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Thread: Fundamental Theorem of Calculus

  1. #1
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    Fundamental Theorem of Calculus

    Theorem. Let $\displaystyle a \in \mathbb{R} $, and let $\displaystyle I $ be an interval containing $\displaystyle a $. Let $\displaystyle I \subseteq K \subseteq \mathbb{R}$. Suppose that the function $\displaystyle f: K \to \mathbb{R} $ is integrable on $\displaystyle [a,x] $ for all $\displaystyle x \in I $ with $\displaystyle x > a $ and on $\displaystyle [x,a] $ for all $\displaystyle x \in I $ with $\displaystyle x < a $. Let $\displaystyle F_a $ be the area accumulation function for $\displaystyle f $ on $\displaystyle I $ based at $\displaystyle a $. Then $\displaystyle F_a $ is uniformly continuous on $\displaystyle I $. If $\displaystyle f $ happens to be continuous on $\displaystyle I $, then $\displaystyle F_a $ is differentiable on $\displaystyle I $ and $\displaystyle F_{a}'(x) = f(x) $ for all $\displaystyle x \in I $.

    Proof. Suppose $\displaystyle a \in \mathbb{R} $, $\displaystyle I $ an interval containing $\displaystyle a $, and let $\displaystyle f: I \to \mathbb{R} $ be a function. The area accumulation function $\displaystyle F_a: I \to \mathbb{R} $ is defined by $\displaystyle F_{a}(x) = \int_{a}^{x} f(t) \ dt $. (1) We want to show that $\displaystyle F_a $ is uniformly continuous on $\displaystyle I $. To do this, would we use a sequence approach? Because $\displaystyle F_a $ is not Lipschitz. (2) Suppose $\displaystyle f $ is continuous on $\displaystyle I $. We want to show that $\displaystyle F_a $ is differentiable at some point $\displaystyle c $. So we look at the following:

    $\displaystyle \frac{F_{a}(x)-F_{a}(c)}{x-c} = \frac{\int_{a}^{x} f(t) \ dt- \int_{a}^{c} f(t) \ dt}{x-c} = \frac{1}{x-c} \int_{x}^{c} f(t) \ dt $

    We want to show that $\displaystyle \lim\limits_{x \to c} \frac{1}{x-c} \int_{c}^{x} f(t) \ dt = f(c) $. And so $\displaystyle f(c) = \frac{1}{x-c} \int_{c}^{x} f(c) \ dt $. The result follows. $\displaystyle \diamond $

    Is the second part correct? In the first part, would the easiest way be to use a sequence approach? Or just the regular $\displaystyle \epsilon-\delta $ definition?
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  2. #2
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    Quote Originally Posted by manjohn12 View Post
    Theorem. Let $\displaystyle a \in \mathbb{R} $, and let $\displaystyle I $ be an interval containing $\displaystyle a $. Let $\displaystyle I \subseteq K \subseteq \mathbb{R}$. Suppose that the function $\displaystyle f: K \to \mathbb{R} $ is integrable on $\displaystyle [a,x] $ for all $\displaystyle x \in I $ with $\displaystyle x > a $ and on $\displaystyle [x,a] $ for all $\displaystyle x \in I $ with $\displaystyle x < a $. Let $\displaystyle F_a $ be the area accumulation function for $\displaystyle f $ on $\displaystyle I $ based at $\displaystyle a $. Then $\displaystyle F_a $ is uniformly continuous on $\displaystyle I $. If $\displaystyle f $ happens to be continuous on $\displaystyle I $, then $\displaystyle F_a $ is differentiable on $\displaystyle I $ and $\displaystyle F_{a}'(x) = f(x) $ for all $\displaystyle x \in I $.

    Proof. Suppose $\displaystyle a \in \mathbb{R} $, $\displaystyle I $ an interval containing $\displaystyle a $, and let $\displaystyle f: I \to \mathbb{R} $ be a function. The area accumulation function $\displaystyle F_a: I \to \mathbb{R} $ is defined by $\displaystyle F_{a}(x) = \int_{a}^{x} f(t) \ dt $. (1) We want to show that $\displaystyle F_a $ is uniformly continuous on $\displaystyle I $. To do this, would we use a sequence approach? Because $\displaystyle F_a $ is not Lipschitz. (2) Suppose $\displaystyle f $ is continuous on $\displaystyle I $. We want to show that $\displaystyle F_a $ is differentiable at some point $\displaystyle c $. So we look at the following:

    $\displaystyle \frac{F_{a}(x)-F_{a}(c)}{x-c} = \frac{\int_{a}^{x} f(t) \ dt- \int_{a}^{c} f(t) \ dt}{x-c} = \frac{1}{x-c} \int_{x}^{c} f(t) \ dt $

    We want to show that $\displaystyle \lim\limits_{x \to c} \frac{1}{x-c} \int_{c}^{x} f(t) \ dt = f(c) $. And so $\displaystyle f(c) = \frac{1}{x-c} \int_{c}^{x} f(c) \ dt $. The result follows. $\displaystyle \diamond $

    Is the second part correct? In the first part, would the easiest way be to use a sequence approach? Or just the regular $\displaystyle \epsilon-\delta $ definition?

    So for (1), would a sequence approach be the best?
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