Linear Function That Comes From Life - Essay Example

 As an example duration for which our phone work is inversely proportional to the number of minutes, we spend on the phone. More time you spend on your phone, the less time the battery last. Again the amount of water needed is directly proportional to weather. More water is needed, to drink, in hot weather, as compared to cold weather. Again probability of success is directly proportional to hard work done. There are many activities, in our life, which can be proven as examples of linear functions in our life.

Module 02 - Case Assignment

1. Suppose you have a lemonade stand, and when you charge $1 per cup of lemonade you sell 50 cups. But when you raise your price to $2 you only sell 25 cups. Write an equation for the number of cups you sell as a function of the price you charge. Denote "C" for several cups, and "P" for the price you charge. Assume the function is linear.

CP=50……(C or P is not equal to zero)

2. Take a look at the table below and write out an equation for f(x).

x -2 -1 0 1 2

f(x) -5 -2 1 4 7

f(x) = 3x+1

3. Which of the following are functions? Explain your reasoning for a, b, and c. Keep the definition of a function strongly in mind as you do this problem, it is not nearly as difficult as it may look. Think about whether the relationship between f(x) and x is consistent with being a function or not and explain your reasoning.

1. f(x) = 2 if x>1 otherwise f(x) = -1

f(x) = 3u(x-1)-1

f(x) is a unit step function whose value for x =< 1 is -1 and for x>1 it is incremented by 3 i.e. -1+3=2. It is a nonlinear function.

1. f(x) = 5 if x>0 or f(x) = -5 if x<0 or f(x) = 5 or -5 if x = 0

f(x) = -5u (-x) +5u(x)

f(x) is a function whose value can be defined by the above function. It has a value of -5 for x<0 and 5 for x>0. at x=0 it can be +5 or -5. Again this is a nonlinear function.

1. f(x) = x/10

f(x) is a linear function of x. It increases linearly with x and has a gain of 0.1. 

4. For each of the relationships below, explain whether you think it is best described by a linear function or a non-linear function. Explain your reasoning thoroughly.

1. The time it takes you to get to work is a function of the speed at which you drive

Time taken by a person is a function of the speed of driving. How much you increase your driving speed, you take less time to cover the required distance. Now the distance between your home and workplace is constant provided you use the same path for driving daily. So the speed of driving is the one of major factors to decide time required to reach to the work. Other factors are traffic, car-related failures, etc which are negligible. Still, if x is variable constant, speed and t time taken to reach can be given by,

T= x/ speed. (Speed is not zero)

Thus T is a linear function of speed.

1. The probability of getting into a car accident as a function of the speed at which you drive

The probability of an accident depends on many factors like traffic, speed of driving, type of road, etc. Still, some sudden incidents cause accidents. Thus the probability of an accident increases as diving speed increases, but it does not have linear relation. Sometimes favorable conditions and alertness in driving allow speed to increase without any accidents. And some times due to unfavorable situations, a minimum speed of the vehicle also results in an accident. Thus other factors are as important as the speed of driving, thus this is a nonlinear function.

1. A person's height as a function of the person's age (from age 0 to 100)

A person’s height increases with increases in age till approximately the age of 21. again in this duration increase is exponential. Till the age of 15, height increases drastically, and between 15 and 21 at a very slow rate. After the age of 21, there is no change in a person’s height. So person’s height has a nonlinear relation with a person’s age.