Calculus Homework Assignment List: Math LPC 1 Sec D08 (21409) Fall 2025

Homework Assignment Overview

This document lists homework assignments for a college-level Calculus course. Each assignment references textbook sections and specific problem numbers, indicating the topics covered and the progression of the course material.

Assignment Structure

• HW 1: Sections 2.1, 2.2, 2.4

• HW 2: Sections 2.5, 2.6, 3.1, 3.2

• HW 3: Section 3.3

Section Topics and Academic Context

Section 2.1: Introduction to Functions

This section typically introduces the concept of a function, its notation, and basic properties.

• Definition: A function is a relation that assigns each element in a set (domain) to exactly one element in another set (codomain).

• Notation: $f(x)$ denotes the value of function f at input x.

• Key Properties:

  • Domain and range

  • Vertical line test for functions

• Example: $f(x) = x^2$ is a function mapping real numbers to their squares.

Section 2.2: Types of Functions and Their Graphs

This section covers different types of functions and how to graph them.

• Types: Linear, quadratic, polynomial, rational, exponential, logarithmic

• Graphing: Understanding intercepts, asymptotes, and transformations

• Example: The graph of $f(x) = x^2$ is a parabola opening upwards.

Section 2.4: Function Operations and Composition

This section discusses how to combine functions using addition, subtraction, multiplication, division, and composition.

• Operations: $(f+g)(x) = f(x) + g(x)$, $(f imes g)(x) = f(x)g(x)$

• Composition: $(f igcirc g)(x) = f(g(x))$

• Example: If $f(x) = x+1$ and $g(x) = x^2$, then $(f igcirc g)(x) = f(g(x)) = x^2 + 1$

Section 2.5: Inverse Functions

This section introduces the concept of inverse functions and how to find them.

• Definition: The inverse function $f^{-1}(x)$ reverses the effect of $f(x)$.

• Finding Inverses: Solve $y = f(x)$ for $x$ in terms of $y$.

• Example: If $f(x) = 2x + 3$, then $f^{-1}(x) = \frac{x-3}{2}$

Section 2.6: Exponential and Logarithmic Functions

This section covers the properties and graphs of exponential and logarithmic functions.

• Exponential Function: $f(x) = a^x$ where $a > 0$

• Logarithmic Function: $f(x) = \log_a(x)$

• Properties:

  • $a^{x+y} = a^x a^y$

  • $\log_a(xy) = \log_a(x) + \log_a(y)$

• Example: $f(x) = 2^x$ grows rapidly as $x$ increases.

Section 3.1: Limits and Their Properties

This section introduces the concept of limits, a foundational idea in calculus.

• Definition: The limit of $f(x)$ as $x$ approaches $a$ is the value $L$ that $f(x)$ gets closer to as $x$ gets closer to $a$.

• Notation: $\lim_{x \to a} f(x) = L$

• Properties:

  • Limit laws (sum, product, quotient)

  • One-sided limits

• Example: $\lim_{x \to 2} (3x+1) = 7$

Section 3.2: Calculating Limits Using Algebra

This section focuses on techniques for evaluating limits algebraically.

• Techniques:

  • Direct substitution

  • Factoring

  • Rationalizing

• Example: $\lim_{x \to 1} \frac{x^2 - 1}{x - 1} = 2$ (by factoring numerator)

Section 3.3: Continuity

This section explains the concept of continuity for functions and how to determine if a function is continuous at a point.

• Definition: A function $f(x)$ is continuous at $x=a$ if $\lim_{x \to a} f(x) = f(a)$

• Types of Discontinuity:

  • Removable

  • Jump

  • Infinite

• Example: $f(x) = \frac{x^2 - 1}{x - 1}$ is not continuous at $x=1$ (removable discontinuity)

Summary Table: Section Topics and Key Concepts

Section	Main Topic	Key Concepts
2.1	Functions	Definition, notation, domain/range
2.2	Types of Functions	Linear, quadratic, graphing
2.4	Function Operations	Addition, multiplication, composition
2.5	Inverse Functions	Finding inverses, notation
2.6	Exponential & Logarithmic	Properties, graphs
3.1	Limits	Definition, notation, properties
3.2	Algebraic Limits	Techniques for calculation
3.3	Continuity	Definition, types of discontinuity

Additional info: Section numbers and problem assignments are inferred to correspond to a standard Calculus textbook, such as Stewart's Calculus or similar. The above academic context is provided to make the notes self-contained and useful for exam preparation.