . , an , an+1 , . . . ,
which is
1
The Real Number System
The rational numbers are beautiful, but are not big enough for various purposes, and the set R of real numbers was constructed in the late nineteenth
century, as a kind of an envelope of Q. (More later on this.) For a nonconstruct
Calculus Of One And Several Variables And Linear Algebra
MA 1a

Fall 2011
1
The Real Number System
The rational numbers are beautiful, but are not big enough for various purposes, and the set R of real numbers was constructed in the late nineteenth
century, as a kind of an envelope of Q. (More later on this.) For a nonconstruct
Calculus Of One And Several Variables And Linear Algebra
MA 1a

Fall 2011
NOTES
The Mathematical Method via
Onevariable Calculus
Ma1a
Fall 2010
DINAKAR RAMAKRISHNAN
TausskyToddLonergan Professor of Mathematics
Oce: 278 Sloan
Extension: 4348
1
0
Proofs in Mathematics
It s not dicult to write mathematical proofs of statements,
Calculus Of One And Several Variables And Linear Algebra
MA 1a

Fall 2011
3
Limits of functions, Continuity
After introducing the basic notions on functions, limits and continuity, we
will go on to Bolzanos theorem, and the Intermediate Value Theorem (IVT)
which follows from it, as well as the Extremal Value Theorem (EVT).
3.1
Calculus Of One And Several Variables And Linear Algebra
MA 1a

Fall 2011
4
Dierential Calculus
4.1
Basic Notions
Denition 4.1 The function f dened on a neighborhood of a R is called
dierentiable at a if the limit
f (a + h) f (a)
h0
h
lim
exists. It is called the derivative of f at a and is denoted as f (a) (or as
df
(a).)
dx
R
8
Approximations, Taylor Polynomials, and
Taylor Series
Polynomials are the nicest possible functions. They are easy to dierentiate
and integrate, which is also true of the basic trigonometric functions, but
more importantly, polynomials can be evaluated
9
9.1
Complex numbers and functions, factoring,
and integration via partial fractions
Complex Numbers
Recall that for every nonzero real number x, its square x2 = x x is always
positive. Consequently, R does not contain the square roots of any negative
n
2
D. Ramakrishnan
0. Proofs in Mathematics
It is not dicult to write mathematical proofs of statements, but
one needs some experience, especially since the high school courses do
not stress this aspect of Math. This is essential for the mathematical metho
1
The Real Number System
The rational numbers are beautiful, but are not big enough for various purposes, and the set R of real numbers was constructed in the late nineteenth century, as a kind of an envelope of Q. (More later on this.) For a nonconstruct
2
Sequences and series
We will rst deal with sequences, and then study innite series in terms of the associated sequence of partial sums.
2.1
Sequences
By a sequence, we will mean a collection of numbers a1 , a2 , a3 , . . . , an , an+1 , . . . , which is
3
Limits of functions, Continuity
After introducing the basic notions on functions, limits and continuity, we will go on to Bolzanos theorem, and the Intermediate Value Theorem (IVT) which follows from it, as well as the Extremal Value Theorem (EVT).
3.1
4
4.1
Dierential Calculus
Basic Notions
Denition 4.1 The function f dened on a neighborhood of a R is called dierentiable at a if the limit f (a + h) f (a) h0 h lim exists. It is called the derivative of f at a and is denoted as f (a) (or as df (a).) dx R
5
Integration
We will rst discuss the question of integrability of bounded functions on closed intervals, followed by the integrability of continuous functions (which are nicer), and then move on to bounded functions with negligible discontinuities. The m
6
Fundamental Theorems, Substitution, Integration by Parts, and Polar Coordinates
So far we have separately learnt the basics of integration and dierentiation. But they are not unrelated. In fact, they are inverse operations. This is what we will try to e
7
Improper Integrals, Exp, Log, Arcsin, and the Integral Test for Series
We have now attained a good level of understanding of integration of nice functions f over closed intervals [a, b]. In practice one often wants to extend the domain of integration an
1
The Real Number System
The rational numbers are beautiful, but are not big enough for various purposes, and the set R of real numbers was constructed in the late nineteenth
century, as a kind of an envelope of Q. (More later on this.) For a nonconstruct
6
Fundamental Theorems, Substitution, Integration by Parts, and Polar Coordinates
So far we have separately learnt the basics of integration and dierentiation.
But they are not unrelated. In fact, they are inverse operations. This
is what we will try to e
7
Improper Integrals, Exp, Log, Arcsin, and
the Integral Test for Series
We have now attained a good level of understanding of integration of nice
functions f over closed intervals [a, b]. In practice one often wants to extend
the domain of integration an
8
Approximations, Taylor Polynomials, and
Taylor Series
Polynomials are the nicest possible functions. They are easy to dierentiate
and integrate, which is also true of the basic trigonometric functions, but
more importantly, polynomials can be evaluated
Calculus Of One And Several Variables And Linear Algebra
MA 1a

Fall 2011
2
Sequences and series
We will rst deal with sequences, and then study innite series in terms of
the associated sequence of partial sums.
2.1
Sequences
By a sequence, we will mean a collection of numbers
a1 , a2 , a3 , . . . , an , an+1 , . . . ,
which is
Calculus Of One And Several Variables And Linear Algebra
MA 1a

Fall 2011
5
Integration
We will rst discuss the question of integrability of bounded functions on
closed intervals, followed by the integrability of continuous functions (which
are nicer), and then move on to bounded functions with negligible discontinuities.
The m
Calculus Of One And Several Variables And Linear Algebra
MA 1a

Fall 2011
HOMEWORK N.6
MA1A FALL 2011
(1) Prove that, for n = 0, 1, 2, . . . ,
1
xn+ 2 ex dx =
0
(2n + 1)!
.
n! 22n+1
(2) Prove that
1
1 x2 dx =
1
.
2
Compute then the integral
2
(x 3) 4 x2 dx.
2
(3) Let f (x) = x x2 and g (x) = x. Find the value of such
that the
Calculus Of One And Several Variables And Linear Algebra
MA 1a

Fall 2011
HOMEWORK N.7
MA1A FALL 2011
(1) Show that the following orthogonality relations between trigonometric functions hold: for all integers n and m with n = m
2
sin(nx) cos(mx) dx = 0
0
2
sin(nx) sin(mx) dx = 0
0
2
cos(nx) cos(mx) dx = 0
0
while for all n = 0