STEM Quest - Calculus Learning Platform

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Master the fundamentals of derivatives, integrals, and their applications in this comprehensive course.

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Calculus Course

Course Overview

This course will take you through the fundamental concepts of calculus, from limits and derivatives to integrals and their applications. By the end, you'll have a solid understanding of how calculus is used to model change and solve real-world problems.

Estimated completion time: 6-8 weeks

Course Timeline

Functions and Limits

Introduction to Functions

Understanding Limits

Continuity and Limits

Quiz: Functions and Limits

Derivatives

Introduction to Derivatives

Derivative Rules

Derivatives and Graphs

Applications of Derivatives

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Unit 1: Functions and Limits

Lesson 3 of 4

Continuity and Limits

A function is continuous at a point if three conditions are met:

The function is defined at the point.
The limit exists at that point.
The limit equals the function value at that point.

Mathematically, for a function \( f(x) \) to be continuous at \( x = a \):

\[ \lim_{x \to a} f(x) = f(a) \]

Let's look at an example:

Consider the function:

\[ f(x) = \begin{cases} x^2 & \text{if } x \leq 2 \\ x + 2 & \text{if } x > 2 \end{cases} \]

Is \( f(x) \) continuous at \( x = 2 \)?

Interactive Exploration

Current x-value: 1.0

Function value: 1.0

Check Your Understanding

1. Which of the following best describes continuity at a point?

The function has no breaks or jumps at that point

The function is differentiable at that point

The limit exists and equals the function value at that point

The function is defined at that point

2. Consider the function \( f(x) = \frac{x^2 - 1}{x - 1} \). What is \( \lim_{x \to 1} f(x) \)?

Undefined

Calculus Formulas

Key Calculus Formulas

Basic Derivatives

Constant Rule

d/dx [c] = 0

Power Rule

d/dx [xⁿ] = n·xⁿ⁻¹

Exponential Rule

d/dx [eˣ] = eˣ

Sum Rule

d/dx [f(x) + g(x)] = f'(x) + g'(x)

Basic Integrals

Power Rule

∫xⁿ dx = xⁿ⁺¹/(n+1) + C (n ≠ -1)

Exponential Rule

∫eˣ dx = eˣ + C

Trigonometric Rules

∫sin(x) dx = -cos(x) + C

∫cos(x) dx = sin(x) + C

Limit Laws

Sum Law

lim [f(x) + g(x)] = lim f(x) + lim g(x)

Product Law

lim [f(x)·g(x)] = lim f(x) · lim g(x)

Quotient Law

lim [f(x)/g(x)] = lim f(x) / lim g(x) (if lim g(x) ≠ 0)

Power Law

lim [f(x)]ⁿ = [lim f(x)]ⁿ

Fundamental Theorems

FTC Part 1

d/dx ∫ₐˣ f(t) dt = f(x)

Differentiation undoes integration

FTC Part 2

∫ₐᵇ f(x) dx = F(b) - F(a)

Where F'(x) = f(x)

Unit Circle Reference

Key Trigonometric Values

Angle (θ)	sin θ	cos θ	tan θ
0°	0	1	0
30°	1/2	√3/2	√3/3
45°	√2/2	√2/2	1
60°	√3/2	1/2	√3
90°	1	0	Undefined

Trigonometric Identities

\[ \sin^2 \theta + \cos^2 \theta = 1 \]

\[ 1 + \tan^2 \theta = \sec^2 \theta \]

\[ \sin(2\theta) = 2 \sin \theta \cos \theta \]

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Available Courses

Calculus

Linear Algebra

Probability & Stats

Calculus Course

Course Overview

Course Timeline

Functions and Limits

Introduction to Functions

Understanding Limits

Continuity and Limits

Quiz: Functions and Limits

Derivatives

Introduction to Derivatives

Derivative Rules

Derivatives and Graphs

Applications of Derivatives

Upload Your Own Material

Unit 1: Functions and Limits

Continuity and Limits

Interactive Exploration

Professor Nova Explains:

Check Your Understanding

Quiz Results

Calculus Formulas

Key Calculus Formulas

Basic Derivatives

Basic Integrals

Limit Laws

Fundamental Theorems

Unit Circle Reference

Key Trigonometric Values

Trigonometric Identities