# real world applications

#### tmd1979

##### New member
I have to come up with a real world example using jump discontinuity and one using infinite discontinuity. I do not even know where to begin. I looked for applications in my text book to no avail. The examples can be in business applications (manufacturing or economics) or in science applications (physics, biology, or chemistry). Please help! I am stuck!

Hello, tmd1979!

I think I can help you with part of your problem.

I have to come up with a real world example using jump discontinuity and one using infinite discontinuity.

Suppose a delivery service (UPS, FedEx, etc.) charges, say, $5 per pound. The rates are usually given as: "Five dollars per pound or fraction thereof." This means that they will always "round up". . . $$\displaystyle \begin{array}{c}\text{Any weight over 0 pound up to 1 pound is charged \5.} \\ \text{Any weight over 1 pound up to 2 pounds is charged \10.} \\ \text{Any weight over 2 pounds up to 3 pounds is charged \15.} \\ \text{etc.} \end{array}$$ The graph is a "step function". Code:  |$20 |                 o-----*
|
$15 | o-----* |$10 |     o-----*
|-
\$5 o-----*
|
- + - - * - - * - - * - - * - - -
|     1     2     3     4  (lbs.)

The function has jump discontinuities at every integer value.

Thanks, that helps out! Any ideas for an infinite discontinuity application?

Do you know what an infinite discontinuity is?

My text book shows the graph of 1/x^2 as being one with infinite discontinuity. In the graph, the limit as x approaches 0 is infinity from both the positive and negatives sides. That is all my book says about it. They are very vague in their description of each and there are no examples in there about them. Usually they have real qorld applications as exercises at the end of each section, but not about this. I am stumped.

$$\displaystyle Improper \ Integrals \ are \ loaded \ with \ infinite \ discontinuities.$$

$$\displaystyle Example: \ Use \ the \ formula \ for \ arc \ length \ to \ show \ the \ circumference \ of \ the \ circle$$

$$\displaystyle x^2+y^2 \ = \ 1 \ is \ 2\pi.$$

$$\displaystyle To \ simplify \ the \ work, \ we \ consider \ the \ quarter \ circle, \ y \ = \ \sqrt{1-x^2}, \ 0 \ \le \ x \ \le \ 1.$$

$$\displaystyle s \ = \ \int_{0}^{1}\sqrt{1+(y')^2}dx \ = \ \int_{0}^{1}\frac{dx}{\sqrt{1-x^2}}$$

$$\displaystyle This \ integral \ is \ improper, \ since \ it \ has \ an \ infinite \ discontinuity \ at \ x \ = \ 1.$$

$$\displaystyle Thus, \ s \ = \ \int_{0}^{1}\frac{dx}{\sqrt{1-x^2}} \ = \ \lim_{b\to1^-}\bigg[arcsin(x)\bigg]_{0}^{b} \ = \ \pi/2-0 \ = \ \pi/2.$$

$$\displaystyle Hence, \ 4s \ = \ 2\pi.$$

tmd1979 said:
the limit as x approaches 0 is infinity from both the positive and negatives sides.

That's one example. So, in general, an infinitie discontinuity occurs where the function value blows up as x approaches some specific value. (Think of vertical asymptotes; y goes to plus or minus infinity, as x get closer to some specific value.)

The Salary Theorem is another example. This theorem states that people earn more money the less they know.

It is given that knowledge is power.

k = p

We also know that time is money.

t = m

Power is defined as work/time.

p = w/t

and by substitution

m = w/k

This last relationship makes it clear that as knowledge approaches zero, money approaches infinity, regardless of the amount of work done.

Do you have a textbook in which you could look up real-life examples of vertical asymptotes?

Thanks a lot! Yes I do have some other math sources I can look at. Thank you for clearing it up for me. It is greatly appreciated.