Paper 4: Fundamentals of Business Mathematics & Statistic

3.56 I FUNDAMENTALS OF BUSINESS MATHEMATICS AND STATISTICS

Calculus

8. If y = 5x find

2

d ydx 2 [Ans. 5x (log 5) (^2) ]

9. If f(x) =2x^3 + 3x^2 – 12x for what value of x will f′(x) = 0 [Ans. 1, –2)


10. If y = x^3 , evaluate

2

2

1 d y

dx

+^

when x =–1 [Ans. –5]

11. Find dydx for 2x = t^2 and 3y = t^3 [Ans. t]


12. If x = at, y =

2

2

a d yfind

t dx [Ans.^3

2

at ]

13. If x = at^2 , y = 2at find

2

d ydx 2 [Ans. 3

1

2at

− ]

14. If y = x^1 x find dydx [Ans.
    x.^1 x1 logx
    x

− ]

15. If y = xlog x find dydx [Ans. x .logx2logxx ]


16. If e3x^2 +5x-2 find dydx [Ans. e3x^2 +5x-2, (6x +5)]


17. Given y = 2x2 – x + 1 find whether y is increasing, decreasing or stationary at x = 41.
    [Ans. Stationary at x = 41 ]


18. For what value of x, y = x^3 – 3x^2 – 9x + 5 is minimum. [Ans. 3]


19. In the above example, for what value of x the functions is maximum? [Ans.–1]


20. If the total cost function is c = q^2 – 2q + 5q find MC [Ans. 3q^2 – 4q +5]


21. The average cost function (AC) for certain commodity is AC = −2x 1− +^50 x in terms of output x.
    Find the MC [Ans. 4x –1]


22. In the above example, find the slope of MC [Ans. 4]


23. The average cost function (AC) for certain commodity is AC=x 2 − +^421 x find the slope of MC.
    [Ans. x–4]


24. Examine f(x) = x^3 – 6x^2 + 9x – 18 for maximum or minimum values. [Ans. max. at x = 1, min. at x =– 3]


25. If y = Aekx + be–kx evaluate y^2 – k^2 y [Ans. 0]


26. If the cost functions is
    q^3
    c= 3 −2q 12+ find average variable cost. [Ans.^13 q 2^2 − ]


27. The cost function (c) of a firm is as follows:
    c q q 4q 2.2 3^32
    =3 2− + + Is the slope of AC =

1

q (MC – AC). [Ans. Yes]