Understanding Change: Peer Into the World of Calculus
Calculus is often considered one of the most challenging branches of mathematics. But don't let that scare you! With the right approach & a bit of exploration, anyone (Yes, you!) can get a grasp on many of its concepts, including derivatives. In this blog post, we're going to peer into the world of calculus & learn something fun about derivatives in an interactive way. As always, don't worry if you're not sure what you're looking at, the point is to challenge yourself to think outside the box & explore.
First, what is a derivative? A derivative is like a ruler that tells us how fast something is moving. For example, if you throw a ball, a derivative can tell us how fast the ball is moving at any moment. We can also use derivatives to understand how things change over time, like how a plant grows or how a car slows down. Understanding that a derivative simply helps us to understand change, we will use the three equations provided here to anchor our exploration: a simple equation, its first derivative, & its second derivative.
Begin by observing the left side of the equations (Highlighted here). Take a moment to study the way each is written & jot down what you notice & what you wonder about. As you explore, consider the following question:
Are you able to determine which of the three is the simple equation, which is its first derivative, & which is its second derivative? How do you think you know?
Now look at the first expression of each equation (Highlighted here). Take a moment to study the way each is written & jot down what you notice & what you wonder about. As you explore, consider the following questions:
Look at the leading coefficients of each (2, 6, & 12). Why does the 2 become a 6? Why does the 6 become a 12?
Look at the exponent of each (3, 2, & ?). Why does the 3 become a 2? What happened to the third exponent?
What patterns or structure do you notice from the simple equation, to its first derivative, to its second derivative?
Nicely done! You are now equipped with the knowledge of noticings, wonderings, & conclusions regarding Calculus’ derivatives! Now, spend some time playing around with the rest of the expressions. Do your noticings, wonderings, & conclusions apply everywhere? Try them out! Do they work? Why or why not? As you explore, consider the following questions:
Do your observations apply to positive numbers, negative numbers, or both? How do you know?
Why does the + & - sign change sometimes, but not always?
Where did that “+3” disappear to after the first derivative?
Where did that “-8” come with an “x” from in the second derivative?
Congratulations! You’ve successfully learned a bit about derivatives using simple, careful observation & proven that you don’t have to be a Mathematician in order to take a peek into the world of calculus. Simply by exercising our ability to explore & discover, we were able to see the patterns & structure of the derivative in a way that I hope was fun to comprehend & interact with. Remember, the aim of this journey was not to find the perfect answers, but to foster curiosity & the desire to discover; so don't be afraid to think differently & challenge yourself. Until next time, keep exploring & keep learning!
