Math 441

Calculus Review

Basic Notation

  • \(\mathbb R\): the set of all real numbers
  • \(\mathbb N = \left\{0, 1, 2, \dots\right\}\): the set of all natural numbers
  • \(\mathbb N^* = \left\{1, 2, \dots\right\}\): the set of all positive natural numbers
  • \(\mathbb Z = \left\{dots, -2, -1, 0, 1, 2, \dots\right\}\): the set of all integers
  • \(a \in A\): the object \(a\) is an “element” or “member” of the set \(A\)
  • \(A \subset B\): the set \(A\) is a subset of the set \(B\) (they could be equal)
  • \(A \subsetneq B\): the set \(A\) is a proper subset of the set \(B\) (not equal)
  • \(f\colon A \to B\) means \(f\) is a function whose domain is a subset of \(A\), with range that is a subset of \(B\).
  • Interval notation

Limits

Let \(f: \mathbb R \to \mathbb R\) be defined on an open interval \(I\) and let \(x_0 \in I\).

We say that \(\displaystyle \lim_{x\to x_0} f(x) = L\) if for any \(\varepsilon > 0\), there exists \(\delta > 0\) such that \(\left\lvert f(x) - L\right\rvert < \varepsilon\) whenever \(0 < \left\lvert x - x_0\right\rvert < \delta\).

Let \(\displaystyle\left\{x_n\right\}_{n \in \mathbb N}\) be an infinite sequence of real numbers. We say that \(\displaystyle\lim_{n\to\infty} = x\) if, for any \(\varepsilon > 0\), there exists \(N \in \mathbb N\) such that \(\left\lvert x_n - x\right\rvert < \varepsilon\) whenever \(n > N\).

We also say that \(x_n\) converges to \(x\) as approaches infinity, in symbols \(x_n \to x\) as \(n \to \infty\).

The following two statements are equivalent:

  1. \(\displaystyle\lim_{x\to x_0} = L\)
  2. For any sequence \(\displaystyle\left\{x_n\right\}_{n \in \mathbb N} \subset I\) such that \(x_n \to x_0\) as \(n\to \infty\), the sequence \(\displaystyle\left\{f\left(x_n\right)\right\}_{n\in\mathbb N} \to L\) as \(n \to \infty\).

Continuity

Let \(f: \mathbb R \to \mathbb R\) be defined on an open interval \(I\) and let \(x_0 \in I\).

We say that \(f\) is continuous at \(x_0\) if \(\displaystyle\lim_{x\to x_0} = f(x_0)\).

The following two statements are equivalent:

  1. \(f\) is continuous at \(x_0\).
  2. For any sequence \(\displaystyle\left\{x_n\right\}_{n \in \mathbb N} \subset I\) such that \(x_n \to x_0\) as \(n\to \infty\), the sequence \(\displaystyle\left\{f\left(x_n\right)\right\}_{n\in\mathbb N} \to f(x_0)\) as \(n \to \infty\).

We say that \(f\) is continuous on a set \(A\) if it is continuous at every point \(x_0 \in A\).

Notation: The set of all functions continuous on a set \(A\) is denoted \(\mathcal{C}(A)\), or \(\mathcal{C}^0(A)\).

The Intermediate Value Theorem (IVT)

Let \(f \in \mathcal{C}([a,b])\) and let either \(f(a) < K < f(b)\) or \(f(a) > K > f(b)\). Then there is a \(c\in (a,b)\) such that \(f(c) = K\).

In other words, the equation \(f(x) = K\) has at least one solution on the interval \((a,b)\).

Derivatives

Let \(f: \mathbb R \to \mathbb R\) be defined on an open interval \(I\) and let \(x_0 \in I\).

The derivative of \(f\) at \(x_0\) is

\[f'(x_0) = \lim_{x \to x_0} \frac{f(x) - f(x_0)}{x - x_0} = \lim_{h\to 0}\frac{f(x_0 + h) - f(x_0)}{h}\]

The function \(f'\) is called the derivative of \(f\).

Notation: The set of all functions whose derivative is continuous on a set \(A\) is denoted \(\mathcal{C}^1(A)\).

These functions are called (one time) continuously differentiable on \(A\).

Notation: The set of all functions \(f\) such that \(f' \in \mathcal{C}^n(A)\) is denoted \(\mathcal{C}^{n+1}(A)\).

Notation: The set \(\mathcal{C}^\infty (A)\) is the intersection of all sets \(\mathcal{C}^{n}(A)\) for \(n = 0, 1, \dots\).

These are called infinitely differentiable functions.

The Mean Value Theorem (MVT)

Rolle’s Theorem: Let \(f \in \mathcal{C}([a,b])\) such that \(f\) is differentiable on \((a,b)\), and \(f(a) = f(b)\). Then there is a point \(c \in (a,b)\) such that \(f'(c) = 0\).

The MVT: Let \(f \in \mathcal{C}([a,b])\) such that \(f\) is differentiable on \((a,b)\). Then there is a point \(c \in (a,b)\) such that \[f'(c) = \frac{f(b) - f(a)}{b - a}.\]

\[f(b) - f(a) = f'(c)(b-a)\]

\[f(b) = f(a) + f'(c)(b-a)\]

The Taylor’s Theorem

Let \(f\in \mathcal{C}^n([a,b])\), and suppose \(f^{(n+1)}\) exists in \((a,b)\), and \(x_0 \in (a,b)\).

Then for any \(x \in (a,b)\)

\[f(x) = P_n(x) + R_n(x)\]

where \(P_n\) is the \(n\)th order Taylor polynomial

\[P_n(x) = f(x_0) + f'(x_0)(x-x_0) + \frac{f''(x_0)}{2}(x - x_0)^2 + \cdots + \frac{f^{(n)}(x_0)}{n!}(x - x_0)^n\]

and \(R_n\) is the remainder term

\[R_n(x) = \frac{f^{(n+1)}(\xi(x))}{(n+1)!}(x - x_0)^{n+1}\]

for some (generally unknown) \(\xi(x)\) between \(x\) and \(x_0\).