<p>Can somebody please explain the Fundamental Theorem of Calculus, Part 1?</p>
<p>if f is continuous on the interval from a to b, then the function has a derivative at every point x in that interval.</p>
<p>i don't get why the derivative of F(x) would equal f(x)...</p>
<p>Different books define the two parts of the Fundamental Theorem of Calculus differently. Even the order is inconsistent between different books.</p>
<p>I’m not entirely sure which part you’re talking about. I’ll talk about both, and hopefully, one of them is the one you’re thinking of.</p>
<p>One says, if f(x) is continuous on [a, b], and has an antiderivative equal to F(x), then the integral from a to b of f(x) dx = F(b) - F(a). The proof of this typically involves Riemann sums, but I think you can make a better intuitive connection using your knowledge of position and velocity functions.</p>
<p>You should already know that the velocity function v(t) is equal to the derivative of the position function, x(t), or simply that v(t) = x’(t). If you wanted to find the net distance traveled from time t = a to time t = b using the position function, you’d simply subtract the positions at the two times, or x(b) - x(a). If you wanted to find the net distance traveled from time t = a to time t = b using the velocity function, you’d simply find the definite integral from a to b of v(t) dt. Since these distances must be equal, you have that the integral from a to b of v(t) dt = x(b) - x(a), where x represents the antiderivative of v.</p>
<p>The usefulness of this first part is to show that you can use antiderivatives to evaluate definite integrals.</p>
<p>The second part of the Fundamental Theorem of Calculus shows that if you take the derivative of a definite integral, with the upper limit of integration equal to a variable x, you should get back to where you started. Symbolically, d/dx[integral from a to x of f(t) dt] = f(x).</p>
<p>This follows from the first part:</p>
<p>The integral from a to x of f(t) dt = F(x) - F(a), where F is an antiderivative of f, from the first part that we looked at.
d/dx[the integral from a to x of f(t) dt] = d/dx[F(x) - F(a)]
Now the derivative of F(x) = f(x), since F(x) is defined as an antiderivative. It’s very similar to how adding 4 and subtracting 4 are inverse operations of one another.
On the other hand, since F(a) is a constant, the derivative of F(a) is 0.
So d/dx[F(x) - F(a)] = f(x).</p>
<p>I hope that helps, but I can go into more detail if need be…</p>