Calculus Study Guide: Continuity, Limits, Derivatives, and Differentiation Rules

2.5 Continuity

Finding Domains and Intervals of Continuity

Continuity of a function describes where the function is defined and does not have any breaks, jumps, or holes. The domain is the set of all input values (x-values) for which the function is defined.

• Domain: The set of all real numbers x for which f(x) is defined.

• Interval of Continuity: The set of all intervals where the function is continuous.

• Example: For $f(x) = \sqrt{x/5}$, the domain is $x \geq 0$ and the function is continuous on $[0, \infty)$.

Types of Discontinuities

Discontinuities occur where a function is not continuous. They can be classified as:

• Removable Discontinuity: A 'hole' in the graph; the limit exists but the function is not defined or is defined differently at that point.

• Nonremovable Discontinuity: Includes jump and infinite discontinuities; the limit does not exist at the point.

• Left/Right Continuity: A function is left-continuous at a point if the left-hand limit equals the function value, and right-continuous if the right-hand limit equals the function value.

• Example: For $f(x) = \begin{cases} x^2 - 2, & x < 1 \\ \sqrt{x}, & x \geq 1 \end{cases}$, check continuity at $x=1$ by comparing left and right limits.

Intermediate Value Theorem (IVT)

The IVT states that if a function is continuous on a closed interval $[a, b]$ and $N$ is any number between $f(a)$ and $f(b)$, then there exists at least one $c$ in $(a, b)$ such that $f(c) = N$.

• Application: Used to show the existence of roots or solutions within an interval.

• Example: If $f(x) = \sqrt{x}$ on $[-1, 1]$, IVT does not apply since $f(x)$ is not defined for $x < 0$.

2.6 Limits at Infinity; Horizontal Asymptotes

Limits at Infinity

Limits at infinity describe the behavior of a function as $x$ approaches $\infty$ or $-\infty$.

• Horizontal Asymptote: A horizontal line $y = L$ that the graph of a function approaches as $x \to \infty$ or $x \to -\infty$.

• Vertical Asymptote: A vertical line $x = a$ where the function grows without bound as $x$ approaches $a$.

• Example: For $f(x) = \frac{3x^2 + 4x}{5x^2 + 6x}$, as $x \to \infty$, $f(x) \to \frac{3}{5}$.

Evaluating Limits

• Use algebraic simplification, factoring, or L'Hospital's Rule (if applicable) to evaluate limits.

• If the degree of the numerator and denominator are equal, the limit at infinity is the ratio of leading coefficients.

• If the numerator's degree is less, the limit is 0; if greater, the limit is $\infty$ or $-\infty$.

Applications to Population and Growth Models

• Population models often use limits to describe long-term behavior.

• Example: For $p(t) = \frac{2500t}{t+1}$, as $t \to \infty$, $p(t) \to 2500$.

2.7 The Derivative and the Tangent Line

Average and Instantaneous Rate of Change

The average rate of change of $f(x)$ over $[a, b]$ is $\frac{f(b) - f(a)}{b - a}$. The instantaneous rate of change at $x = a$ is the derivative $f'(a)$.

• Tangent Line: The line that touches the curve at a point and has the same slope as the curve at that point.

• Equation: $y = f(a) + f'(a)(x - a)$

• Example: For $f(x) = \sqrt{x}$ at $x = 1$, $f'(x) = \frac{1}{2\sqrt{x}}$, so $f'(1) = \frac{1}{2}$.

Torricelli's Law (Application)

• Describes the rate at which water drains from a tank: $V(t) = 6000\left(1 - \frac{t}{60}\right)^2$.

• To find the rate at which water is flowing out, compute $V'(t)$.

2.8 The Derivative as a Function

Graphical Interpretation of the Derivative

The derivative $f'(x)$ gives the slope of the tangent line to the graph of $f(x)$ at each point. The graph of $f'(x)$ can be sketched by analyzing the slopes of $f(x)$.

• Where $f(x)$ is increasing, $f'(x) > 0$; where $f(x)$ is decreasing, $f'(x) < 0$.

• Where $f(x)$ has a maximum or minimum, $f'(x) = 0$.

• Points where $f(x)$ is not smooth (sharp corners, cusps, vertical tangents) or not continuous, $f'(x)$ does not exist.

Physical Meaning of the Derivative

• In applications, the derivative can represent velocity, rate of change, or other physical rates.

• Example: If $C(t)$ is the percentage of battery capacity, $C'(t)$ is the rate at which the battery is charging or discharging.

3.2 The Product and Quotient Rules

Product Rule

Used to differentiate products of two functions:

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

• Example: $\frac{d}{dx}[x^2 \ln|x|] = 2x \ln|x| + x^2 \frac{1}{x} = 2x \ln|x| + x$

Quotient Rule

Used to differentiate quotients of two functions:

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

• Example: $\frac{d}{dx}\left[\frac{x^2}{x^2-1}\right] = \frac{2x(x^2-1) - x^2(2x)}{(x^2-1)^2}$

Applications to Population and Growth

• Biomass: $B(t) = N(t)M(t)$, where $N(t)$ is the number of individuals and $M(t)$ is the average mass.

• Use the product rule to find $B'(t)$.

3.4 The Chain Rule

Chain Rule

Used to differentiate composite functions:

• $\frac{d}{dx}[f(g(x))] = f'(g(x)) \cdot g'(x)$

• Example: $\frac{d}{dx}[\sin(4x^2 + 7)] = \cos(4x^2 + 7) \cdot 8x$

Hyperbolic Functions

• $\sinh(x) = \frac{e^x - e^{-x}}{2}$

• $\cosh(x) = \frac{e^x + e^{-x}}{2}$

• $\frac{d}{dx}[\sinh(x)] = \cosh(x)$

• $\frac{d}{dx}[\cosh(x)] = \sinh(x)$

Applications: Atmospheric Pressure and Gas Laws

• Atmospheric temperature and pressure can be related by formulas involving differentiation.

• Van der Waals equation for gases: $\left(P + \frac{a}{V^2}\right)(V - nb) = nRT$

3.5 Implicit Differentiation

Implicit Differentiation

Used when a function is not given explicitly as $y = f(x)$, but rather as an equation involving $x$ and $y$.

• Differentiating both sides with respect to $x$, treating $y$ as a function of $x$.

• Apply the chain rule to terms involving $y$.

• Example: For $x^2 + y^2 = 6$, $2x + 2y \frac{dy}{dx} = 0 \implies \frac{dy}{dx} = -\frac{x}{y}$

Identifying Differentiation Methods

Function	Method
$\frac{e^x}{x^2-1}$	Quotient Rule
$x^2 e^{x^2}$	Product Rule
$\sin(3x^2+1)$	Chain Rule
$e^{\tan^{-1}(x)}$	Chain Rule
$x^2 - y^2 = 5$	Implicit Differentiation

Summary Table: Differentiation Rules

Rule	Formula
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	$\frac{d}{dx}[f(g(x))] = f'(g(x)) \cdot g'(x)$
Implicit Differentiation	Differentiating both sides, treating $y$ as a function of $x$

Additional info: These notes are based on a set of calculus homework and exam questions covering continuity, limits, derivatives, and differentiation rules, with applications to real-world problems such as population growth, physics, and atmospheric science.