M athem atical analysis I

Course Name
Semester Syllabus
We
ek
Topic details
Total
hours
Lecture
hours
Practic 
e hours
The algebra o f the limits, examples. Standard limits. 
Standard limits. Examples. Bounded sequences.' 
The N eper number. Example.
4
Lim its o f functions when x tends to infinity. 
Exam ples. Lim its o f functions for x\to-infty. Limits 
o f functions for x \to a. Examples.
Powers, Exponentials, Hyperbolic functions and 
inverse hyperbolic functions.
Com parison theorem . Limits and sequences. 
A lgebra o f limits.
10
6
4
5
Lim its o f elem entary functions. Examples. Limit 
Sin(x)/x, as x tends to 0. Fundamental limits and 
Landau symbols.
The fundam ental lim its using the Landau symbols. 
O ther fundam ental limits. How to use the Landau 
symbols to com pute limits. Infinite and 
infinitesim al functions.
10
6
4
6
A symptotes. Examples. Continuity o f a function. 
The algebra o f continuous functions-Examples. 
Examples. Converging subsequences. M axim a and 
m inim a o f continuous functions on a closed 
interval. The W eierstrass theorem.
The intermediate value theorem -Exam ples-Inverse 
o f continuous function. Examples and exercises.
10
6
4
7
T he derivative o f a function. Geometrical and 
kinem atical interpretation o f the derivative. 
Continuity o f differentiable functions.
Basic rules o f the derivation. Derivative o f 
com posite functions. Derivatives o f elem entary 
functions.
Derivative o f inverse trigonom etric and inverse 
hyperbolic functions. Rolle theorem and the mean 
value theorem.
10
6
4
8
Taylor theorem and its consequences. Examples. 
Taylor and M aclaurin expansions and their 
properties. Taylor expansions o f basic elem entary 
functions. Local m axim a and local minima. Critical 
points. Flex points with horizontal tangent lines. 
Characterizations o f critical points with derivatives. 
Examples.
10
6
4
9
Global m axim a and minim a o f differentiable 
functions on a closed interval. L ’hopital theorem. 
Examples.
P ro o f o f L ’hopital theorem . Tangent line,
10
6
4
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