Have you ever wondered how physicists, economists, engineers, and many others use maths to solve real world problems? One answer lies in a powerful branch of mathematics called Differential Calculus. In just one minute, we now introduce you to the fundamental concept behind it.
Choose two points on a curve.
Draw a line connecting the two points and turn your focus to the slope of the (blue) line.
Bring one of the points on the curve closer to the other point observing how the slope (of the blue line) changes.
Notice how the slope of the (blue) line connecting the points begins approximating a tangent (a tangent is a line that touches a curve at exactly one point).
In fact, as the distance between our two points closes in on zero the two points become effectively one; providing us with the tangent — in our example case, the tangent to (0,0).
You may be surprised, but we have accomplished the underpinning idea to what is known as “differentiation from first principles”; the backbone of Differential Calculus.
In short, when we want to understand how something is changing at a specific point, we examine the slope of the tangent at that point. This groundbreaking concept, discovered by Isaac Newton in the 1600s, is truly remarkable. It enables us to understand how things change and how to measure that change.
For example, on a distance-time graph differential calculus gives us the instantaneous velocity at any point in time. By taking the derivative a second time, we can even determine the acceleration.
Why does any of this matter? Well physicists use differential calculus to analyse motion. Economists use differential calculus to optimise profit. Engineers use differential calculus to examine stress and strain on materials, as well as to design structures and the list goes on.
Welcome to Differential Calculus.
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