Reference Materials for Mathematics

Section 4.11 Theorems

• The elementary functions (polynomails, rational functions, algebraic function, trig, inverse trig, exponentials, logarithms, hyperbolics) are continuous on their domains.
• (Intermediate Value Theorem) If \(f\) is continuous on \([a,b]\) and \(k\) is a number such that \(f(a) \lt k \lt f(b)\) then \(\exists c \in (a,b)\) such that \(f(c) = k\)
• If \(f\) and \(g\) are continuous function such that \(f(x) \geq g(x)\) near \(a\) (that is, in some open interval containing \(a\)), then

  \begin{equation*} \lim_{x \rightarrow a} f(x) \geq \lim_{x \rightarrow a} g(x) \text{.} \end{equation*}

  Likewise, if \(f\) and \(g\) are continuous function such that \(f(x) \leq g(x)\) near \(a\) then

  \begin{equation*} \lim_{x \rightarrow a} f(x) \leq \lim_{x \rightarrow a} g(x) \text{.} \end{equation*}
• (The Squeeze Theorem) If \(f\) and \(g\) are continuous functions, \(h\) is any function such that \(f(x) \leq h(x) \leq g(x)\) near \(a\text{,}\) and

  \begin{equation*} \lim_{x \rightarrow a} f(x) = \lim_{x \rightarrow a} g(x) \text{,} \end{equation*}

  then the limit of \(h(x)\) at \(x=a\) exists and satisfies

  \begin{equation*} \lim_{x \rightarrow a} h(x) = \lim_{x \rightarrow a} f(x) = \lim_{x \rightarrow a} g(x) \text{.} \end{equation*}
• (Rolle’s Theorem) If \(f\) is continuous on \([a,b]\) and differentiable on \((a,b)\text{,}\) and if \(f(a) = f(b)\text{,}\) then \(\exists c \in (a,b)\) such that \(f^\prime(c) = 0\text{.}\)
• (Mean Value Theorem) If \(f\) is continuous on \([a,b]\) and differentiable on \((a,b)\text{,}\) then \(\exists c \in (a,b)\) such that

  \begin{equation*} f^\prime(c) = \frac{f(b) - f(a)}{b-a} \text{.} \end{equation*}
• (MVT Corollary) If \(f\) is continuous on \([a,b]\text{,}\) differentiable on \((a,b)\) and satisfies \(f^\prime(x) = 0\text{,}\) then \(f\) must be a constant function.
• (MVT Corollary) If \(F\) and \(G\) are both antiderivatives of \(f\text{,}\) then they must differ by a constant, i.e., \(\exists c \in \RR\) such that \(F(x) - G(x) = c\text{.}\)
• (Fundamental Theorem of Calculus I) If \(f\) is continuous on \([a,b]\text{,}\) then

  \begin{equation*} \frac{d}{dx} \int_a^x f(t) dt = f(x)\text{.} \end{equation*}
• (Fundamental Theorem of Calculus II) If \(f\) is continuous on \([a,b]\text{,}\) then

  \begin{equation*} \int_a^b f(x) dx = F(b) - F(a) \end{equation*}

  where \(F\) is any antiderivative of \(f\) on \([a,b]\text{.}\)
• (Monotonic Convergence Theorem) If \(\{a_n\}_{n=1}^\infty\) is a bounded monotonic sequence, then it is convergent.
• (Taylor’s Theorem) If \(f(x)\) is infinitely differentiable at \(c \in (a,b)\text{,}\) then for some radius of convergence around \(c\text{,}\) \(f\) can be expressed as a power series around \(c\) in the following form:

  \begin{equation*} f(x) = \sum_{n=0}^\infty \frac{f^{(n)}(c)}{n!} (x-c)^n\text{.} \end{equation*}