Question

Polynomial and rational functions can be used to model a wide variety of phenomena of science, technology, and everyday life.

Choose one of these sectors and give an example of a polynomial or rational function modeling a situation in that sector. [Hint: see the examples and exercises in the book.]

Go to www.desmos.com/calculator, write your equation, or function, and develop your explanation using the properties of graphs.

Your Discussion should be a minimum of 250 words in length and not more than 750 words.

Dear expert, I will submit this assignment after an hour, please don't copy from internet and add APA text citation if needed.

Answer 1

Polynomial functions or Rational functions are definitely of great use while modelling. One of the reasons we say that, is it's property of continuity over R. Polynomial functions are continuous everywhere so as Rational functions (except where denominator function is zero) over R. We also have result that says any continuous function on a compact interval can be aproximated by polynomial functions.

So Now let's choose a phenomena in physics. We will discuss about projectile motion. We wish to throw a ball from point A at an angle 45 degree with respect to horizontal base with with initial velocity of m/s. Further we can ask the range covered by the ball or the maximum height it attained or when it reached the ground.   Now to model this problem we use Newtonian mechanics formula, to get the equation of it's trajectory. The equation of trajectory with respect to time that we get is-

.

Here t represents time and y represents the height attainted after time t.

(Obtained in the following way using newton's formula-)

Here we see this polynomial function has 2 real values t=0 and t=5 at which the y value becomes zero. That means at t=0 the ball attained height 0 as it was initial position, then at t=5sec the ball reached the ground after the shown trajectory it undertook.

As we can see that vertical height attained by the ball after time t is expressed in terms of polynomial function. Here the continuity property offers the idea that if the height of the ball attained is suppose 30m after 2 second then it can't attain height arbitarily to any value, suppose 1064m, in the next mili second.

Similarly, There are other areas like signal processing (collecting data when we are singing or speaking) we require functions that will make the inputs smoother or continuous., economics (data analysis with probability distribution functions), or engineering (like making curves to design various components ) where polynomial functions are being used regularly.