Elementary Analysis I: Complete Study Guide

Limits and Continuity

An Intuitive Approach to Limits

The limit is the foundational concept of calculus. It describes the behavior of a function $f(x)$ as the independent variable $x$ approaches a specific value $a$, without necessarily reaching $a$.

To distinguish between the value of a function at a real number $a$ (denoted $f(a)$) and its limit as $x$ approaches $a$:

• A function $f$ may be undefined at $a$, yet its behavior near $a$ can be fully described.
• Consider three distinct functions:
  1. $f(x) = 3x - 1$
  2. $g(x) = \frac{3x^2 - 4x + 1}{x - 1} = \frac{(3x - 1)(x - 1)}{x - 1}$ for $x \neq 1$
  3. $h(x) = \begin{cases} 3x - 1, & x \neq 1 \\ 0, & x = 1 \end{cases}$

Evaluating these functions at values near $x = 1$:

• For $f(x)$: as $x$ takes values such as $0, 0.5, 0.9, 0.99, 0.99999$ from the left, $f(x)$ produces $-1, 0.5, 1.7, 1.97, 1.99997$. From the right, for $x = 2, 1.5, 1.1, 1.001, 1.00001$, $f(x)$ produces $5, 3.5, 2.3, 2.003, 2.00003$. Thus, $f(x)$ approaches $2$.
• For $g(x)$: the function is undefined at $x = 1$. However, for all $x \neq 1$, $g(x) = 3x - 1 = f(x)$. As $x \to 1$, $g(x)$ approaches $2$.
• For $h(x)$: $h(1) = 0$, but for all $x \neq 1$, $h(x) = f(x)$. As $x \to 1$, $h(x)$ approaches $2$.

In all three cases, as $x$ gets closer to $1$, the values of the function approach $2$.

Informal Definition of Limit: Let $f$ be a function defined on an open interval $I$ containing $a$, except possibly at $a$ itself. The limit of $f(x)$ as $x$ approaches $a$ is $L \in \mathbb{R}$, denoted by: $\lim_{x \to a} f(x) = L$ if the values of $f(x)$ get closer and closer to $L$ as $x$ assumes values going closer and closer to $a$ without reaching $a$.

Alternatively, (\lim_{x \to a} f(x) = L) if $f(x)$ can be made as close to $L$ as desired by choosing $x$ sufficiently close to $a$ (where $x \neq a$).

Key Limit Remarks:

• Finding (\lim_{x \to a} f(x)) requires analyzing values of $x$ near $a$, not at $a$. The limit can exist even if $f(a)$ is undefined.
• If both (\lim_{x \to a} f(x)) and $f(a)$ exist, they are not required to be equal (e.g., $h(1) = 0$ while (\lim_{x \to 1} h(x) = 2)).
• If $f(x)$ does not approach a single real number as $x \to a$, the limit does not exist (dne).
• The Heaviside step function $H(x)$, defined by:   $H(x) = \begin{cases} 1, & x \ge 0 \\ 0, & x < 0 \end{cases}$   has no limit as $x \to 0$. Approaching $0$ from the right yields $1$, while approaching from the left yields $0$. Thus, (\lim_{x \to 0} H(x)) does not exist.

Theorem (Limit Theorems): Let $f(x)$ and $g(x)$ be functions defined on an open interval containing $a$, except possibly at $a$

1. Uniqueness: If (\lim_{x \to a} f(x)) exists, it is unique.
2. Constant Rule: For $c \in \mathbb{R}$, (\lim_{x \to a} c = c).
3. Identity Rule: (\lim_{x \to a} x = a).
4. Algebraic Rules: Suppose (\lim_{x \to a} f(x) = L_1) and (\lim_{x \to a} g(x) = L_2), where $L_1, L_2 \in \mathbb{R}$, and $c \in \mathbb{R}$.
   • Sum/Difference Rule: (\lim_{x \to a} [f(x) \pm g(x)] = L_1 \pm L_2)
   • Constant Multiple Rule: (\lim_{x \to a} [c f(x)] = c L_1)
   • Product Rule: (\lim_{x \to a} [f(x) g(x)] = L_1 L_2)
   • Quotient Rule: (\lim_{x \to a} \frac{f(x)}{g(x)} = \frac{L_1}{L_2}), provided $g(x) \neq 0$ near $a$ and $L_2 \neq 0$
   • Power Rule: (\lim_{x \to a} [f(x)]^n = (L_1)^n$for$n \in \mathbb{N}\n - Root Rule: \(\lim_{x \to a} \sqrt[n]{f(x)} = \sqrt[n]{L_1} for $n \in \mathbb{N}$, $n > 1$, provided $L_1 > 0$ when $n$ is even.

Theorem (Limits of Polynomial and Rational Functions): If $f$ is a polynomial or rational function and $a \in \text{dom}\,f$, then: $\lim_{x \to a} f(x) = f(a)$

Indeterminate Forms of Type (\left(\frac{0}{0}\right)): If (\lim_{x \to a} f(x) = 0) and (\lim_{x \to a} g(x) = 0), then (\lim_{x \to a} \frac{f(x)}{g(x)}) is called an indeterminate form of type (\left(\frac{0}{0}\right)).

• Indeterminate forms may or may not exist.
• Methods to evaluate (\left(\frac{0}{0}\right)) limits include factoring and cancelling common factors, as well as multiplying by conjugates to rationalize radical expressions.

One-Sided Limits

When evaluating (\lim_{x \to a} f(x)), the function is observed from both sides of $a$. One-sided limits handle cases where behavior differs on either side of $a$, such as piecewise functions or functions with restricted domains (e.g., $f(x) = \sqrt{x}$ near $x = 0$).

Definitions of One-Sided Limits:

• Left-Hand Limit: Let $f$ be defined on an open interval $(c, a)$. The limit of $f(x)$ as $x$ approaches $a$ from the left is $L$, denoted:   $\lim_{x \to a^-} f(x) = L$   if $f(x)$ approaches $L$ as $x$ approaches $a$ through values less than $a$.
• Right-Hand Limit: Let $f$ be defined on an open interval $(a, c)$. The limit of $f(x)$ as $x$ approaches $a$ from the right is $L$, denoted:   $\lim_{x \to a^+} f(x) = L$   if $f(x)$ approaches $L$ as $x$ approaches $a$ through values greater than $a$.

Theorem (Relationship Between One-Sided and Two-Sided Limits): $\lim_{x \to a} f(x) = L \iff \lim_{x \to a^-} f(x) = L = \lim_{x \to a^+} f(x)$ If the left-hand and right-hand limits exist but are unequal, the two-sided limit does not exist.

Approaching Zero Through Positive/Negative Values: If (\lim_{x \to a} f(x) = 0):

• We write $f(x) \to 0^+$ if $f(x)$ approaches $0$ through positive values.
• We write $f(x) \to 0^-$ if $f(x)$ approaches $0$ through negative values.

Rules for Roots of Limits Approaching Zero:

• If $f(x) \to 0^+$ as $x \to a$ and $n$ is even, then (\lim_{x \to a} \sqrt[n]{f(x)} = 0).
• If $f(x) \to 0^-$ as $x \to a$ and $n$ is even, then (\lim_{x \to a} \sqrt[n]{f(x)}) does not exist in $\mathbb{R}$.

Limits Involving Infinity

Infinite Limits: Let $f$ be a function defined on an open interval containing $a$, except possibly at $a$.

• Positive Infinite Limit: (\lim_{x \to a} f(x) = +\infty) if $f(x)$ increases without bound as $x$ approaches $a$.
• Negative Infinite Limit: (\lim_{x \to a} f(x) = -\infty) if $f(x)$ decreases without bound as $x$ approaches $a$.

Note: $\pm\infty$ are not real numbers. Stating (\lim_{x \to a} f(x) = \pm\infty) means the limit does not exist, but describes the specific unbounded growth or decay of the function near $a$.

Theorem (Infinite Limits of Quotients): Let $c \neq 0$ be a real number. Suppose (\lim_{x \to a} f(x) = c) and (\lim_{x \to a} g(x) = 0).

1. If $c > 0$:
   • If $g(x) \to 0^+$, then (\lim_{x \to a} \frac{f(x)}{g(x)} = +\infty).
   • If $g(x) \to 0^-$, then (\lim_{x \to a} \frac{f(x)}{g(x)} = -\infty).
2. If $c < 0$:
   • If $g(x) \to 0^+$, then (\lim_{x \to a} \frac{f(x)}{g(x)} = -\infty).
   • If $g(x) \to 0^-$, then (\lim_{x \to a} \frac{f(x)}{g(x)} = +\infty).

Vertical Asymptote: The line $x = a$ is a vertical asymptote of the graph of $y = f(x)$ if at least one of the following holds: $\lim_{x \to a^-} f(x) = +\infty, \quad \lim_{x \to a^-} f(x) = -\infty, \quad \lim_{x \to a^+} f(x) = +\infty, \quad \lim_{x \to a^+} f(x) = -\infty$

Indeterminate Forms Involving Infinity:

• Type (\infty - \infty): Occurs when (\lim_{x \to a} f(x) = +\infty) and (\lim_{x \to a} g(x) = +\infty), evaluating (\lim_{x \to a} [f(x) - g(x)]).
• Type (0 \cdot \infty): Occurs when (\lim_{x \to a} f(x) = 0) and (\lim_{x \to a} g(x) = \pm\infty), evaluating (\lim_{x \to a} [f(x) g(x)]).
• Type (\frac{\infty}{\infty}): Occurs when both numerator and denominator approach $\pm\infty$. These forms require algebraic re-expression (such as finding a common denominator or multiplying by conjugates) to be evaluated.

Limits at Infinity:

• (\lim_{x \to +\infty} f(x) = L): $f(x)$ approaches $L$ as $x$ increases without bound.
• (\lim_{x \to -\infty} f(x) = L): $f(x)$ approaches $L$ as $x$ decreases without bound.

Theorem (Powers at Infinity):

1. (\lim_{x \to \pm\infty} x^n = +\infty) if $n$ is an even positive integer.
2. (\lim_{x \to \pm\infty} x^n = \pm\infty) if $n$ is an odd positive integer.
3. (\lim_{x \to \pm\infty} \frac{1}{x^n} = 0) for any positive integer n$.\n4. If \(\lim_{x \to +\infty} f(x) = c\) (c \in \mathbb{R}) and \(\lim_{x \to +\infty} g(x) = \pm\infty\), then \(\lim_{x \to +\infty} \frac{f(x)}{g(x)} = 0\).\n\nHorizontal Asymptote:\nThe line y = L$is a horizontal asymptote of the graph of$y = f(x) if:\n\lim_{x \to +\infty} f(x) = L \quad \text{or} \quad \lim_{x \to -\infty} f(x) = L\n\n\n## The Formal Definition of a Limit\n\nFormal \varepsilon-\delta Definition of a Limit:\nLet f(x)$be defined on an open interval containing$a$, except possibly at$a. Then:\n\lim_{x \to a} f(x) = L\nif and only if for every \varepsilon > 0$, there exists a$\delta > 0 such that:\n|f(x) - L| < \varepsilon \quad \text{whenever} \quad 0 < |x - a| < \delta\n\nGeometric Interpretation:\n- |f(x) - L| < \varepsilon \iff L - \varepsilon < f(x) < L + \varepsilon$(an open interval of radius$\varepsilon$centered at$L on the y-axis).\n- 0 < |x - a| < \delta \iff a - \delta < x < a + \delta$with$x \neq a$(a punctured open interval of radius$\delta$centered at$a on the x-axis).\n- The formal definition asserts that for any chosen target interval I_y = (L - \varepsilon, L + \varepsilon)$, an interval$J_x = (a - \delta, a + \delta)$can be constructed such that every$x \in J_x$(except possibly$x = a$) maps into$I_y\n\nProving Limits Using \varepsilon-\delta:\n1. Scratchwork (Finding \delta$): Express$|f(x) - L|$in terms of$|x - a|$. Find a relationship$|f(x) - L| \le C |x - a|$. Set$C \delta = \varepsilon$, which suggests choosing$\delta = \frac{\varepsilon}{C}$. If$C$depends on$x$, restrict$\delta \le \delta_0$(e.g.,$\delta_0 = 1) to bound the $x$-dependent terms.\n2. Formal Proof: State: "Given \varepsilon > 0$, choose$\delta = \min{\delta_0, \frac{\varepsilon}{C}}$. If$0 < |x - a| < \delta$, then..." and show step-by-step that$|f(x) - L| < \varepsilon.\n\n\n## Continuity of Functions and the Intermediate Value Theorem\n\nContinuity at a Point:\nA function f$is continuous at$x = a if all three of the following conditions are satisfied:\n1. f(a)$is defined ($a \in \text{dom}\,f).\n2. \(\lim_{x \to a} f(x)\) exists.\n3. \(\lim_{x \to a} f(x) = f(a)\).\nOtherwise, f$is discontinuous at$x = a.\n\nClassification of Discontinuities:\n1. Removable Discontinuity: \(\lim_{x \to a} f(x)\) exists, but either f(a) is undefined or \(\lim_{x \to a} f(x) \neq f(a)\). The discontinuity can be removed by redefining f(a) = \lim_{x \to a} f(x).\n2. Essential Discontinuity: \(\lim_{x \to a} f(x)\) does not exist.\n - Jump Essential Discontinuity: Both one-sided limits \(\lim_{x \to a^-} f(x)\) and \(\lim_{x \to a^+} f(x)\) exist as real numbers, but \(\lim_{x \to a^-} f(x) \neq \lim_{x \to a^+} f(x)\).\n - Infinite Essential Discontinuity: At least one of the one-sided limits \(\lim_{x \to a^-} f(x)\) or \(\lim_{x \to a^+} f(x)\) is \pm\infty.\n\nContinuity Properties:\n- Polynomial and rational functions are continuous at every point in their domains.\n- If f$and$g$are continuous at$x = a$and$c \in \mathbb{R}$, then$f + g$,$f - g$,$f g$,$c f$, and$\frac{f}{g}$(provided$g(a) \neq 0$) are continuous at$x = a\n- One-Sided Continuity:\n - Continuous from the left at a if \(f(a) = \lim_{x \to a^-} f(x)\).\n - Continuous from the right at a if \(f(a) = \lim_{x \to a^+} f(x)\).\n\nContinuity on Intervals:\n- Continuous everywhere: Continuous at every x \in \mathbb{R}.\n- Continuous on [a, b]$: Continuous on$(a, b)$, continuous from the right at$a$, and continuous from the left at$b.\n\nComposite Function Theorems:\n- If \(\lim_{x \to a} g(x) = b\) and f$is continuous at$b, then:\n  \lim_{x \to a} f(g(x)) = f\left(\lim_{x \to a} g(x)\right) = f(b)\n- If g$is continuous at$a$and$f$is continuous at$g(a)$, then$(f \circ g)(x) = f(g(x))$is continuous at$a\n\nIntermediate Value Theorem (IVT):\nLet f$be continuous on a closed interval$[a, b]$with$f(a) \neq f(b)$. For every real number$k$between$f(a)$and$f(b)$, there exists at least one number$c \in (a, b) such that:\nf(c) = k\n\nApplication to Root-Finding:\nIf f$is continuous on$[a, b]$and$f(a)$and$f(b)$have opposite signs (meaning$f(a) f(b) < 0$), then$k = 0$lies between$f(a)$and$f(b)$. By the IVT, there exists at least one$c \in (a, b)$such that$f(c) = 0.\n\n\n## Trigonometric Functions: Limits, Continuity, and the Squeeze Theorem\n\nThe Squeeze Theorem (Sandwich Theorem):\nLet f(x)$,$g(x)$, and$h(x)$be defined on an open interval$I$containing$a$, except possibly at$x = a, such that:\nf(x) \le g(x) \le h(x) \quad \text{for all } x \in I \setminus {a}\nIf \(\lim_{x \to a} f(x) = L\) and \(\lim_{x \to a} h(x) = L\), then:\n\lim_{x \to a} g(x) = L\n\nSpecial Trigonometric Limits:\n1. \(\lim_{x \to 0} \frac{\sin x}{x} = 1\) (and reciprocally, \(\lim_{x \to 0} \frac{x}{\sin x} = 1\))\n2. \(\lim_{x \to 0} \frac{1 - \cos x}{x} = 0\)\n3. \(\lim_{x \to 0} \sin x = 0\)\n4. \(\lim_{x \to 0} \cos x = 1\)\n\nNote: Trigonometric limit formulas require x to be measured in radians.\n\nContinuity of Trigonometric Functions:\n- \(\lim_{x \to a} \sin x = \sin a\) and \(\lim_{x \to a} \cos x = \cos a\) for all a \in \mathbb{R}.\n- All six fundamental trigonometric functions (\sin x$,$\cos x$,$\tan x$,$\cot x$,$\sec x$,$\csc x) are continuous at every point in their respective domains.\n\n\n## Transcendental and Inverse Functions: Limits and Continuity\n\nInverse Functions:\n- A function f$has an inverse$f^{-1}$if and only if$f is one-to-one (passes the Horizontal Line Test).\n- Domain and Range Relations: \text{dom}\,f^{-1} = \text{ran}\,f$and$\text{ran}\,f^{-1} = \text{dom}\,f\n- Cancellation Equations: f^{-1}(f(x)) = x$for all$x \in \text{dom}\,f$, and$f(f^{-1}(x)) = x$for all$x \in \text{dom}\,f^{-1}.\n- The graph of f^{-1}$is the reflection of the graph of$f$across the line$y = x\n\nExponential and Logarithmic Functions:\n- Exponential Function: f(x) = a^x$($a > 0, a \neq 1$). Domain:$\mathbb{R}$, Range:$(0, +\infty).\n - If a > 1$:$f is increasing, \(\lim_{x \to +\infty} a^x = +\infty\), \(\lim_{x \to -\infty} a^x = 0\).\n - If 0 < a < 1$:$f is decreasing, \(\lim_{x \to +\infty} a^x = 0\), \(\lim_{x \to -\infty} a^x = +\infty\).\n- Logarithmic Function: f(x) = \log_a x$($a > 0, a \neq 1$). Inverse of$a^x$. Domain:$(0, +\infty)$, Range:$\mathbb{R}.\n - If a > 1$:$f is increasing, \(\lim_{x \to +\infty} \log_a x = +\infty\), \(\lim_{x \to 0^+} \log_a x = -\infty\).\n - If 0 < a < 1$:$f is decreasing, \(\lim_{x \to +\infty} \log_a x = -\infty\), \(\lim_{x \to 0^+} \log_a x = +\infty\).\n- Euler's Number e: Defined by the limit:\n  e = \lim_{h \to 0} (1 + h)^{1/h} \approx 2.718281828459045\n- Natural Exponential Function: f(x) = e^x\n- Natural Logarithm Function: f(x) = \ln x = \log_e x\n- Change of Base Formulas: a^x = e^{x \ln a}$and$\log_a x = \frac{\ln x}{\ln a}.\n\nInverse Circular Functions:\n1. Inverse Sine: y = \sin^{-1} x \iff x = \sin y$, where$x \in [-1, 1]$and$y \in [-\frac{\pi}{2}, \frac{\pi}{2}].\n2. Inverse Cosine: y = \cos^{-1} x \iff x = \cos y$, where$x \in [-1, 1]$and$y \in [0, \pi].\n3. Inverse Tangent: y = \tan^{-1} x \iff x = \tan y$, where$x \in \mathbb{R}$and$y \in (-\frac{\pi}{2}, \frac{\pi}{2}).\n - Limits at infinity: \(\lim_{x \to +\infty} \tan^{-1} x = \frac{\pi}{2}\), \(\lim_{x \to -\infty} \tan^{-1} x = -\frac{\pi}{2}\).\n4. Inverse Cotangent: y = \cot^{-1} x \iff x = \cot y$, where$x \in \mathbb{R}$and$y \in (0, \pi).\n5. Inverse Secant: y = \sec^{-1} x \iff x = \sec y$, where$x \in (-\infty, -1] \cup [1, +\infty)$and$y \in [0, \frac{\pi}{2}) \cup [\pi, \frac{3\pi}{2}).\n - Limits at infinity: \(\lim_{x \to +\infty} \sec^{-1} x = \frac{\pi}{2}\), \(\lim_{x \to -\infty} \sec^{-1} x = \frac{3\pi}{2}\).\n6. Inverse Cosecant: y = \csc^{-1} x \iff x = \csc y$, where$x \in (-\infty, -1] \cup [1, +\infty)$and$y \in (-\pi, -\frac{\pi}{2}] \cup (0, \frac{\pi}{2}].\n - Limits at infinity: \(\lim_{x \to +\infty} \csc^{-1} x = 0\), \(\lim_{x \to -\infty} \csc^{-1} x = -\pi\).\n\nHyperbolic Functions:\n1. Hyperbolic Sine: \sinh x = \frac{e^x - e^{-x}}{2}$, Domain:$\mathbb{R}$, Range:$\mathbb{R}\n2. Hyperbolic Cosine: \cosh x = \frac{e^x + e^{-x}}{2}$, Domain:$\mathbb{R}$, Range:$[1, +\infty)\n3. Hyperbolic Tangent: \tanh x = \frac{\sinh x}{\cosh x} = \frac{e^x - e^{-x}}{e^x + e^{-x}}$, Domain:$\mathbb{R}$, Range:$(-1, 1)\n4. Hyperbolic Cotangent: \coth x = \frac{\cosh x}{\sinh x} = \frac{e^x + e^{-x}}{e^x - e^{-x}}$, Domain:$\mathbb{R} \setminus {0}$, Range:$(-\infty, -1) \cup (1, +\infty)\n5. Hyperbolic Secant: \text{sech}\,x = \frac{1}{\cosh x} = \frac{2}{e^x + e^{-x}}$, Domain:$\mathbb{R}$, Range:$(0, 1]\n6. Hyperbolic Cosecant: \text{csch}\,x = \frac{1}{\sinh x} = \frac{2}{e^x - e^{-x}}$, Domain:$\mathbb{R} \setminus {0}$, Range:$\mathbb{R} \setminus {0}$

Fundamental Hyperbolic Identities:

• $\cosh x + \sinh x = e^x$
• $\cosh x - \sinh x = e^{-x}$
• $\cosh^2 x - \sinh^2 x = 1$
• $1 - \tanh^2 x = \text{sech}^2 x$
• $1 - \coth^2 x = -\text{csch}^2 x$
• $\sinh(x \pm y) = \sinh x \cosh y \pm \cosh x \sinh y$
• $\cosh(x \pm y) = \cosh x \cosh y \pm \sinh x \sinh y$
• $\sinh(2x) = 2 \sinh x \cosh x$
• $\cosh(2x) = \cosh^2 x + \sinh^2 x = 2 \cosh^2 x - 1 = 1 + 2 \sinh^2 x$

Inverse Hyperbolic Functions Expressed as Logarithms:

1. $\sinh^{-1} x = \ln\left(x + \sqrt{x^2 + 1}\right)$ for $x \in \mathbb{R}$
2. $\cosh^{-1} x = \ln\left(x + \sqrt{x^2 - 1}\right)$ for $x \ge 1$
3. $\tanh^{-1} x = \frac{1}{2} \ln\left(\frac{1 + x}{1 - x}\right)$ for $|x| < 1$
4. $\coth^{-1} x = \frac{1}{2} \ln\left(\frac{x + 1}{x - 1}\right)$ for $|x| > 1$
5. $\text{sech}^{-1} x = \ln\left(\frac{1 + \sqrt{1 - x^2}}{x}\right)$ for $x \in (0, 1]$
6. $\text{csch}^{-1} x = \ln\left(\frac{1}{x} + \frac{\sqrt{1 + x^2}}{|x|}\right)$ for $x \neq 0$

Derivatives and Differentiation

Slopes, Tangent Lines, and the Definition of the Derivative

Tangent Line Definition: Let $P(x_0, f(x_0))$ be a point on the graph of $y = f(x)$. Let $Q(x_0 + \Delta x, f(x_0 + \Delta x))$ be another point on the curve. The secant line $\overleftrightarrow{PQ}$ has slope: $m_{PQ} = \frac{f(x_0 + \Delta x) - f(x_0)}{\Delta x}$ As $Q \to P$ ($\Delta x \to 0$), the secant line approaches the tangent line $\ell$ at $P$.

The slope $m$ of the tangent line at $P(x_0, f(x_0))$ is: $m = \lim_{\Delta x \to 0} \frac{f(x_0 + \Delta x) - f(x_0)}{\Delta x}$ provided this limit exists.

• If (\lim_{\Delta x \to 0} \frac{f(x_0 + \Delta x) - f(x_0)}{\Delta x} = \pm\infty), the tangent line is vertical, given by $x = x_0$.

Normal Line Definition: The normal line to the graph of $f$ at $P(x_0, f(x_0))$ is the line perpendicular to the tangent line at $P$. Its slope is $m_{\text{normal}} = -\frac{1}{m}$, provided $m \neq 0$.

Definition of the Derivative: The derivative of a function $f$, denoted $f'$, is the function defined by: $f'(x) = \lim_{\Delta x \to 0} \frac{f(x + \Delta x) - f(x)}{\Delta x}$ at all points $x$ in the domain of $f$ where the limit exists.

Alternative Definition at a Point $x_0$: $f'(x_0) = \lim_{x \to x_0} \frac{f(x) - f(x_0)}{x - x_0}$

Notations for Derivative: $y', \quad f'(x), \quad \frac{dy}{dx}, \quad \frac{d}{dx}[f(x)], \quad D_x[f(x)]$

Basic Differentiation Rules and Derivatives of Trigonometric Functions

Differentiation Rules: Let $f$ and $g$ be differentiable functions, and $c, n \in \mathbb{R}$.

1. Constant Rule: $\frac{d}{dx}[c] = 0$
2. Power Rule: $\frac{d}{dx}[x^n] = n x^{n-1}$
3. Constant Multiple Rule: $\frac{d}{dx}[c f(x)] = c f'(x)$
4. Sum/Difference Rule: $\frac{d}{dx}[f(x) \pm g(x)] = f'(x) \pm g'(x)$
5. Product Rule: $\frac{d}{dx}[f(x) g(x)] = f'(x) g(x) + f(x) g'(x)$
6. Quotient Rule: $\frac{d}{dx}\left[\frac{f(x)}{g(x)}\right] = \frac{g(x) f'(x) - f(x) g'(x)}{[g(x)]^2}$ (where $g(x) \neq 0$)

Derivatives of Trigonometric Functions:

1. $\frac{d}{dx}[\sin x] = \cos x$
2. $\frac{d}{dx}[\cos x] = -\sin x$
3. $\frac{d}{dx}[\tan x] = \sec^2 x$
4. $\frac{d}{dx}[\cot x] = -\csc^2 x$
5. $\frac{d}{dx}[\sec x] = \sec x \tan x$
6. $\frac{d}{dx}[\csc x] = -\csc x \cot x$

The Chain Rule, One-Sided Derivatives, and Differentiability

The Chain Rule: If $g$ is differentiable at $x$ and $f$ is differentiable at $g(x)$, then the composite function $(f \circ g)(x) = f(g(x))$ is differentiable at $x$, and: $(f \circ g)'(x) = f'(g(x)) \cdot g'(x)$ In Leibniz notation, if $y = f(u)$ and $u = g(x)$, then: $\frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dx}$

One-Sided Derivatives:

• Left-Hand Derivative at $x_0$:   $f'_{-}(x_0) = \lim_{x \to x_0^-} \frac{f(x) - f(x_0)}{x - x_0}$
• Right-Hand Derivative at $x_0$:   $f'_{+}(x_0) = \lim_{x \to x_0^+} \frac{f(x) - f(x_0)}{x - x_0}$
• $f$ is differentiable at $x_0$ if and only if $f'_{-}(x_0)$ and $f'_{+}(x_0)$ both exist and $f'_{-}(x_0) = f'_{+}(x_0) = f'(x_0)$.

Theorem (Differentiability Implies Continuity): If $f$ is differentiable at $x = x_0$, then $f$ is continuous at $x = x_0$.

• The converse is false: continuity does not guarantee differentiability (e.g., $f(x) = |x|$ at $x = 0$ is continuous, but $f'_{-}(0) = -1 \neq 1 = f'_{+}(0)$).
• A function fails to be differentiable at $x_0$ if:
  1. $f$ is discontinuous at $x_0$
  2. The graph has a sharp corner, edge, or cusp at $x_0$
  3. The graph has a vertical tangent line at $x_0$

Higher Order Derivatives and Implicit Differentiation

Higher Order Derivatives:

• Second derivative: $f''(x) = \frac{d^2 y}{dx^2} = \frac{d}{dx}[f'(x)]$
• Third derivative: $f'''(x) = \frac{d^3 y}{dx^3} = \frac{d}{dx}[f''(x)]$
• $n$-th derivative: $f^{(n)}(x) = \frac{d^n y}{dx^n} = \frac{d}{dx}\left[f^{(n-1)}(x)\right]$

Implicit Differentiation: When an equation defines $y$ implicitly as a function of $x$:

1. Treat $y$ as an implicit differentiable function of $x$ (applying the chain rule: $\frac{d}{dx}[y^n] = n y^{n-1} \frac{dy}{dx}$).
2. Differentiate both sides of the equation with respect to x$.\n3. Collect all terms containing \frac{dy}{dx}$on one side and factor out$\frac{dy}{dx}.\n4. Solve algebraically for \frac{dy}{dx}.\n\n\n## Derivatives of Exponential and Logarithmic Functions\n\nDerivatives of Logarithmic Functions:\n1. \frac{d}{dx}[\ln x] = \frac{1}{x}$(for$x > 0)\n2. \frac{d}{dx}[\log_a x] = \frac{1}{x \ln a}$(for$x > 0$,$a > 0, a \neq 1)\n3. \frac{d}{dx}[\ln |x|] = \frac{1}{x}$(for$x \neq 0)\n\nLogarithmic Differentiation:\nUsed for functions involving complex products, quotients, and powers, or functions of the form y = [f(x)]^{g(x)}:\n1. Take the absolute value and then natural logarithm of both sides: \ln |y| = \ln |f(x)|.\n2. Use properties of logarithms to expand products into sums, quotients into differences, and exponents into factors.\n3. Differentiate both sides implicitly with respect to x$:$\frac{1}{y} \frac{dy}{dx} = \frac{d}{dx}[\text{expanded side}].\n4. Multiply by y$to solve for$\frac{dy}{dx}$, substituting the original expression for$y\n\nDerivatives of Exponential Functions:\n1. \frac{d}{dx}[e^x] = e^x\n2. \frac{d}{dx}[a^x] = a^x \ln a$($a > 0, a \neq 1)\n\nGeneral Power Rule:\nFor any real number r \in \mathbb{R}$and$x > 0:\n\frac{d}{dx}[x^r] = r x^{r-1}\n\nDerivative of Variable Base and Exponent [f(x)]^{g(x)}$($f(x) > 0):\nRewrite as y = e^{g(x) \ln f(x)}, then apply the chain rule:\n\frac{d}{dx}\left[[f(x)]^{g(x)}\right] = [f(x)]^{g(x)} \left[ g'(x) \ln f(x) + g(x) \frac{f'(x)}{f(x)} \right]\n\n\n## Derivatives of Inverse Circular, Hyperbolic, and Inverse Hyperbolic Functions\n\nDerivatives of Inverse Circular Functions:\n1. \frac{d}{dx}[\sin^{-1} x] = \frac{1}{\sqrt{1 - x^2}}\n2. \frac{d}{dx}[\cos^{-1} x] = -\frac{1}{\sqrt{1 - x^2}}\n3. \frac{d}{dx}[\tan^{-1} x] = \frac{1}{1 + x^2}\n4. \frac{d}{dx}[\cot^{-1} x] = -\frac{1}{1 + x^2}\n5. \frac{d}{dx}[\sec^{-1} x] = \frac{1}{x \sqrt{x^2 - 1}}\n6. \frac{d}{dx}[\csc^{-1} x] = -\frac{1}{x \sqrt{x^2 - 1}}\n\nDerivatives of Hyperbolic Functions:\n1. \frac{d}{dx}[\sinh x] = \cosh x\n2. \frac{d}{dx}[\cosh x] = \sinh x\n3. \frac{d}{dx}[\tanh x] = \text{sech}^2 x\n4. \frac{d}{dx}[\coth x] = -\text{csch}^2 x\n5. \frac{d}{dx}[\text{sech}\,x] = -\text{sech}\,x \tanh x\n6. \frac{d}{dx}[\text{csch}\,x] = -\text{csch}\,x \coth x\n\nDerivatives of Inverse Hyperbolic Functions:\n1. \frac{d}{dx}[\sinh^{-1} x] = \frac{1}{\sqrt{x^2 + 1}}\n2. \frac{d}{dx}[\cosh^{-1} x] = \frac{1}{\sqrt{x^2 - 1}}$(for$x > 1)\n3. \frac{d}{dx}[\tanh^{-1} x] = \frac{1}{1 - x^2}$(for$|x| < 1)\n4. \frac{d}{dx}[\coth^{-1} x] = \frac{1}{1 - x^2}$(for$|x| > 1)\n5. \frac{d}{dx}[\text{sech}^{-1} x] = -\frac{1}{x \sqrt{1 - x^2}}$(for$x \in (0, 1))\n6. \frac{d}{dx}[\text{csch}^{-1} x] = -\frac{1}{|x| \sqrt{x^2 + 1}}$(for$x \neq 0)\n\n\n## Indeterminate Forms and L'Hôpital's Rule\n\nL'Hôpital's Rule:\nLet f$and$g$be differentiable on an open interval$I$containing$a$(except possibly at$a$), and$g'(x) \neq 0$for all$x \in I \setminus {a}.\nIf \(\lim_{x \to a} \frac{f(x)}{g(x)}\) is an indeterminate form of type \(\left(\frac{0}{0}\right)\) or \(\left(\frac{\infty}{\infty}\right)\), then:\n\lim_{x \to a} \frac{f(x)}{g(x)} = \lim_{x \to a} \frac{f'(x)}{g'(x)}\nprovided \(\lim_{x \to a} \frac{f'(x)}{g'(x)}\) exists as a real number, or is \pm\infty.\n- Applies equally to one-sided limits (x \to a^+$,$x \to a^-$) and limits at infinity ($x \to +\infty$,$x \to -\infty).\n- Note: L'Hôpital's Rule requires differentiating the numerator and denominator separately (\frac{f'}{g'}), not using the quotient rule.\n\nConverting Other Indeterminate Forms:\n1. Type 0 \cdot \infty: If \(\lim f(x) = 0\) and \(\lim g(x) = \pm\infty\), rewrite f(x) g(x)$as$\frac{f(x)}{1/g(x)}$(type$\frac{0}{0}$) or$\frac{g(x)}{1/f(x)}$(type$\frac{\infty}{\infty}).\n2. Type \infty - \infty$: Combine terms using a common denominator, factoring, or rationalization to form$\frac{0}{0}$or$\frac{\infty}{\infty}.\n3. Types 1^\infty, 0^0, \infty^0$: Let$y = [f(x)]^{g(x)}$. Take natural logarithms:$\ln y = g(x) \ln[f(x)]$, which converts the exponent into type$0 \cdot \infty$. Evaluate$L = \lim \ln y$, then the original limit is$e^L.\n\n\n## Rolle's Theorem and the Mean Value Theorem\n\nRolle's Theorem:\nLet f be a function that satisfies:\n1. f$is continuous on the closed interval$[a, b]\n2. f$is differentiable on the open interval$(a, b)\n3. f(a) = f(b)\nThen there exists at least one number c \in (a, b) such that:\nf'(c) = 0\n\nThe Mean Value Theorem (MVT):\nLet f be a function that satisfies:\n1. f$is continuous on the closed interval$[a, b]\n2. f$is differentiable on the open interval$(a, b)\nThen there exists at least one number c \in (a, b) such that:\nf'(c) = \frac{f(b) - f(a)}{b - a}\n\nGeometric Interpretation:\nThere is at least one point (c, f(c))$on the graph where the tangent line is parallel to the secant line passing through$(a, f(a))$and$(b, f(b)).\n\n\n## Relative Extrema, Monotonicity, and the First Derivative Test\n\nDefinitions of Extrema:\n- Relative Maximum: f$has a relative maximum at$x = c$if there exists an open interval$I$containing$c$such that$f(x) \le f(c)$for all$x \in I\n- Relative Minimum: f$has a relative minimum at$x = c$if there exists an open interval$I$containing$c$such that$f(x) \ge f(c)$for all$x \in I\n\nCritical Numbers:\nA number c \in \text{dom}\,f$is a critical number of$f$if either$f'(c) = 0$or$f'(c) is undefined.\n\nTheorem (Fermat's Theorem on Stationary Points):\nIf f$has a relative extremum at$x = c$, then$c$must be a critical number of$f.\n\nMonotonicity Test:\nLet f$be continuous on$[a, b]$and differentiable on$(a, b).\n1. If f'(x) > 0$for all$x \in (a, b)$, then$f$is strictly increasing on$[a, b].\n2. If f'(x) < 0$for all$x \in (a, b)$, then$f$is strictly decreasing on$[a, b].\n3. If f'(x) = 0$for all$x \in (a, b)$, then$f$is constant on$[a, b].\n\nThe First Derivative Test for Relative Extrema:\nLet c$be a critical number of$f$, and$f$be continuous at$c.\n1. If f'(x) > 0$on$(a, c)$and$f'(x) < 0$on$(c, b)$, then$f$has a relative maximum at$x = c\n2. If f'(x) < 0$on$(a, c)$and$f'(x) > 0$on$(c, b)$, then$f$has a relative minimum at$x = c\n3. If f'(x)$has the same sign on both sides of$c$, then$f$has no relative extremum at$x = c\n\n\n## Concavity, Points of Inflection, and the Second Derivative Test\n\nConcavity:\n1. Concave Up: The graph of f$lies above its tangent lines on an interval$I$. This occurs when$f'(x)$is increasing, which corresponds to$f''(x) > 0$on$I\n2. Concave Down: The graph of f$lies below its tangent lines on an interval$I$. This occurs when$f'(x)$is decreasing, which corresponds to$f''(x) < 0$on$I\n\nTest for Concavity:\n- If f''(x) > 0$for all$x \in (a, b)$, the graph of$f$is concave up on$(a, b).\n- If f''(x) < 0$for all$x \in (a, b)$, the graph of$f$is concave down on$(a, b).\n\nPoint of Inflection:\nA point P(c, f(c))$on the graph of$f$is a point of inflection if$f$is continuous at$c$and the graph changes concavity at$P.\n- If P(c, f(c))$is a point of inflection, then either$f''(c) = 0$or$f''(c) is undefined.\n\nThe Second Derivative Test for Relative Extrema:\nLet f$be a function such that$f'(c) = 0$and$f''$is continuous near$c\n1. If f''(c) < 0$, then$f$has a relative maximum at$x = c\n2. If f''(c) > 0$, then$f$has a relative minimum at$x = c\n3. If f''(c) = 0, the test is inconclusive (the First Derivative Test must be used).\n\n\n## Graph Sketching and Asymptotes\n\nAsymptotes Summary:\n- Vertical Asymptote: x = a if \(\lim_{x \to a^{\pm}} f(x) = \pm\infty\).\n- Horizontal Asymptote: y = L if \(\lim_{x \to \pm\infty} f(x) = L\).\n- Oblique (Slant) Asymptote: y = m x + b$($m \neq 0) if \(\lim_{x \to \pm\infty} [f(x) - (m x + b)] = 0\). For a rational function \frac{P(x)}{Q(x)}$, an oblique asymptote exists when$\text{deg}(P) = \text{deg}(Q) + 1, obtained by polynomial long division.\n\nComprehensive Graph Sketching Procedure:\n1. Domain and Intercepts: Determine \text{dom}\,f$, x-intercepts ($f(x) = 0$), y-intercept ($f(0)).\n2. Symmetry and Asymptotes: Check for even/odd symmetry. Find vertical, horizontal, and slant asymptotes.\n3. First Derivative Analysis: Compute f'(x)$. Find critical numbers. Construct a sign table for$f'(x) to identify intervals of increase/decrease and relative extrema.\n4. Second Derivative Analysis: Compute f''(x)$. Find candidates for inflection points. Construct a sign table for$f''(x) to identify intervals of concavity and inflection points.\n5. Plotting: Plot intercepts, relative extrema, inflection points, and draw asymptotes. Sketch the curve following monotonicity and concavity.\n\n\n# Applications of Differentiation\n\n## Rectilinear Motion\n\nPosition, Velocity, and Acceleration:\nFor a particle moving along a coordinate line with position function s(t)$at time$t \ge 0:\n- Position: s(t)\n- Instantaneous Velocity: v(t) = s'(t) = \frac{ds}{dt}\n- Instantaneous Speed: |v(t)| = |s'(t)|\n- Instantaneous Acceleration: a(t) = v'(t) = s''(t) = \frac{d^2s}{dt^2}\n\nInterpretation of Signs in Motion:\n- Velocity:\n - v(t) > 0: Particle is moving in the positive direction (right / upward).\n - v(t) < 0: Particle is moving in the negative direction (left / downward).\n - v(t) = 0: Particle is momentarily at rest or changing direction.\n- Speeding Up vs. Slowing Down:\n - Speeding Up: v(t)$and$a(t)$have the same sign ($v(t) a(t) > 0).\n - Slowing Down: v(t)$and$a(t)$have opposite signs ($v(t) a(t) < 0).\n- Total Distance Traveled: Calculate the sum of absolute displacements between consecutive time turning points (where v(t) = 0).\n\n\n## Rates of Change and Related Rates\n\nRates of Change:\n- Average rate of change of y = f(x)$on$[x_0, x]$:$\frac{\Delta y}{\Delta x} = \frac{f(x) - f(x_0)}{x - x_0}\n- Instantaneous rate of change of y$with respect to$x$at$x_0$:$f'(x_0) = \lim_{\Delta x \to 0} \frac{\Delta y}{\Delta x}\n\nRelated Rates Procedure:\n1. Draw a diagram representing the situation for any time t > 0.\n2. Assign variables to all quantities that change with time t. Label constants with their numerical values.\n3. State given rates of change as derivatives with respect to t$(e.g.,$\frac{dx}{dt}$,$\frac{dV}{dt}). Assign negative signs to quantities that decrease with time.\n4. Identify the target rate of change to be found at a specific instant.\n5. Write an equation relating the variables valid for all t > 0.\n6. Differentiate implicitly with respect to time t.\n7. Substitute the specific values corresponding to the instant of interest into the resulting equation and solve for the unknown rate of change.\n8. State the final result with correct units.\n\n\n## Local Linear Approximation and Differentials\n\nDifferentials:\nLet y = f(x)$be differentiable at$x\n- The differential dx$of the independent variable$x$is an arbitrary increment$dx = \Delta x\n- The differential dy$of the dependent variable$y is defined by:\n  dy = f'(x) dx\n- While \Delta y = f(x + \Delta x) - f(x)$represents the actual change in$y$,$dy represents the estimated change along the tangent line.\n\nLocal Linear Approximation:\nThe linear function L(x)$approximating$f(x)$near$x_0 is the tangent line equation:\nL(x) = f(x_0) + f'(x_0)(x - x_0)\nFor x = x_0 + dx$(where$dx = \Delta x is small):\nf(x_0 + dx) \approx f(x_0) + f'(x_0) dx\n\Delta y \approx dy = f'(x_0) dx\n\n\n## Absolute Extrema and Optimization Problems\n\nAbsolute Extrema Definitions:\n- Absolute Maximum: f(x_0) \ge f(x)$for all$x \in I\n- Absolute Minimum: f(x_0) \le f(x)$for all$x \in I\n\nThe Extreme Value Theorem (EVT):\nIf f$is continuous on a closed, bounded interval$[a, b]$, then$f$attains both an absolute maximum and an absolute minimum on$[a, b].\n\nClosed Interval Method for Absolute Extrema on [a, b]:\n1. Find all critical numbers of f$in the open interval$(a, b).\n2. Evaluate f$at each critical number$c_1, c_2, \dots, c_m\n3. Evaluate f$at the endpoints$a$and$b\n4. The largest of these values is the absolute maximum value; the smallest is the absolute minimum value.\n\nSingle Critical Number Theorem for Open/Unbounded Intervals:\nSuppose f$is continuous on an interval$I$containing a single critical number$x_0\n- If f$has a relative maximum at$x_0$, then$f(x_0)$is the absolute maximum of$f$on$I\n- If f$has a relative minimum at$x_0$, then$f(x_0)$is the absolute minimum of$f$on$I\n\nOptimization Word Problems - Step-by-Step:\n1. Draw a diagram and assign variable names.\n2. Formulate the primary objective equation for the quantity q to be optimized.\n3. Express q$as a function of a single variable$x using constraint equations.\n4. Determine the feasible domain of x based on physical/geometric limits.\n5. Find the absolute maximum or minimum using the Closed Interval Method or First/Second Derivative Tests.\n\n\n# Integration and Its Applications\n\n## Antidifferentiation and Indefinite Integrals\n\nAntiderivatives:\nA function F$is an antiderivative of$f$on an interval$I$if$F'(x) = f(x)$for all$x \in I.\n- If F$is an antiderivative of$f$, the general antiderivative is$F(x) + C$, where$C \in \mathbb{R} is an arbitrary constant.\n- The indefinite integral is denoted by:\n  \int f(x) dx = F(x) + C\n\nBasic Integration Formulas:\n1. \int dx = x + C\n2. \int a f(x) dx = a \int f(x) dx\n3. \int [f(x) \pm g(x)] dx = \int f(x) dx \pm \int g(x) dx\n4. \int x^n dx = \frac{x^{n+1}}{n + 1} + C$(for$n \neq -1)\n5. \int \frac{1}{x} dx = \ln |x| + C\n6. \int e^x dx = e^x + C\n7. \int a^x dx = \frac{a^x}{\ln a} + C$($a > 0, a \neq 1)\n8. \int \sin x dx = -\cos x + C\n9. \int \cos x dx = \sin x + C\n10. \int \sec^2 x dx = \tan x + C\n11. \int \csc^2 x dx = -\cot x + C\n12. \int \sec x \tan x dx = \sec x + C\n13. \int \csc x \cot x dx = -\csc x + C\n14. \int \sinh x dx = \cosh x + C\n15. \int \cosh x dx = \sinh x + C\n16. \int \text{sech}^2 x dx = \tanh x + C\n17. \int \text{csch}^2 x dx = -\coth x + C\n18. \int \text{sech}\,x \tanh x dx = -\text{sech}\,x + C\n19. \int \text{csch}\,x \coth x dx = -\text{csch}\,x + C\n\nIntegrals Yielding Inverse Trigonometric and Inverse Hyperbolic Functions (a > 0):\n1. \int \frac{du}{\sqrt{a^2 - u^2}} = \sin^{-1}\left(\frac{u}{a}\right) + C\n2. \int \frac{du}{a^2 + u^2} = \frac{1}{a} \tan^{-1}\left(\frac{u}{a}\right) + C\n3. \int \frac{du}{u \sqrt{u^2 - a^2}} = \frac{1}{a} \sec^{-1}\left(\frac{u}{a}\right) + C\n4. \int \frac{du}{\sqrt{u^2 + a^2}} = \sinh^{-1}\left(\frac{u}{a}\right) + C = \ln\left(u + \sqrt{u^2 + a^2}\right) + C\n5. \int \frac{du}{\sqrt{u^2 - a^2}} = \cosh^{-1}\left(\frac{u}{a}\right) + C = \ln\left(u + \sqrt{u^2 - a^2}\right) + C$(for$u > a)\n6. \int \frac{du}{a^2 - u^2} = \frac{1}{2a} \ln\left|\frac{a + u}{a - u}\right| + C\n\n\n## Integration by Substitution\n\nTheorem (Substitution Rule):\nIf u = g(x)$is a differentiable function whose range is an interval$I$, and$f$is continuous on$I, then:\n\int f(g(x)) g'(x) dx = \int f(u) du\n\nIntegrals of Remaining Trigonometric Functions:\n1. \int \tan x dx = \ln |\sec x| + C\n2. \int \cot x dx = \ln |\sin x| + C\n3. \int \sec x dx = \ln |\sec x + \tan x| + C\n4. \int \csc x dx = \ln |\csc x - \cot x| + C\n\nIntegrals of Remaining Hyperbolic Functions:\n1. \int \tanh x dx = \ln(\cosh x) + C\n2. \int \coth x dx = \ln |\sinh x| + C\n3. \int \text{sech}\,x dx = 2 \tan^{-1}(e^x) + C = \tan^{-1}(\sinh x) + C\n4. \int \text{csch}\,x dx = \ln |\text{csch}\,x - \coth x| + C\n\n\n## Particular Antiderivatives and Rectilinear Motion\n\nInitial Value Problems:\nA particular antiderivative is obtained when an initial condition (e.g., y(x_0) = y_0$) is provided to solve for the constant of integration$C\n\nApplication to Rectilinear Motion:\nGiven acceleration a(t)$, initial velocity$v(0) = v_0$, and initial position$s(0) = s_0:\n1. v(t) = \int a(t) dt + C_1$, solve$C_1$using$v(0) = v_0\n2. s(t) = \int v(t) dt + C_2$, solve$C_2$using$s(0) = s_0\n- Constant Acceleration under Gravity (g):\n - a(t) = -g\n - v(t) = -gt + v_0\n - s(t) = -\frac{1}{2}gt^2 + v_0 t + s_0\n\n\n## Area of a Plane Region and the Definite Integral\n\nSigma Notation and Properties:\n\sum_{i=1}^n F(i) = F(1) + F(2) + \dots + F(n)\n1. \sum_{i=1}^n c = c n\n2. \sum_{i=1}^n c F(i) = c \sum_{i=1}^n F(i)\n3. \sum_{i=1}^n [F(i) \pm G(i)] = \sum_{i=1}^n F(i) \pm \sum_{i=1}^n G(i)\n4. \sum_{i=1}^n i = \frac{n(n + 1)}{2}\n5. \sum_{i=1}^n i^2 = \frac{n(n + 1)(2n + 1)}{6}\n6. \sum_{i=1}^n i^3 = \frac{n^2(n + 1)^2}{4}\n\nRiemann Sums and Area Definition:\nLet f$be continuous and non-negative on$[a, b].\n1. Divide [a, b]$into$n$subintervals of equal width$\Delta x = \frac{b - a}{n}.\n2. Choose sample points x_i^$in each subinterval$[x_{i-1}, x_i].\n3. The area A_R of the region is defined as the limit of Riemann sums:\n   A_R = \lim_{n \to \infty} \sum_{i=1}^n f(x_i^) \Delta x\n\nThe Definite Integral:\nLet f$be defined on$[a, b]$. A partition$P$divides$[a, b]$into$n$subintervals of length$\Delta x_i = x_i - x_{i-1}$. The mesh size is$\max \Delta x_i.\n\int_a^b f(x) dx = \lim_{\max \Delta x_i \to 0} \sum_{i=1}^n f(x_i^*) \Delta x_i\nprovided the limit exists.\n- If f$is continuous on$[a, b]$, then$f$is integrable on$[a, b].\n\nGeometric Interpretation:\n\int_a^b f(x) dx$represents the net-signed area between the graph of$y = f(x)$and the x-axis on$[a, b] (area above the x-axis minus area below the x-axis).\n\nProperties of the Definite Integral:\n1. \int_a^b f(x) dx = -\int_b^a f(x) dx\n2. \int_a^a f(x) dx = 0\n3. \int_a^b c dx = c (b - a)\n4. \int_a^b c f(x) dx = c \int_a^b f(x) dx\n5. \int_a^b [f(x) \pm g(x)] dx = \int_a^b f(x) dx \pm \int_a^b g(x) dx\n6. \int_a^b f(x) dx = \int_a^c f(x) dx + \int_c^b f(x) dx$(for any$c\n\n\n## The Fundamental Theorems of Calculus\n\nThe First Fundamental Theorem of Calculus (FTC 1):\nLet f$be continuous on$[a, b]$. The function$F defined by:\nF(x) = \int_a^x f(t) dt\nis continuous on [a, b]$, differentiable on$(a, b), and its derivative is:\nF'(x) = \frac{d}{dx} \left[ \int_a^x f(t) dt \right] = f(x)\n\nLeibniz Extension of FTC 1:\n\frac{d}{dx} \left[ \int_{a(x)}^{b(x)} f(t) dt \right] = f(b(x)) b'(x) - f(a(x)) a'(x)\n\nThe Second Fundamental Theorem of Calculus (FTC 2):\nLet f$be continuous on$[a, b]$. If$F$is any antiderivative of$f$on$[a, b], then:\n\int_a^b f(x) dx = F(b) - F(a)\nWe denote F(b) - F(a)$by$\left[ F(x) \right]a^b.\n\nSubstitution in Definite Integrals:\n\int_a^b f(g(x)) g'(x) dx = \int{g(a)}^{g(b)} f(u) du\n\n\n## Area of Plane Regions Between Curves\n\nVertical Rectangles (dx Approach):\nIf f$and$g$are continuous on$[a, b]$with$f(x) \ge g(x)$for all$x \in [a, b]$, the area$A_R$bounded above by$y = f(x)$, below by$y = g(x)$, and laterally by$x = a$and$x = b is:\nA_R = \int_a^b [f(x) - g(x)] dx = \int_a^b h(x) dx\nwhere h(x) = y_{\text{upper}} - y_{\text{lower}}.\n\nHorizontal Rectangles (dy Approach):\nIf u$and$v$are continuous on$[c, d]$with$v(y) \ge u(y)$for all$y \in [c, d]$, the area$A_R$bounded on the right by$x = v(y)$, on the left by$x = u(y)$, and horizontally by$y = c$and$y = d is:\nA_R = \int_c^d [v(y) - u(y)] dy = \int_c^d l(y) dy\nwhere l(y) = x_{\text{right}} - x_{\text{left}}.\n\n\n## Arc Length of Plane Curves\n\nSmooth Curve Definition:\nA curve is smooth on [a, b]$if its derivative$f'$is continuous on$[a, b].\n\nArc Length Formulas:\n1. Function of x$on$[a, b]:\n   L = \int_a^b \sqrt{1 + \left(\frac{dy}{dx}\right)^2} dx = \int_a^b \sqrt{1 + [f'(x)]^2} dx\n2. Function of y$on$[c, d]:\n   L = \int_c^d \sqrt{1 + \left(\frac{dx}{dy}\right)^2} dy = \int_c^d \sqrt{1 + [u'(y)]^2} dy\n\n\n## Volumes of Solids of Revolution and Volumes by Slicing\n\nDisk and Washer Methods (Rectangles Perpendicular to Axis of Revolution):\n1. Disk Method (No gap between region and axis of revolution):\n - Horizontal Axis of Revolution (y = y_0):\n     V = \int_a^b \pi [r(x)]^2 dx\n - Vertical Axis of Revolution (x = x_0):\n     V = \int_c^d \pi [r(y)]^2 dy\n2. Washer Method (Gap exists between region and axis of revolution):\n - Horizontal Axis of Revolution (y = y_0):\n     V = \int_a^b \pi \left( [r_2(x)]^2 - [r_1(x)]^2 \right) dx\n     where r_2(x)$is the outer radius and$r_1(x) is the inner radius.\n - Vertical Axis of Revolution (x = x_0):\n     V = \int_c^d \pi \left( [r_2(y)]^2 - [r_1(y)]^2 \right) dy\n\nCylindrical Shell Method (Rectangles Parallel to Axis of Revolution):\n1. Vertical Axis of Revolution (x = x_0$) using Vertical Rectangles ($dx):\n   V = \int_a^b 2\pi r(x) h(x) dx\n   where r(x)$is the distance from$x$to the axis of revolution ($|x - x_0|$) and$h(x) = y_{\text{upper}} - y_{\text{lower}}.\n2. Horizontal Axis of Revolution (y = y_0$) using Horizontal Rectangles ($dy):\n   V = \int_c^d 2\pi r(y) h(y) dy\n   where r(y) = |y - y_0|$and$h(y) = x_{\text{right}} - x_{\text{left}}.\n\nVolume by Slicing (Cross-Sectional Area):\nFor a solid S$extending from$x = a$to$x = b$whose cross-sectional area perpendicular to the x-axis is$A(x):\nV = \int_a^b A(x) dx\nFor a solid extending from y = c$to$y = d$whose cross-sectional area perpendicular to the y-axis is$A(y):\nV = \int_c^d A(y) dy\n\n\n## Mean Value Theorem for Integrals and Average Value of a Function\n\nDomination and Bounding Theorems:\n1. If f(x) \le g(x)$for all$x \in [a, b], then:\n   \int_a^b f(x) dx \le \int_a^b g(x) dx\n2. If m \le f(x) \le M$for all$x \in [a, b], then:\n   m (b - a) \le \int_a^b f(x) dx \le M (b - a)\n\nMean Value Theorem for Integrals:\nIf f$is continuous on the closed interval$[a, b]$, then there exists at least one number$c \in [a, b] such that:\n\int_a^b f(x) dx = f(c) (b - a)\n\nAverage Value of a Function:\nIf f$is integrable on$[a, b]$, its average value$f_{\text{ave}}$on$[a, b] is:\nf_{\text{ave}} = \frac{1}{b - a} \int_a^b f(x) dx$$