Paper 4: Fundamentals of Business Mathematics & Statistic

3.62 I FUNDAMENTALS OF BUSINESS MATHEMATICS AND STATISTICS

Calculus

Example 126:If dxdy=x 3x 1^3 −^2 + and y = 2 when x = 1, find y.

or,

3 2 x x^43

y x dx 3 x dx dx= − + = 4 3−3 x c+ +

or,^2 = 41 − + +1 1 c or,^2 = 41 +cor, c 2= −^14 =7/ 4

∴

x^437

y= 4 −x x+ + 4

SELF EXMINATION QUESTIONS

1. (i) x^11 (ii) x^410 (iii) x (iv) xa (v) a (vi) 2.
   x^12 x^411 x^2 x 1a
   Ans.(i) 12 (ii) 411 (iii) (iv)x a 1++ (v) ax (vi) 2x;+c in all the case^


2. (i) x–11 (ii) x–n (iii) x^1 n (iv) x^12 (v)^1 x (vi) x–1

x^10 xn 1 xn 1 1

Ans.(i) 10 (ii) n 1 (iii) n 1 (iv) (v) and (vi) logxx

− − + − + −

(^) − − + − + (^)

3. (i) x3/2 (ii) x^5

(^25) /2 3 4 5/2 x^2
(^) Ans.(i) x (ii)x x 5 + 5 + 2
Integrate :

4. (i) x xdx (ii) ( )
   2
   x x dx

(^2) 5/2 3 4 5/2 x^2
(^) Ans. (i) x (ii) x x 5 + 5 + 2

5. (i)

4 2

3

x 2x 1dx

x

+ +

(ii)

(x 3 )^2 dx

x

−

(iii)

( )

2

2 x 1

dx

x x

+

(^25) /2 3/2
2
Ans. (i) 2logxx^1 (ii) x 4x 18 x (iii) 4 x 4logx^24
2 2x (^5) x
(^) + − − + + −
(^)

6. (i)
   x 1^2

(^) x 1−− dx (ii)
x 1^3
(^) x 1−− dx (iii)
x 3x 2^2
+x 1+ + dx (iv)
x^2 x 6
(^) x 2− −+ dx.
x^2 x x^32 x^2 x^2
Ans. (i) x (ii) 2 + 3 2+ (iii) 2x (iv) 3x 2 + 2 −