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The Intermediate Value Theorem (IVT), for example, is a mathematical concept. In mathematical form, the IVT can be expressed in the following way:
If arithmetic isn’t your strong suit, you might find that term a little difficult to comprehend at first. Consider trying to apply the IVT to real-world situations; you may discover that this theorem is something you already instinctively understand!
Example 1: A period of rapid growth.
When I was in 7th grade, I was a little over 5 feet tall (5’1″), to be precise. By the time I reached the eleventh grade, I stood at 6 feet tall.
As a result of this, I’d like to know if there was a moment between 7th grade and 11th grade when I was 5’10” tall.
Now, let’s look at another scenario that college students may be familiar with:
Example 2: The “Freshman Fifteen” is a group of fifteen college freshmen.
This is merely a hypothetical illustration. This is purely speculative.
Assume that I weighed 175 pounds on my first day of college and that by the conclusion of my freshman year, I had increased my weight to 190 pounds. Following the Intermediate Value Theorem, which of the following weights did I 100% without a doubt achieve at some time during my freshman year, based on my own personal experience?
A) 168 pounds
B) 178 pounds
C) 188 pounds
D) 198 pounds
B and C are the correct answers. For example, if my starting weight was 175 pounds and my ending weight was 190 pounds, I must have weighed every possible number between 175 and 190 pounds at some time throughout the year to arrive at this result.
The rationale behind that example should be understandable to you at this point, and you’ll be well on your way to learning one of the core theorems of calculus!
Conclusion
I hope that these little anecdotes help to demonstrate that the Intermediate Value Theorem is not an abstract concept that must be memorised in order to be able to regurgitate it on a test. Instead, it may be described in layman’s terms, and you don’t have to be an expert in mathematics to get it.
In calculus, not every theorem you come across can be described in this basic manner. A creative and knowledgeable teacher, on the other hand, will be able to come up with examples that will assist bring abstract mathematics down to earth.
