What good is calculus?

No, this isn't a math course on doing calculus. My goal is to show the value of calculus. I like the definition of calculus that says that it's the mathematical study of change. Isaac Newton and Gottfried Wilhelm Leibniz developed this amazing branch of math in the 17th century. That's way back then!

Calculus is broken down into Differential calculus and Integral calculus.

A differential of a function (a math equation that defines a result based on variables and constants) is mathematical way of showing how a function changes with a change in a variable. In other words, how a plot of the function changes versus the variable. If you think about it, a derivative is like determining the slope (ratio of the rise to the run) of a tangent line on the function at any given point. A derivative is sort of like a rate of how a function changes as a variable is changed.

A plot or graph of a function looks like a smooth curved line, but if you draw a tangent to any point on the curve, the resulting straight line has a slope, which is a change in the function caused by a change in the variable (df/dx). If the variable is time, then the derivative is the change of the function over time (df/dt).

Why do we need this calculus math? Think of this way. If all you wanted to know is how far a car would go if it was moving at constant speed for a given amount of time you would simply multiply the speed and the time. However, if the speed changes with time, then what? You can't use a simple multiplication.

To answer this more complicated question, you need to integrate the function that represents the change in the distance verses the change in speed of the car. If you plot the easy problem, you get a graph that looks like a box. The distance is the area under the plot, and it's simple to calculate. However, if you plot the new more complicated case, you get a graph that has a curvy line. You can't easily figure out the area under this curved graph. One way would be to take small slices under the curve and approximate them as thin rectangles and add the areas of these rectangles. This would give you an approximation of the area. However, a better solution is to determine the integral of the function and then you end up with a much better result. An integral would be equivalent to taking an infinite number of rectangular slices under the curve. To do that manually would take you forever.

Let's say you have a function that says that y equals x squared (y=x2). This means that as x is increased y changes to the square of x. The derivative of this function is the change in y over the change in x becomes (is equal to) 2 times x. If we integrate this '2 times x' derivative we end up with the original x squared plus a constant. The constant is because of the curve not beginning at zero.

Physics uses calculus because of problems dealing with time such as Newton's second law, which states that force is equal to mass times acceleration. The acceleration is change in velocity with time. That's perfect for differential calculus because it can derive the path or trajectory of an object that's accelerating.

The good thing about calculus is that there are shortcuts for determining what is known as a definite integral. A definite integral is integration over a given range of values. One calculus theorem says that an integral of a function times dx (derivative x) is equal to the value of the function at the upper limit of integration minus the value of the function at the lower limit.
Another trick is to use an Anti-derivative list of formulas. Essentially, this uses backwards differentiation to come up with formulas to calculate definite integrals.

A good way to explain integration is: the expression to the right of the integration symbol is a little bit of something, and integrating the expression is accomplished by adding the little pieces between the lower limit and the upper limit. I realize that this is not easy to understand, but it is important.

This is just a tiny fraction of the complexities of calculus. Calculus has made modern science and engineering possible. Without it we would still be back in the age of stone knives and bearskins.

Thanks for reading.