When dancing with the beautiful girl or an elegant boy called Integration on the prompt night, which theme is Calculus, you may figure out by now that they don’t always dress according to the dress code. And your job during this party is to discover what is going on with the dress your partner decided to shock you with.
Partial fractions have proven to be a vital tool for solving complex problems. This blog post will explore partial fractions, how they work, and why they’re so important.
Imagine you get yourself in front of an integral that is wearing a rational function that you can’t decipher using U-substitution, integration by Parts, or table of integration (and if you tell yourself, “I will not be in front of such a thing,” check the applications example in this post and think again). When an interesting problem is placed in the way of people with a growth mindset, we find a way around it. Partial Fractions are the mathematical tool that helps us to convert such monstrous rational expressions into a sum of simpler ones that we can solve by any method learned before; we transform it into something familiar and DONE!
Partial Fractions are the mathematical tool that helps us to convert such monstrous rational expressions into a sum of simpler ones that we can solve by any method learned before; we transform it into something familiar and DONE!
Yes, yes; the two expressions inside the integration symbols in the image above are the same. But don’t worry, if you didn’t see it, let me show you:
Now, it’s easy to go from LHS to RHS (now you remember your algebra, don’t you?). But how in EARTH do we go from RHS to LHS? Since the integration problems are proposed in the RHS form. My dear friend, Partial Fractions are here to save your head again. Explaining partial fractions rules is out of this post’s scope, but you can let me know if you need an extra hand, and I will share my notes on the topic with you (contact me).
BTW Maybe after this, you still can say, “I will not be in front of such a thing.” Then you are safe, but now you know which tool economists, engineers, mathematicians, and analysts use to solve that situation.
How did I say It works? Oh, I haven't
Since now we have the big idea behind this amazing tool called Partial Fractions, let me tell you broadly how it works…
The decomposition of a rational function into partial fractions involves several steps, including long polynomial division and solving a system of linear equations. The resulting partial fractions are expressed in terms of the coefficients of the numerator and denominator polynomials, and they provide valuable information about the behavior of the rational function.
Remind me why are Partial Fractions important again
Partial fractions are important because they provide a way to simplify complicated functions, making it easier to analyze and solve problems. Some applications of partial fractions are:
Economic
Econometric models: partial fraction decomposition is used to estimate econometric models. It helps you decompose a system of simultaneous equations into a set of simpler equations that can be calculated separately. The resulting estimates can then be combined to obtain an overall system estimate. Analysis of production functions(APF): APF describes the relationship between inputs and outputs in a production process. Partial fraction decomposition gives you a hand when decomposing a production function into its constituent components, which can then be analyzed separately. Offer and demand: partial fraction decomposition can be used in estimating demand functions, which describe how the quantity of a good that consumers are willing to purchase depends on its price and other factors.
Engineering
Mathematics and statistics
In conclusion, partial fractions are a powerful mathematical tool that has been used for centuries to solve complex problems in various fields. They provide a way to simplify complicated functions and provide valuable information about the behavior of rational functions. If you’re a mathematics student, engineer, economist, or statistician, partial fractions are an important concept to understand and master.
Do you want to solve some real problems related to the examples above to test your Partial Fractions skills and see if you decipher your way out of this Calculus Party?
For teachers
Stay checking because Im adding to this post a project based on the topics you can apply in your classroom if you are a teacher or play with if you are a student or enthusiast. You will see situations like this:
A company produces a good that can be sold for a price of p dollars per unit. The company's production function is given by:
where Q is the quantity of the good produced, K is a constant, L is the amount of labor used, and a and b are parameters. Use partial fractions to determine the relationship between labor and the quantity of the goods produced.
For students and parents
Interested in learning with me? You will dive into a lot of fun and "Ahhhhh" moments. Contact me for tutoring or project consulting. You will get the following:
A PDF version of all of our lesson notes.
A personal cloud folder with your notes, resources, and more.
A live version of the OneNote of your digital notebook (sample below)
Recording of the lesson (following client approval).
Access to Discovery and Lab projects to apply your knowledge and improve your critical thinking and discussion skills.
Access and training on using Ed. Tech such as Desmos, WolframAlpha, OneNote, Wacom writing pad, etc.
References:
Oppenheim, A. V., & Schafer, R. W. (1999). Discrete-time signal processing (2nd ed.). Prentice Hall.
Chen, W. (2017). Principles and applications of electrical engineering (6th ed.). McGraw-Hill Education.
Kachroo, P. (2017). Control systems engineering (7th ed.). John Wiley & Sons.