Theorems from Calculus

Appendix A Theorems from Calculus

Phyllis explained to him, trying to give of her deeper self, “Don’t you find it so beautiful, math? Like an endless sheet of gold chains, each link locked into the one before it, the theorems and functions, one thing making the next inevitable. It’s music, hanging there in the middle of space, meaning nothing but itself, and so moving...”
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―John Updike (1932–2009)
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Here we collect a few theorems from real calculus that we make use of in the course of the text.

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Theorem A.0.1. Extreme-Value Theorem.

Suppose \(K \subset \R^n\) is closed and bounded and \(f: K \to \R\) is continuous. Then \(f\) has a minimum and maximum value, i.e.,

\begin{equation*} \min_{ x \in K } f(x) \qquad \text{ and } \qquad \max_{ x \in K } f(x) \end{equation*}

exist in \R

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Theorem A.0.2. Mean-Value Theorem.

Suppose \(I \subseteq \R\) is an interval, \(f: I \to \R\) is differentiable, and \(x, \, x + \D x \in I\text{.}\) Then there exists \(0\lt a \lt 1\) such that

\begin{equation*} \frac{f(x + \D x) - f(x)}{ \D x } = f'(x + a \, \D x) \, \text{.} \end{equation*}

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Many of the most important results of analysis concern combinations of limit operations. The most important of all calculus theorems combines differentiation and integration (in two ways):

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Theorem A.0.3. Fundamental Theorem of Calculus.

Suppose \(f: [a,b] \to \R\) is continuous.

The function \(F: [a,b] \to \R\) defined by \(F(x)=\int_a^x f(t)\,\diff{t}\) is differentiable and \(F'(x)=f(x)\text{.}\)
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If \(F\) is any antiderivative of \(f\text{,}\) that is, \(F' = f\text{,}\) then \(\int_a^b f(x) \, \diff{x} = F(b) - F(a)\text{.}\)
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Theorem A.0.4.

If \(f, g : [a,b] \to \R\) are continuous functions and \(c \in \R\) then

\begin{equation*} \int_a^b \bigl( f(x) + c \, g(x) \bigr) \, \diff{x} \ = \ \int_a^b f(x) \, \diff{x} + c \int_a^b g(x) \, \diff{x} \,\text{.} \end{equation*}

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Theorem A.0.5.

If \(f, g : [a,b] \to \R\) are continuous functions then

\begin{equation*} \left| \int_a^b f(x) \, g(x) \, \diff{x} \right| \ \le \ \int_a^b \left| f(x) \, g(x) \right| \diff{x} \ \le \ \left( \max_{ a \le x \le b } |f(x)| \right) \int_a^b \left| g(x) \right| \diff{x} \,\text{.} \end{equation*}

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Theorem A.0.6.

If \(g: [a,b] \to \R\) is differentiable, \(g'\) is continuous, and \(f : [g(a), g(b)] \to \R\) is continuous then

\begin{equation*} \int_a^b f(g(t)) \, g'(t) \, \diff{t} \ = \ \int_{ g(a) }^{ g(b) } f(x) \, \diff{x} \,\text{.} \end{equation*}

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For functions of several variables we can perform differentiation/integration operations in any order, if we have sufficient continuity:

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Theorem A.0.7.

If the mixed partials \(\mderiv fxy\) and \(\mderiv fyx\) are defined on an open set \(G \subseteq \R^2\) and are continuous at a point \((x_0,y_0) \in G\text{,}\) then they are equal at \((x_0,y_0)\text{.}\)

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Theorem A.0.8.

If \(f\) is continuous on \([a,b] \times [c,d] \subset \R^2\) then

\begin{equation*} \int_a^b\!\int_c^d f(x,y)\,\diff{y}\,\diff{x} \ = \ \int_c^d\!\int_a^b f(x,y)\,\diff{x}\,\diff{y} \,\text{.} \end{equation*}

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We can apply differentiation and integration with respect to different variables in either order:

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Theorem A.0.9. Leibniz’s Rule.

Suppose \(f\) is continuous on \([a,b] \times [c,d] \subset \R^2\) and the partial derivative \(\fderiv fx\) exists and is continuous on \([a,b] \times [c,d]\text{.}\) Then

\begin{equation*} \frac{ d }{ \diff{x} } \int_c^d f(x,y) \, \diff{y} \ = \ \int_c^d \fderiv fx(x,y) \, \diff{y} \,\text{.} \end{equation*}

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Leibniz’s Rule follows from the Fundamental Theorem of Calculus (Theorem A.0.3). You can try to prove it, e.g., as follows: Define \(F(x)=\int_c^d f(x,y)\,\diff{y}\text{,}\) get an expression for \(F(x)-F(a)\) as an iterated integral by writing \(f(x,y)-f(a,y)\) as the integral of \(\fderiv fx\text{,}\) interchange the order of integrations, and then diafferentiate using Theorem A.0.3.

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Theorem A.0.10. Green’s Theorem.

Let \(C\) >be a positively oriented, piecewise smooth, simple, closed path in \(\R^2\) and let \(D\) be the set bounded by \(C\text{.}\) If \(f(x,y)\) and \(g(x,y)\) have continuous partial derivatives on an open region containing \(D\) then

\begin{equation*} \int_C f \, \diff{x} + g \, \diff{y} \ = \ \int_D \frac{ \partial g }{ \partial x } - \frac{ \partial f }{ \partial y } \, \diff{x} \, \diff{y} \,\text{.} \end{equation*}

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Theorem A.0.11. L’Hôspital’s Rule.

Suppose \(I\subset\R\) is an open interval and either \(c\) is in \(I\) or \(c\) is an endpoint of \(I\text{.}\) Suppose \(f\) and \(g\) are differentiable functions on \(I\setminus\{\,c\}\) with \(g'(x)\) never zero. Suppose

\begin{equation*} \lim_{x\to c} f(x) = 0,\quad \lim_{x\to c} g(x) = 0,\quad \lim_{x\to c} \frac{f'(x)}{g'(x)} = L\,\text{.} \end{equation*}

Then

\begin{equation*} \lim_{x\to c} \frac{f(x)}{g(x)} = L\,\text{.} \end{equation*}

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There are many extensions of L’Hôspital’s rule. In particular, the rule remains true if any of the following changes are made:

\(L\) is infinite..
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\(I\) is unbounded and \(c\) is an infinite endpoint of \(I\text{.}\)
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\(\lim_{x\to c}f(x)\) and \(\lim_{x\to c}g(x)\) are both infinite.
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