AIOU Course Code 1429-2 Solved Assignment Spring 2022

Course: Business Mathematics (1429)

Semester: Spring, 2022

ASSIGNMENT No. 2

1. 1

(a)    An economy depends on two basic products, wheat and oil. To produce 1 metric ton of wheat requires .25 metric tons of wheat and .33 metric tons of oil. Production of 1 metric ton of oil consumes .08 metric tons of wheat and .11 metric tons of oil. Find the production that will satisfy a demand of 500 metric tons of wheat and 1000 metric tons of oil.

(b)    Given the matrices: ,  ,   . Verify the following statements:

1. i) (PX)T = P(XT) (Associative Property)

PX(T) =

P(XT)=

1. ii) P(X + T) = PX + PT (Distributive Property)

P(X + T) =

 

PX + PT =

PX + PT =

1. 2

(a)    Determine the inverse of   by using the matrix of cofactor approach.

(b)    Solve the given system of equations by using Cramer’s rule.

x₁ – 4x₂ + x₃ = 12

7x₁  –6x₃  = 18

2x₂ + 5x₃ = 7

1. 3

(a)    Find the following limits:                                                                                 

1. i)
2. ii)

iii)    

(b)    Living standards are defined by the total output of goods and services divided by the total population. In the United States during the 1980s, living standards were closely approximated by

Where x=0 corresponds to 1981. Find the derivative of f. Use the derivative to find the rate of change in living standards in the following years.

• 1981

x = 0

f(x) = 11.6

• 1988

f’(x) =

x = 0

f’(x) = -0.4

• 1989

f’’(x)

x = 0

f’’(x) = 0.6

 

• 1990

f’’’(x) = – 69 /500 = -0.138

• What do your answers to part (i) – (iv) tell you about living standards in those years?

From above result, 1981 is best for living standards.

1. 4

(a)    Find critical points of the given functions. Use second derivate test on each critical point to determine whether it leads to a relative maximum or minimum.

f(x) = x⁴, –32x² + 7

                                                       

(b)    Determine the location and values of the absolute maximum and absolute minimum for the given function.

f(x) = (–x + 2)⁴, where 0 £ x £ 3

1. 5

(a)    Find f_x ,  f_y , f_z , f_yx , f_xz , f_zy , f_y , (2, –1, 3), f_yz, (–1, 1, 0). if                             

f(x, y, z) =           

f_x =

f_{y =}

f_{z =}

 

f_{yz =}

f_{yx =}

f_{zy =}

(b)    Suppose z = f (x, y) describes the cost to build a certain structure, where x represents the labor costs and y represents the cost of materials. Describe what f_x and f_y represent.

Basically fx and fy represents the marginal labour cost and marginal cost of materials respectively. fx and fy are calculated by taking partial derivatives of z.