What are f(x) and f(t) in fundamental theorem of calculus

Question

I know the proof of part 1 of fundamental theorem but still if x and t are on same horizontal axis then how come we say f(x) and f(t) are sort of same functions. i mean not equal but we substitute x in place of t.

for eg. say we have,

We Know, F'(x) = d/dx INTEGRAL[From a to x] f(t) dt = f(x).

then

y = d/dx INTEGRAL [from a to x] (t^3 + 1) dt = x^3 + 1

Also,why do we write integral from [a to x] f(t) dt. Why do we use f(t) dt?

i know we get the original function back,is it like f(t) and f(x) are same but their intervals or say boundaries differ.

x and t are on the same axis and f(x) and f(t) are function values of x and t resp.

Here f(t) and f(x) are different functions, right?? what are f(t) and f(x) what's the relation between them?



Answer 1

Hello,

I can answer your question about what t and x represent in the Fundamental Theorem of Calculus, Pt. 1. I attempted to write a detailed explanation with pictures in this forum, but it contained too many characters.

Basically, t is what is called a "dummy variable" and represents all numbers between a and b in INTEGRAL[From a to b] f(t) dt. If we allow b to vary, then we will replace b with x and get a new function

F(x) = INTEGRAL[From a to x] f(t) dt.

Recall now that INTEGRAL is the area underneath the curve of f(t). So, in your example of f(t) = t^3 + 1, if I were to graph that function and shade in different regions below that curve, fixing the starting point a, I get different areas. Those different areas are the values of F(x).

The Fundamental Theorem of Calculus, Pt. 1 says this function F(x) is a differentiable function and the derivative of this function (i.e. how much the area below f(t) changes) is no different than just plugging in x for t in f(t). So if I define F(x) = INTEGRAL[From a to x] t^3 + 1 dt and want to find the derivative of this "area function", I can just plug in x for t. So, d/dx F(x) = x^3 +1. This avoids needing to use that limit definition of derivative.

This theorem also shows that computing area and computing speed are opposite operations, like how addition and subtraction are opposite operations.

I think my explanation is better explained with pictures or animations. But in this page, it would take too many characters. If you'd like to discuss this more with me, please feel free to schedule an appointment. I am happy to go over the Fundamental Theorem of Calculus and its proof in greater detail.