Study Notes: Sequences, Limits, and Derivatives (Calculus I)

Sequences

Types of Sequences

Sequences are ordered lists of numbers following specific patterns or rules. Understanding different types of sequences is fundamental in calculus and mathematical analysis.

• Arithmetic Sequence: Each term is obtained by adding a constant difference to the previous term.

  • General formula: $a_{n+1} = a_n + d$

  • Example: 2, 5, 8, 11, ... (here, $d = 3$)

• Geometric Sequence: Each term is obtained by multiplying the previous term by a constant ratio.

  • General formula: $g_{n+1} = r g_n$, or $g_n = r^{n-1} g_1$

  • Example: 3, 6, 12, 24, ... (here, $r = 2$)

• Fibonacci Sequence: Each term is the sum of the two preceding terms.

  • General formula: $f_{n+2} = f_{n+1} + f_n$

  • Example: 0, 1, 1, 2, 3, 5, 8, ...

• Recursively Defined Sequence: A sequence where the first few terms are given, and subsequent terms are defined using a formula involving previous terms.

  • Examples include Fibonacci, arithmetic, and geometric sequences.

Summation Notation

Summation notation is used to represent the sum of a sequence of terms.

• General form: $a_1 + a_2 + a_3 + \cdots + a_n = \sum_{k=1}^n a_k$

• Index of summation (k): Indicates where to start and end the sum.

• Behaves like normal algebraic expressions (e.g., $2(x+y) = 2x + 2y$)

Properties of Summation

• $\sum_{k=1}^n (c a_k) = c \sum_{k=1}^n a_k$

• $\sum_{k=1}^n (a_k + b_k) = \sum_{k=1}^n a_k + \sum_{k=1}^n b_k$

• $\sum_{k=1}^n (a_k - b_k) = \sum_{k=1}^n a_k - \sum_{k=1}^n b_k$

• $\sum_{k=j}^n a_k = \sum_{k=1}^n a_k - \sum_{k=1}^{j-1} a_k$ (where $0 < j < n$)

Common Summation Formulas

• $\sum_{k=1}^n k = 1 + 2 + 3 + \cdots + n = \frac{n(n+1)}{2}$

• $\sum_{k=1}^n k^2 = 1^2 + 2^2 + 3^2 + \cdots + n^2 = \frac{n(n+1)(2n+1)}{6}$

• $\sum_{k=1}^n k^3 = 1^3 + 2^3 + 3^3 + \cdots + n^3 = \left(\frac{n(n+1)}{2}\right)^2$

• Sum of a finite geometric sequence:

  • $\sum_{k=1}^n r^{k-1} g_1 = g_1 \frac{1 - r^n}{1 - r}$, when $r \neq 1$

Limits

Limit Properties

Limits describe the behavior of a function as its input approaches a particular value. The following are fundamental limit laws:

• Sum Rule: $\lim_{x \to c} [f(x) + g(x)] = L + M$

• Difference Rule: $\lim_{x \to c} [f(x) - g(x)] = L - M$

• Constant Multiple Rule: $\lim_{x \to c} [k f(x)] = kL$

• Product Rule: $\lim_{x \to c} [f(x) g(x)] = L \cdot M$

• Quotient Rule: $\lim_{x \to c} \frac{f(x)}{g(x)} = \frac{L}{M}$, $M \neq 0$

• Power Rule: $\lim_{x \to c} [f(x)]^n = L^n$, $n$ a positive integer

• Root Rule: $\lim_{x \to c} \sqrt[n]{f(x)} = \sqrt[n]{L}$, $n$ a positive integer (if $n$ is even, assume $\lim_{x \to c} f(x) > 0$)

Continuity

A function is continuous at a number $a$ if:

• $f(a)$ is defined

• $\lim_{x \to a} f(x)$ exists

• $\lim_{x \to a} f(x) = f(a)$

A function is continuous on an open interval $(a, b)$ if it is continuous at every value in $(a, b)$.

Algebraic Limit Examples

• $(x + y)^2 = x^2 + 2xy + y^2$

• $(x - y)^2 = x^2 - 2xy + y^2$

• $(x + y)(x - y) = x^2 - y^2$

• $(x + a)(x + b) = x^2 + (a + b)x + ab$

• $(x + y + z)^2 = x^2 + y^2 + z^2 + 2xy + 2yz + 2zx$

• $(x + y)^3 = x^3 + 3x^2y + 3xy^2 + y^3$

• $(x - y)^3 = x^3 - 3x^2y + 3xy^2 - y^3$

• $x^3 + y^3 + z^3 - 3xyz = (x + y + z)(x^2 + y^2 + z^2 - xy - yz - zx)$ $x^3 + y^3 + z^3 = 3xyz$ if $x + y + z = 0$

Derivatives

Basic Derivative Rules

The derivative measures the instantaneous rate of change (IROC) of a function. The following are the foundational rules for differentiation:

• Constant Rule: $\frac{d}{dx}c = 0$

• Constant Multiple Rule: $\frac{d}{dx}[cf(x)] = c f'(x)$

• Power Rule: $\frac{d}{dx}x^n = n x^{n-1}$

• Sum Rule: $\frac{d}{dx}[f(x) + g(x)] = f'(x) + g'(x)$

• Difference Rule: $\frac{d}{dx}[f(x) - g(x)] = f'(x) - g'(x)$

• Product Rule: $\frac{d}{dx}[f(x)g(x)] = f(x)g'(x) + f'(x)g(x)$

• Quotient Rule: $\frac{d}{dx}\left[\frac{f(x)}{g(x)}\right] = \frac{f'(x)g(x) - f(x)g'(x)}{[g(x)]^2}$

• Chain Rule: $\frac{d}{dx}f(g(x)) = f'(g(x))g'(x)$

Derivative Rules for Common Functions

Function Type	Derivative Rule
Exponential Functions	$\frac{d}{dx}e^x = e^x$ $\frac{d}{dx}a^x = a^x \ln a$
Logarithmic Functions	$\frac{d}{dx}\ln x = \frac{1}{x}$ $\frac{d}{dx}\log_a x = \frac{1}{x \ln a}$
Trigonometric Functions	$\frac{d}{dx}\sin x = \cos x$ $\frac{d}{dx}\cos x = -\sin x$ $\frac{d}{dx}\tan x = \sec^2 x$ $\frac{d}{dx}\cot x = -\csc^2 x$ $\frac{d}{dx}\sec x = \sec x \tan x$ $\frac{d}{dx}\csc x = -\csc x \cot x$
Inverse Trigonometric Functions	$\frac{d}{dx}\arcsin x = \frac{1}{\sqrt{1 - x^2}}$ $\frac{d}{dx}\arccos x = -\frac{1}{\sqrt{1 - x^2}}$ $\frac{d}{dx}\arctan x = \frac{1}{1 + x^2}$ $\frac{d}{dx}\arccot x = -\frac{1}{1 + x^2}$ $\frac{d}{dx}\arcsec x = \frac{1}{|x|\sqrt{x^2 - 1}}$ $\frac{d}{dx}\arccsc x = -\frac{1}{|x|\sqrt{x^2 - 1}}$
Hyperbolic Functions	$\frac{d}{dx}\sinh x = \cosh x$ $\frac{d}{dx}\cosh x = \sinh x$ $\frac{d}{dx}\tanh x = \text{sech}^2 x$ $\frac{d}{dx}\coth x = -\text{csch}^2 x$ $\frac{d}{dx}\text{sech} x = -\text{sech} x \tanh x$ $\frac{d}{dx}\text{csch} x = -\text{csch} x \coth x$
Inverse Hyperbolic Functions	$\frac{d}{dx}\text{arcsinh} x = \frac{1}{\sqrt{x^2 + 1}}$ $\frac{d}{dx}\text{arccosh} x = \frac{1}{\sqrt{x^2 - 1}}$ $\frac{d}{dx}\text{arctanh} x = \frac{1}{1 - x^2}$ $\frac{d}{dx}\text{arccoth} x = \frac{1}{1 - x^2}$ $\frac{d}{dx}\text{arcsech} x = -\frac{1}{x\sqrt{1 - x^2}}$ $\frac{d}{dx}\text{arccsch} x = -\frac{1}{|x|\sqrt{1 + x^2}}$

Derivative is IROC: The derivative at a point gives the Instantaneous Rate of Change of the function at that point.