Issac Newton created calculus in order to derive his mechanics equations. So what exactly is calculus? Calculus is actually not too hard to understand. There are basically two main concepts in calculus: derivatives and integrals.
Derivatives are used to find the slope of any point on a curve. So what is that useful for? Amazingly, it is useful for many things. Remember how we needed to find the slope of the displacement versus time graph to find the velocity? Well, if the graph was a straight line, it wouldn’t be that hard. It would simply be:
We have to use the bar over v to indicate that it is an average velocity because we cannot use it to find the instantaneous velocity of an equation that is non-linear (not a straight line).
This is where we need calculus. With a derivative, you can find the exact slope of any point on any equation. If we decrease the change in time, the average slope becomes closer to the instantaneous slope. So, what if we decreased Dt until it is infinitely small? Well, then the value would become exactly the instantaneous value. When Dt becomes really small, we denote it as dt. This is how we write the velocity in terms of a derivative of distance with respect to time:
Notice how we did not need to use the bar over the v.
This is a table of common derivatives:
Well, let’s give an example of what this can be used for. Let’s say an object is moving and its displacement is given by the equation:
s = 5t2 + 4t + 9
Well, since velocity is the derivative of displacement with respect to time, we should take the derivative of both sides of the equation (also called differentiation):
Now we have to simplify it:
v = 5(2t) + 4 + 0
v = 10t + 4
Now, to find the velocity at any time, you would just plug it into the equation.
Also, just like the derivative of displacement is velocity, the derivative of velocity is acceleration.
To differentiate the product or quotient of two functions (f and g):
f’ and g’ denote the derivative of f and g, respectively.
Another thing that you should know in differentiation is the chain rule (what to do if you have functions within functions):
Integrals are used to find the area under a curve. This is useful to find the displacement in a certain time if you had the equation for the velocity, or the velocity in a certain time if you had the equation for the acceleration.
Okay, so how can we get the area of something under a curve? Well, we can try to approximate it with rectangles as shown in the illustration to the left. In the illustration, the area under the curve shown between points a and b is approximated by 6 rectangles. As you can see, the approximation is not too accurate. However, we can get more accurate if we used even smaller rectangles.
The illustration to the left shows that our approximation is more accurate with smaller rectangles. However, the approximation is still not the exact value. With integrals, we can get the exact value.
Integrals are based on the same rectangular sum method. If the rectangles have a width of Dx, we can say that the area under the curve is the sum of all of the rectangles with width Dx:
Again, we when the width of the rectangles get smaller and smaller, we get more accurate. When the width is infinitely small, we call it dx and we can write the summation in integral format:
The a and b are used to denote the limits of integration. In other words, between which points we want to do the integration. If we have no limits of integration, the integral is called an indefinite integral, and a constant (usually C) is added since it is unknown where the summation begins. For example, if we are integrating velocity with respect to time, we would get displacement, and if the integral is indefinite, the constant would be the initial displacement (s0).
This is a table of common integrals: