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

Test 2 Study Guide: Limits, Continuity, and the Derivative

Overview

This study guide covers essential calculus concepts from sections 2.5, 2.6, 2.7, and 2.8, focusing on limits, continuity, and the derivative. Mastery of these topics is foundational for understanding calculus and succeeding on Test 2.

Evaluating Logarithmic and Exponential Expressions

Logarithms and Exponents

• Logarithmic Expressions: The logarithm base b of x is the exponent to which b must be raised to yield x. Written as $\log_b(x)$.

• Negative and Rational Exponents: Negative exponents represent reciprocals, and rational exponents represent roots. For example, $a^{-n} = \frac{1}{a^n}$ and $a^{1/n} = \sqrt[n]{a}$.

• Evaluating Expressions: Simplify using exponent and logarithm rules.

Example: Evaluate $27^{2/3}$.

Solution: $27^{2/3} = (27^{1/3})^2 = (3)^2 = 9$.

Functions: Domains, Intervals, and Evaluation

Domain and Interval Notation

• Domain: The set of all input values (x) for which a function is defined.

• Interval Notation: Used to describe domains, e.g., $(-\infty, 2]$.

• Evaluating Functions: Substitute values into the function and simplify.

Example: Find the domain of $f(x) = \frac{1}{x-3}$.

Solution: $x-3 \neq 0 \implies x \neq 3$. Domain: $(-\infty, 3) \cup (3, \infty)$.

Continuity and Discontinuities

Definition of Continuity

• A function f is continuous at a point $x=a$ if:

  • $f(a)$ is defined

  • $\lim_{x \to a} f(x)$ exists

  • $\lim_{x \to a} f(x) = f(a)$

• Discontinuity: Occurs when any of the above conditions fail.

Example: $f(x) = \frac{x^2-1}{x-1}$ is not defined at $x=1$, but $\lim_{x \to 1} f(x) = 2$.

Types of Discontinuities

• Removable Discontinuity: Limit exists, but function is not defined or not equal to the limit at that point.

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

• Infinite Discontinuity: Function approaches infinity near the point.

Intermediate Value Theorem (IVT)

Statement and Application

• If f is continuous on $[a, b]$ and $N$ is between $f(a)$ and $f(b)$, then there exists $c$ in $(a, b)$ such that $f(c) = N$.

• Used to show that equations have solutions within an interval.

Example: If $f(1) = -2$ and $f(3) = 4$, then $f(x) = 0$ for some $x$ in $(1, 3)$.

Limits and Asymptotes

Limits at Infinity and Continuity

• Limit at Infinity: $\lim_{x \to \infty} f(x)$ describes the behavior as $x$ grows large.

• Continuity: A function is continuous where its limit equals its value.

Horizontal and Vertical Asymptotes

• Horizontal Asymptote: $y = L$ if $\lim_{x \to \infty} f(x) = L$ or $\lim_{x \to -\infty} f(x) = L$.

• Vertical Asymptote: $x = a$ if $\lim_{x \to a^+} f(x) = \pm \infty$ or $\lim_{x \to a^-} f(x) = \pm \infty$.

• A function can cross a horizontal asymptote but not a vertical one.

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

Graphical Analysis of Functions

Limits, Continuity, and Differentiability from Graphs

• Identify where a function is continuous or discontinuous by looking for breaks, jumps, or holes in the graph.

• Differentiability requires the function to be continuous and have no sharp corners or cusps.

Sketching Graphs with Specified Properties

• Given properties (e.g., limits, continuity, differentiability), sketch a function that satisfies all conditions.

Derivatives

Limit Definition of the Derivative

• The derivative of $f$ at $x=a$ is:

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

• Represents the slope of the tangent line at $x=a$.

Example: For $f(x) = x^2$, $f'(a) = \lim_{h \to 0} \frac{(a+h)^2 - a^2}{h} = 2a$.

Interpreting the Derivative

• Slope of the Tangent Line: The derivative at a point gives the slope of the tangent to the curve at that point.

• Instantaneous Rate of Change: The derivative represents how fast the function is changing at a specific value of $x$.

Equation of the Tangent Line

• The equation of the tangent line to $f$ at $x=a$ is:

$y = f(a) + f'(a)(x - a)$

Relating Graphs of Functions and Their Derivatives

• The graph of $f'(x)$ shows where $f(x)$ is increasing (when $f'(x) > 0$) or decreasing (when $f'(x) < 0$).

• Critical points occur where $f'(x) = 0$ or is undefined.

Summary Table: Types of Discontinuities

Type	Description	Graphical Feature
Removable	Limit exists, function not defined or not equal to limit at point	Hole in the graph
Jump	Left and right limits exist but are not equal	Jump in the graph
Infinite	Function approaches infinity near the point	Vertical asymptote

Study and Exam Preparation Tips

• Practice problems from each section, including homework, worksheets, and in-class activities.

• Review objectives and ensure you can explain each concept.

• Work through examples without notes to test understanding.

• Show all work and reasoning on the exam; unsupported answers will not receive credit.

• Write answers in exact form unless otherwise specified.

Additional info: This guide synthesizes the exam objectives and expands on each topic with definitions, examples, and key formulas to provide a comprehensive review for Test 2 in Calculus I.