Calculus I Exam 1 Study Guide: Limits, Continuity, and the Derivative

Exam 1 Overview and Preparation

This study guide covers key concepts and problem types for a first Calculus exam, focusing on limits, continuity, asymptotes, and the introduction to derivatives. The material is based on tex tbook sections 2.1–2.6 and 3.1, and includes both conceptual questions and computational techniques.

Exam Logistics and Coverage

• Exam Duration: 100 minutes

• Allowed Materials: No notes, books, phones, electronics, or calculators

• Sections Covered: 2.1, 2.2, 2.3, 2.4, 2.5, 2.6, 3.1

• Recommended Practice: Worksheets 1–4, textbook exercises from each section

Limits and Their Properties

Average and Instantaneous Rate of Change

The average rate of change of a function f(x) over an interval [a, b] is the change in function value divided by the change in input:

• Formula: $\frac{f(b) - f(a)}{b - a}$

• Interpretation: Represents the slope of the secant line between points (a, f(a)) and (b, f(b)).

• Instantaneous rate of change: The limit of the average rate of change as the interval shrinks to a point; this is the derivative.

Example: For $f(x) = x^2$, the average rate of change from x = 1 to x = 3 is $\frac{3^2 - 1^2}{3 - 1} = \frac{9 - 1}{2} = 4$.

Limits: Definitions and Techniques

Limits describe the behavior of a function as the input approaches a particular value.

• Direct Substitution: If $f(x)$ is continuous at $x = a$, then $\lim_{x \to a} f(x) = f(a)$.

• Indeterminate Forms: Expressions like $\frac{0}{0}$ or $\frac{\infty}{\infty}$ require algebraic manipulation (factoring, rationalizing, etc.) to resolve.

• Techniques: Factor and cancel, rationalize, use conjugates, or apply the Squeeze Theorem.

Example: $\lim_{x \to 2} \frac{x^2 - 4}{x - 2}$ Factor numerator: $\frac{(x-2)(x+2)}{x-2} = x+2$ for $x \neq 2$. Thus, the limit is 4.

Special Limit Forms and the Squeeze Theorem

• Squeeze Theorem: If $g(x) \leq f(x) \leq h(x)$ near $a$ and $\lim_{x \to a} g(x) = \lim_{x \to a} h(x) = L$, then $\lim_{x \to a} f(x) = L$.

• Indeterminate Forms: $\frac{0}{0}$, $\frac{\infty}{\infty}$, $0 \cdot \infty$, $\infty - \infty$, $0^0$, $1^\infty$, $\infty^0$

Vertical and Horizontal Asymptotes

Vertical asymptotes occur where a function grows without bound as x approaches a certain value.

• Definition: If $\lim_{x \to a} f(x) = \pm \infty$, then $x = a$ is a vertical asymptote.

Horizontal asymptotes describe the end behavior of a function as $x \to \pm \infty$.

• Definition: If $\lim_{x \to \infty} f(x) = L$ or $\lim_{x \to -\infty} f(x) = L$, then $y = L$ is a horizontal asymptote.

Example: $f(x) = \frac{2x^2 + 1}{x^2 - 3}$ has a horizontal asymptote at $y = 2$.

Graphing and Analyzing Functions

• Be able to sketch graphs of $f(x) = \ln(x)$, $g(x) = h(x)^3$, $y = \sqrt{x}$, $y = \tan^{-1} x$, $y = x^2$.

• Identify asymptotes, intercepts, and points of discontinuity.

Continuity and Types of Discontinuity

Definition of Continuity

A function $f(x)$ is continuous at $x = a$ if:

1. $f(a)$ is defined

2. $\lim_{x \to a} f(x)$ exists

3. $\lim_{x \to a} f(x) = f(a)$

Types of Discontinuity:

• Removable Discontinuity: The limit exists, but $f(a)$ is not defined or $f(a) \neq \lim_{x \to a} f(x)$.

• Jump Discontinuity: The left and right limits exist but are not equal.

• Infinite Discontinuity: The function approaches infinity at $x = a$.

Table: Types of Discontinuity

Type	Description	Example
Removable	Limit exists, function value missing or not equal to limit	$f(x) = \frac{x^2 - 1}{x - 1}$ at $x = 1$
Jump	Left and right limits exist but are not equal	Piecewise function with different values on each side
Infinite	Function approaches $\infty$ or $-\infty$	$f(x) = \frac{1}{x}$ at $x = 0$

The Intermediate Value Theorem (IVT)

The Intermediate Value Theorem states that if $f(x)$ is continuous on $[a, b]$ 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 show that equations have solutions within an interval.

Introduction to the Derivative

Rate of Change and Tangent Lines

The derivative of $f(x)$ at $x = a$ is the instantaneous rate of change of $f$ at $a$, or the slope of the tangent line to the graph at $(a, f(a))$.

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

• Equation of Tangent Line: $y - f(a) = f'(a)(x - a)$

Example: For $f(x) = x^2$ at $x = 1$, $f'(1) = 2$, so the tangent line is $y - 1 = 2(x - 1)$.

Secant vs. Tangent Lines

• Secant Line: Passes through two points on the curve; slope is average rate of change.

• Tangent Line: Touches the curve at one point; slope is instantaneous rate of change (the derivative).

Factorization and Limits

• Be able to factor polynomials to simplify limits, e.g., $x^2 - 9 = (x - 3)(x + 3)$.

• Apply factoring to resolve indeterminate forms.

Practice and Conceptual Understanding

• Redo worksheet and textbook problems for each section.

• Focus on understanding concepts, not just memorizing procedures.

Additional info:

• Some content inferred from textbook images and standard Calculus I curriculum.

• Practice problems and exercises are referenced for further study.