微分中值定理的证明英文书

微分中值定理的证明英文书《The Proof of the Mean Value Theorem in Differential Calculus》
The Mean Value Theorem is a fundamental result in differential calculus, which states that if a function is continuous on a closed interval and differentiable on the open interval, then there exists at least one point in the open interval where the derivative of the function is equal to the average rate of change of the function over the closed interval. The theorem has many important applications in mathematics and science, and its proof is a key milestone in the study of differential calculus.
To prove the Mean Value Theorem, we first need to consider the function f(x) and its derivative f'(x). Since f(x) is continuous on the closed interval [a, b] and differentiable on the open interval (a, b), we can apply the Intermediate Value Theorem to the derivative f'(x) to show that it takes on every value between f'(a) and f'(b) at some point c in the open interval (a, b).
Next, we consider the average rate of change of f(x) over the interval [a, b], which is given by the difference in the function values divided by the difference in the input values:
Average rate of change = (f(b) - f(a))/(b - a)
Now, we can construct a new function g(x) = f(x) - (f(b) - f(a))/(b - a) * x, which represents the difference between f(x) and the line with slope equal to the average rate of change. Notice that g(a) = f(a) and g(b) = f(b), which means g(x) is continuous on the closed interval and differentiable on the open interval.
By applying Rolle's Theorem to the function g(x), we find that there exists at least one point d in the open interval (a, b) where g'(d) = 0. Simplifying g'(x) gives us g'(x) = f'(x) - (f(b) - f(a))/(b - a), and therefore g'(d) = f'(d) - (f(b) - f(a))/(b - a) = 0. Rearranging the terms, we get f'(d) = (f(b) - f(a))/(b - a), which is the desired result of the Mean Value Theorem.
Therefore, we have successfully proven the Mean Value Theorem for the function f(x) over the interval [a, b] by using the properties of derivatives and applying Rolle's Theorem. This result has significant implications for the behavior of differentiable functions and is a fundamental tool in the study of calculus.。