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In this article, we will look closer at the Fundamental Theorem of Calculus and how it ties integration and differentiation together. We will look at both parts 1 and 2 of the theorem, as well as a few examples of how the theorem can be used in practice.
First Part of the Theorem
To get started, we will need to consider a function. So, let us imagine that we have a function, f(t). Our function, f(t), is defined to be continuous on the interval [a,b]. This means that ‘a’ is our lower bound on the interval while ‘b’ is the upper bound of our interval. We don’t want this to be too abstract, so let us try to visualize f(t). Here f(t) will simply be an arbitrary function:
One important thing to note is that our function is a function of ‘t’ and not of ‘x’. The reason for this will become clear soon.
Let us now imagine that we have a number, x, which is inside our interval. Hence, we have that:
We can try to draw this into our graph above:
We can now see that x lies inside our interval, [a,b]. We then want to define a function, which captures the area…
