Course: Business Mathematics (1429)

Semester: Spring, 2022

**ASSIGNMENT No. 2**

**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:**

**i) (PX)T = P(XT) (Associative Property)**

**PX(T) = **

**P(XT)=**

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

**P(X + T) =**

** **

**PX + PT =**

**PX + PT =**

**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.**

**x _{1} – 4x_{2} + x_{3} = 12**

**7x _{1} –6x_{3} = 18**

**2x _{2} + 5x_{3} = 7**

**3**

**(a) Find the following limits: **

**i)****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.

**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) = x ^{4}, –32x^{2} + 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**

**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.