دانلود جزوه زبان تخصصی رشته ریاضی کاربردی

دانلود جزوه زبان تخصصی رشته ریاضی کاربردی

   Season 1:Function
and Limit
An equation of the form y=f(x) is said to define y explicitly as a function of
x (the
function being f), and an equation of the form x=g(y) is said to define x
explicitly as a
function of y (the function being g). For example, y=5x
2
sin x explicitly as a function of x
and x=(7y
3
-2y)2/3 defines x explicitly as a function of y.
An equation the is not of the form y=f(x) but whose graph in the xy-plane
passes the
vertical line test is said to x, and an equation that is not of the form x=g(y)
but whose
graph in the xy-plane passes the horizontal line test is said to define x
implicitly as a
function of y.
In the preceding sections we treated limits informally, interpreting
®ax
lim f(x)=L to mean that the values of f(x)
approaches L as x approaches a from either side (but remains 
different from a). However, the phrases ‘f(x) approaches L’ and ‘x approaches
a’ are    intuitive ideas
without precise mathematical definitions. This means that if we pick anypositive number, say e ,
and construct an open interval on they y-axis that extends eThen is deducing these
limits results from the fact that for each of them the numerator 
and denominator both approach zero as h ® 0. As a result, there are two
conflicting 
influences on the ratio. The numerator approaching 0 drives the magnitude of
the ratio 
toward zero, while the denominator approaching 0 drives the magnitude of the
ratio 
toward + ¥ . The precise way in which these influences offset on another
determines     whether the
limit exists and what its value isIn a limit problem where
the numerator and denominator both approach zero, it is 
sometimes possible to circumvent the difficulty by using algebraic
manipulations to write 
the limit in a different from. However, if that is not possible, as here, other
methods are 
required. One such method is to obtain the limit by ‘squeezing’ the function
between 
simpler functions whose limits are known. For example, suppose that we are
unable to 
show that 
®ax
lim f(x)=L directly, but we are able to find two functions, g and h, that
have 
same limit L as x®a and such that f is ‘squeezing’ between g and h by means of
the 
inequalities g(x) £f(x) £h(x) it is evident geometrically that f(x) must also
approach L as 
x®a because the graph of f lies between the graphs of g and h. 
This idea is formalized in the following theorem, which is called the Squeezing
Theorem
or sometimes the Pinching Theorem تست هاي فصل اول 1) If the domain of a real-valued, continuous function is connected, then the
range is
a. An interval of R it self b. An open set 
c. A compact set- d. A bounded set 
2) A function : ® RAf is said to ……….on A if there exists a constant M > 0
such 
that )( £ Mxf for all Î Ax .
a. be closed b. be bounded 
c. have extremum d. have maximum 
3) A set Í RU is said to be open if for each ÎUx there is ….number a e such
that 
-e + e ),( ÍUxx .
a. A positive real b. a non-zero real 
c. complex d. a negative set 
4) Let e > 0 , then it is easy to see that <- e . Which of the following 
statements is true about f where 
2
a. f is continues at x = 2 b. lim )(
does not exist. 
c. lim )(
=4 d. lim )(
 exist but it is not necessarily 4. 
5) ‘A function : Rf ®is continuous at a point 0
x in R if given e > 0 , there is a 
d > 0such that for all x in R with <- d 0
xx we have <- e 0
xfxf )()( which of the 
following statements is true in general?
a. e is a small number b. d is a small number 
c. d is a function, of 0
x and e d. d is unique 
6) A function is a special case of a……… .
a. derivative b. equality c. polynomial d. relation
   
7) A function f is said to be even if it is defined on a set symmetric with
respect to 
the ……and if it is possesses the property – = xfxf )()( .
a. origin b. x-axis c. y-axis d. open 
8) For any real number x . The …..value of x , denoted by x .
a. absorbency b. absorption c. abstraction d. absolute 
9) For a real function f, the …..of f is the set of all pairs yx ),( in R´ R
such that 
= xfy )( and x is in the domain of the function.
a. curve b. graph c. greatest d. divisor 
10. The graph = xgy )( is an odd function has the ….as a line symmetry.
a. y-axis b. origin c. y=x d. x-axis 
 
پاسخ تست هاي فصل اول     1)a 2)b 3)a 4)c
5)c 6)d 7)c 8)d 9)b 10)d http://stufile.ir/join/invite439    http://stufile.ir/?ref=439