Question: Use inequalities to describe the given interval.

The set of real number which can be represented by line segment on the number line is called as interval and intervals are very useful to describe the inequality.

Consider the interval

which consist all between and, including but excluding. The round bracket indicates the value is included in the interval while round bracket shows the value is excluded in the intervals.

The objective is to describe inequality for given interval of the following figure,

Consider the following figure,

From the above number line, we can directly point out here that

is the only one end point of the interval.

As rounded bracket at

indicate that endpointis excluded from the interval and further lies in right of , which means.

Question: For comfortable modern living, it is estimated that person needs roughly 60 m^2 for housing, 40 m^2 for his or her job, 50 m^2 for public buildings and recreation facilities, 90 m^2 for transportation (e.g. highways), and 4,000 m^2 for production of food.

Switzerland has approximately 11,000 km^{2} of livable space (arable and habitable land). How many people can comfortably live in Switzerland? Look up the actual population of Switzerland. Based on the figures given here, is Switzerland overcrowded or is there still room for comfortable growth?

Solution: Given that a person needs 60 square meters for housing, 40 square meters for his or her job, 50 square meters for public buildings and recreation facilities, 90 square meters for transportation, and 4,000 square meters for the production of food.

Therefore the total area needed per person is,

So, one person requires the area

Then the number of people that can accommodated in the area of 11,000 square kilometers is,

MANUFACTURING At a certain factory, output Q is related to inputs u and v by the equation

If the current levels of input are u = 10 and v = 25, use calculus to estimate the change in input v that should be made to offset a decrease of 0.7 unit in input u so that output will be maintained at its current level.

Solution:

At a certain factory the output Q is related to inputs u and v given as

…… (1)

We have to find the change in input v that should be made to offset a decrease of 0.7 units in input u so that output is maintained at the current level.

To find the change in input v, differentiating both the sides of (1) with respect to input u, we have

…… (2)

Since the output has to be maintained at current level,

Therefore,

From equation (2), we have

…… (3)

From the question, for inputs u = 10, v = 25, from equation (3), we have

…… (4)

Using approximation formula,

…… (5)

Therefore, from equation (4) and (5), we have

Hence decrease the input in v by 21184.30 units to maintain the output at the current level.