The Fundamental Theorem of Calculus is appropriately named because it establishes a connections between the two branches of calculus: differential calculus and integral calculus. Differential calculus arose from the tangent problem, whereas integral calculus arose from a seemingly unrelateed problem, the area problem. Newton's mentor at Cambridge, Isaac Barrow(1630-1677), discovered that these two problems are actually closely related. In fact, he realized that differentiation and integration are inverse processes. The Fundamental Theorem of Calculus gives the precise inverse relationship between the derivative and the integral. It was Newton and Leibniz who exploited this relationship and used it to develop calculus into a systematic mathematical method. In particular, they saw that the Fundamental Theorem of Calculus enabled them to compute areas and integrals very easily without having to compute them as limits of sums.
tangent ['tændʒ(ə)nt] n.切线；正切
limits of sums：极限和，分割→求和→求极限
The first part of the Funcamental Theorem deals with functions defined by an equation of the form
where f is a continuous function on [a,b] and x varies between a and b. Observe that F depend only on x, which appears as the variable upper limit in the integral. If x is a fixed number, then the integral A(x) is a definite number. If we then let x vary, the number A(x) also varies and define a function of x denoted by A(x)
For a continuous function y = f(x) whose graph is plotted as a curve, each value of x has a corresponding area function A(x), representing the area beneath the curve between 0 and x. The function A(x) may not be known, but it is given that it represents the area under the curve.
Look at the graphic below:
The area shaded in red stripes can be estimated as h times f(x). Alternatively, if the function A(x) were known, it could be computed exactly as A(x h) − A(x). These two values are approximately equal, particularly for small h.
The area under the curve between x and x h could be computed by finding the area between 0 and x h, then subtracting the area between 0 and x. In other words, the area of this “strip” would be A(x h) − A(x).
There is another way to estimate the area of this same strip. As shown in the accompanying figure, h is multiplied by f(x) to find the area of a rectangle that is approximately the same size as this strip. So:
A(x h) − A(x) ≈ f(x)·h
In fact, this estimate becomes a perfect equality if we add the red portion of the "excess" area shown in the diagram. So:
A(x h) − A(x) ≈ f(x)·h (Red Excess)
As h approaches 0 in the limit, the last fraction can be shown to go to zero. This is true because the area of the red portion of excess region is less than or equal to the area of the tiny black-bordered rectangle. More precisely,
By the continuity of f, the latter expression tends to zero as h does. Therefore, the left-hand side tends to zero as h does, which implies
This implies f(x) = A′(x). That is, the derivative of the area function A(x) exists and is the original function f(x); so, the area function is simply an antiderivative of the original function. Computing the derivative of a function and “finding the area” under its curve are "opposite" operations. This is the crux of the Fundamental Theorem of Calculus.
crux [krʌks] 症结；十字座；关键；难题
Physical intuition：
Intuitively, the theorem simply states that the sum of infinitesimal changes in a quantity over time (or over some other variable) adds up to the net change in the quantity.
Imagine for example using a stopwatch to mark-off tiny increments of time as a car travels down a highway. Imagine also looking at the car's speedometer as it travels, so that at every moment you know the velocity of the car. To understand the power of this theorem, imagine also that you are not allowed to look out the window of the car, so that you have no direct evidence of how far the car has traveled.
For any tiny interval of time in the car, you could calculate how far the car has traveled in that interval by multiplying the current speed of the car times the length of that tiny interval of time. (This is because distance = speed × time.)
Now imagine doing this instant after instant, so that for every tiny interval of time you know how far the car has traveled. In principle, you could then calculate the total distance traveled in the car (even though you've never looked out the window) by simply summing-up all those tiny distances.
distance traveled = ∑ the velocity at any instant × a tiny interval of time
In other words,
distance traveled = v(t) × △t
On the right hand side of this equation, as △t becomes infinitesimally small, the operation of "summing up" corresponds to integration. So what we've shown is that the integral of the velocity function can be used to compute how far the car has traveled.
Now remember that the velocity function is simply the derivative of the position function. So what we have really shown is that integrating the velocity simply recovers the original position function. This is the basic idea of the theorem: that integration and differentiation are closely related operations, each essentially being the inverse of the other.
In other words, in terms of one's physical intuition, the theorem simply states that the sum of the changes in a quantity over time (such as position, as calculated by multiplying velocity times time) adds up to the total net change in the quantity. Or to put this more generally:
Given a quantity x that changes over some variable t, and
Given the velocity v(t) with which that quantity changes over that variable
then the idea that "distance equals speed times time" corresponds to the statement
dx = v(t)dt
meaning that one can recover the original function x(t) by integrating its derivative, the velocity v(t), over t.
Corollary推论The fundamental theorem is often employed to compute the definite integral of a function f for which an antiderivative F is known. Specifically, if f is a real-valued continuous function on [a,b] and F is an antiderivative of f in [a,b] then
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