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توضیحات محصول :کتاب های خلاصه منابع رشته ریاضی کاربردی همراه بامجموعه تست در هر فصل با پاسخنامه تست

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 any

positive number say e and construct an open interval on they y-axis that extends e

Then 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 is

In 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 های 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 های فصل اول

1)a 2)b 3)a 4)c 5)c 6)d 7)c 8)d 9)b 10)d

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