جزوه زبان تخصصی رشته ریاضی کاربردی

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 اصطلاحات
Function and Limit:Season 1
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پاسخ تست های فصل پنجم Binary Relations:Season 6
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  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=5x2sin x explicitly as a function of xand x=(7y3-2y)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 ®axlim 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 magnitudeof the ratio  
toward + ¥ . The precise way in which these influences offset on anotherdetermines  
   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 algebraicmanipulations 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 findtwo functions, g and h, that have  
same limit L as x®a and such that f is 'squeezing' between g and h bymeans 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

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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  

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  
 

  
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