Paper 4: Fundamentals of Business Mathematics & Statistic

3.6 I FUNDAMENTALS OF BUSINESS MATHEMATICS AND STATISTICS

Calculus

11. Show that

2

2

x 6x 8

x 8x 12

− +

− + is undefined for x = 2 and also find f(6). [Ans. undefined]

12. For what values of x are the following functions not defined?
    (i) x 4−^1 , (ii) x, (iii) x 1,− (iv) (x 1 x 2 ,−)( − ) (v)
    x 1^2
    x 1

−

−

[Ans. (i) 4, (ii) any negative integer, (iii) x < 1, (iv) 1 < x < 2, (v) x = 1]

13. Let f(x) = 2– x – 1 ≤ x < 0
    = 4 0 ≤ x < 1
    = 2x – 1, 1. ≤ x ≤ 3
    Calculate f(– 1), f(0), f(1), f(3). [Ans. 2, 4, 1, 5]


14. A 3-wheeler charges Re. 1 for 1 km or less from start, and at a rate of 50 p. per km or any fraction
    thereof, for additional distance. Express analytically the fare F (in `) as a function of the distance d (in
    km). [Ans. F = 1, 0 < d < 1, F = 1+^12 (d 1 ,− ) d > 1]


15. If g(x) = 2x, then show that g(a), g(b) = g(a + b).


16. If f(x) = epx + q, (p, q are constants) then show that f(a) f(b) f(c) = f(a + b + c) e2q.


17. If f x( )=ax bbx a++ , prove that f x. f( )^1 x =1.


18. If f x( )=2x 12x 1 2 ++ and f (x) = 2f (2x) then find f(2, 5). [Ans. 22/51]

[Hints : ( ) ( )

( )^22

f x 2. 2 2x 1 8x 2

2. 2x 1 8x 1

= + = +

+ + & etc.]

3.2 LIMIT

LIMIT :

Introduction :

Calculus is based, in general, on the idea of a limit. At present this idea including its related concepts,

continuity to mention, will be discussed.

Some definitions:

(i) Meaning of “x tends to a”. When the difference |x–a| (i.e., numerical difference) can be made less

than any positive quantity, however small, we say x tends to a and is written as x→a.

(ii) Meaning of “x tends to zero”. When the value of x goes on decreasing numerically and can be made

numerically less than any positive quantity, however small, we say x tends to zero and is written as

x→0.

(iii) Meaning of “x tends to infinity”. When the value of x goes increasing and can be made greater than

any positive quantity, however, large, we say x tends to infinity written as x→∝.

Neighbourhood or Proximity of a point

Let c be any real number, then any open interval around c is called the neighbourhood of c, e.g.,