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

2.5 Continuity

Domain and Continuity of Functions

Understanding the domain and intervals of continuity is essential for analyzing functions in calculus. The domain is the set of all input values for which the function is defined, while continuity refers to the absence of breaks, jumps, or holes in the graph of the function.

• Domain: The set of all real numbers for which the function produces a real output.

• Continuity: A function is continuous at a point if the limit as x approaches that point equals the function's value at that point.

• Example: $f(x) = \sqrt{x-5}$ is defined for $x \geq 5$ and is continuous on $[5, \infty)$.

Types of Discontinuities

Discontinuities occur when a function is not continuous at a point. They can be classified as removable or nonremovable.

• Removable Discontinuity: Occurs when a function is undefined at a point, but the limit exists. The discontinuity can be 'fixed' by redefining the function at that point.

• Nonremovable Discontinuity: Occurs when the limit does not exist due to a jump or infinite behavior.

• Left/Right Continuity: A function may be continuous from the left or right at a point if the respective one-sided limit equals the function's value.

• Example: $f(x) = \begin{cases} x^2 - 2, & x < 1 \\ \sqrt{x}, & x \geq 1 \end{cases}$ has a potential discontinuity at $x=1$.

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 $c$ in $[a, b]$ such that $f(c) = N$.

• Application: Used to prove the existence of roots within an interval.

• Example: For $f(x) = \sqrt{x}$ 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 infinity or negative infinity.

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

• Example: $\lim_{x \to \infty} \frac{3x^2 + 4x}{5x^2 + 1} = \frac{3}{5}$

Vertical Asymptotes

Vertical asymptotes occur at values of $x$ where the function grows without bound.

• Example: $f(x) = \frac{1}{x-2}$ has a vertical asymptote at $x=2$.

Applications of Limits at Infinity

• Population Models: Functions such as $p(t) = \frac{2500t}{t+1}$ describe population growth and approach a limiting value as $t \to \infty$.

• Example: $\lim_{t \to \infty} \frac{2500t}{t+1} = 2500$

2.7 The Derivative at a Point

Average and Instantaneous Rate of Change

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

• Secant Line: The line through $(a, f(a))$ and $(b, f(b))$ represents the average rate of change.

• Tangent Line: The line through $(a, f(a))$ with slope $f'(a)$ represents the instantaneous rate of change.

• Example: For $f(x) = \sqrt{x}$ on $[1,9]$, average rate is $\frac{\sqrt{9} - \sqrt{1}}{9-1} = \frac{3-1}{8} = \frac{1}{4}$.

Definition of the Derivative

The derivative of $f$ at $x=a$ is defined as:

• $f'(a) = \lim_{h \to 0} \frac{f(a+h) - f(a)}{h}$

• Application: Used to find the slope of the tangent line at a point.

2.8 The Derivative as a Function

Graphical Interpretation

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

• Critical Points: Where $f'(x) = 0$, the function has a horizontal tangent (possible maxima or minima).

• Non-differentiable Points: Points where $f(x)$ has a cusp, corner, or vertical tangent.

3.1 The Derivative and Rates of Change

Motion and Rates of Change

Derivatives are used to analyze motion, such as velocity, acceleration, and jerk.

• Velocity: $v(t) = f'(t)$

• Acceleration: $a(t) = f''(t)$

• Jerk: $j(t) = f'''(t)$

• Example: For $f(t) = 3t^2 - 24t^2 + 5$, $v(t) = 6t - 48t$

Related Rates

Related rates problems involve finding the rate at which one quantity changes with respect to another.

• Example: The rate of change of the volume of a cube $V = x^3$ with respect to $x$ is $\frac{dV}{dx} = 3x^2$.

3.2 The Product and Quotient Rules

Product Rule

The product rule is used to differentiate products of two functions:

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

• Example: $f(x) = x^2 \ln|x|$, $f'(x) = 2x \ln|x| + x$

Quotient Rule

The quotient rule is used to differentiate quotients of two functions:

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

• Example: $f(x) = \frac{e^x}{x^2 + 1}$, $f'(x) = \frac{e^x(x^2 + 1) - 2x e^x}{(x^2 + 1)^2}$

3.4 The Chain Rule

Chain Rule

The chain rule is used to differentiate composite functions:

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

• Example: $f(x) = \tan^{-1}(\sqrt{x^2 - 1})$, $f'(x) = \frac{1}{1 + x^2 - 1} \cdot \frac{x}{\sqrt{x^2 - 1}} = \frac{x}{x^2 \sqrt{x^2 - 1}}$

Differentiation of Exponential and Logarithmic Functions

• Exponential: $\frac{d}{dx}[e^{g(x)}] = e^{g(x)}g'(x)$

• Logarithmic: $\frac{d}{dx}[\ln(g(x))] = \frac{g'(x)}{g(x)}$

3.5 Implicit Differentiation

Implicit Differentiation

Used when functions are defined implicitly rather than explicitly.

• Method: Differentiate both sides of the equation with respect to $x$, treating $y$ as a function of $x$.

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

3.6 Differentiation Methods Table

Classification of Differentiation Methods

The following table summarizes which differentiation method applies to each function:

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

Additional info:

• Some questions involve sketching graphs and interpreting graphical information, which is essential for understanding the behavior of functions and their derivatives.

• Applications include population models, rates of change in physical systems, and concentration models in biology and chemistry.

• Students should be familiar with the formal definitions and graphical interpretations of continuity, limits, and derivatives.