Understanding the Fundamental Theorem of Calculus: Bridging Derivatives and Integrals

Why Introduce Two Fundamental Theorems?

When studying calculus, it's essential to understand the two Fundamental Theorems of Calculus (FTC). These theorems provide a profound connection between derivatives and integrals and lay the groundwork for grasping and analyzing the nature of change. The applications of these theorems in fields like science, engineering, and economics help us solve many real-world problems. Therefore, delving into both theorems will enhance our comprehension of mathematics and its practical applications.

The First Fundamental Theorem of Calculus

The first fundamental theorem can be stated as: \[\int_a^b f'(x) \, dx = f(b) - f(a).\]

Intuitive Understanding

Intuitively, the Fundamental Theorem of Calculus states that "the total change is the sum of all the little changes." The term \( f'(x)dx \) represents a tiny change in the value of \( f \). You add up all these tiny changes to get the total change \( f(b) - f(a) \).

Detailed Breakdown

In more detail, chop up the interval \([a,b]\) into tiny pieces:

\[a = x_0 < x_1 < \cdots < x_N = b.\]

Note that the total change in the value of \( f \) across the interval \([a,b]\) is the sum of the changes in the value of \( f \) across all the tiny subintervals \([x_i,x_{i+1}]\):

\[f(b) - f(a) = \sum_{i=0}^{N-1} f(x_{i+1}) - f(x_i).\]

(The total change is the sum of all the little changes.) But,

\[f(x_{i+1}) - f(x_i) \approx f'(x_i)(x_{i+1} - x_i).\]

Thus,

\[f(b) - f(a) \approx \sum_{i=0}^{N-1} f'(x_i) \Delta x_i \approx \int_a^b f'(x) \, dx,\]

where \( \Delta x_i = x_{i+1} - x_i \).

We can convert this intuitive argument into a rigorous proof. It helps a lot that we can use the Mean Value Theorem to replace the approximation

\[f(x_{i+1}) - f(x_i) \approx f'(x_i)(x_{i+1} - x_i)\] with the exact equality \[f(x_{i+1}) - f(x_i) = f'(c_i)(x_{i+1} - x_i)\] for some \( c_i \in (x_i, x_{i+1}) \). This gives us

\[f(b) - f(a) = \sum_{i=0}^{N-1} f'(c_i) \Delta x_i.\]

Given \( \epsilon > 0 \), it's possible to partition \([a,b]\) finely enough that the Riemann sum

\[\sum_{i=0}^{N-1} f'(c_i) \Delta x_i \] is within \( \epsilon \) of \[\int_a^b f'(x) \, dx.\]

(This is one definition of Riemann integrability.) Since \( \epsilon > 0 \) is arbitrary, this implies that \[f(b) - f(a) = \int_a^b f'(x) \, dx.\]

The Fundamental Theorem of Calculus is a perfect example of a theorem where:

1. The intuition is extremely clear;

2. The intuition can be converted directly into a rigorous proof.

Background knowledge: The approximation

\[f(x_{i+1}) - f(x_i) \approx f'(x_i)(x_{i+1} - x_i)\] is just a restatement of what I consider to be the most important idea in calculus: if \( f \) is differentiable at \( x \), the 

\[f(x + \Delta x) \approx f(x) + f'(x) \Delta x.\]

The approximation is good when \( \Delta x \) is small. This approximation is essentially the definition of \( f'(x) \):

\[f'(x) = \lim_{\Delta x \to 0} \frac{f(x + \Delta x) - f(x)}{\Delta x}.\]

If \( \Delta x \) is a tiny nonzero number, then we have

\[\Rightarrow f'(x) \approx \frac{f(x + \Delta x) - f(x)}{\Delta x} \quad \text{and} \quad f(x + \Delta x) \approx f(x) + f'(x) \Delta x.\]

Indeed, the whole point of \( f'(x) \) is to give us a local linear approximation to \( f \) at \( x \). The essence of calculus is to study functions that are "locally linear" in the sense that a good linear approximation exists. The term "differentiable" could even be replaced with the more descriptive term "locally linear."

With this perspective on what calculus is, we see that calculus and linear algebra are connected at the most basic level. To define "locally linear" in the case where \( f: \mathbb{R}^n \to \mathbb{R}^m \), we first have to introduce linear transformations. To understand the local linear approximation to \( f \) at \( x \), which is a linear transformation, we must apply linear algebra concepts.

The Second Fundamental Theorem of Calculus

The second fundamental theorem is stated as: \[\frac{d}{dx} \int_a^x f(y) \, dy = f(x)\] for continuous \( f \).

Intuitive Understanding

This latter form is easier to grasp. If you replace \( x \) with \( x + \Delta x \) for a small positive \( \Delta x \), you are adding an area that is "well-approximated" by a rectangle of height \( f(x) \) and width \( \Delta x \). This can be intuitively justified with just a sketch. In the rigorous proof, however, one must carefully manage the error to ensure the property holds.

Riemanm Sum Perspective

The FTC presentation earlier is slightly more challenging to understand. One way to approach it is to consider the Riemann sum

\[\sum_{i=0}^{n-1} f'(a + i \frac{b - a}{n}) \cdot (b-a)/n.\]

On one hand, this is a Riemann sum for \[\int_a^b f'(x) \, dx.\]

On the other hand, it adds up approximations of the change in \( f \) over \([a,b]\) by following the tangent line at \( n \) points. Since the tangent line is the best possible linear approximation, we can reasonably expect that this approximation should be quite accurate, especially as \( n \) becomes large. Once again, the rigorous proof requires careful attention to error bounds to ensure that as \( n \to \infty \), we indeed obtain \( f(b) - f(a) \).

Conclusion: The Importance of Understanding the FTC

In conclusion, the Fundamental Theorem of Calculus is more than just a formula in mathematics. It serves as a key to understanding how functions change, bridging the concepts of derivatives and integrals. Mastering these theorems will empower us to analyze and comprehend changes in various phenomena, whether in the natural sciences, economics, or engineering. Learning the FTC equips us with the tools to tackle complex problems and broadens our perspective within mathematics and its applications.