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P Series: Convergence, Divergence, and Applications

Calculus: Master Limits, Derivatives & Integrals

Limits

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Intermediate value theorem

Proving the Existence of Real Roots Using IVT

Intermediate Value Theorem

Application of Intermediate Value Theorem

Proving the Existence of a Real Root Using IVT

Finding limits from graphs  

Continuity

Discontinuities of Rational Functions (denominator=0)

Finding limits algebraically - direct substitution 

Finding limits algebraically - when direct substitution is not possible

Infinite limits - vertical asymptotes  

Limits at infinity - horizontal asymptotes

Limit Evaluation

Limits at Infinity of Exponential Functions

Squeeze theorem

Quadratic Function with Integer Vertex

Limit laws

Evaluating Limits of Functions

limit laws

Evaluating Limits with specific limits given

Introduction to Calculus - Limits

Derivatives

Estimating derivatives from a table

Slope and equation of tangent line

Chain rule

Chain Rule

Differentiation of Trigonometric Function

Differentiate Radical Functions with Chain Rule

Differentiate: Trigonometric Functions

Chain rule: Differentiate Trigonometric Functions

Differentiation of Exponential Function

Differentiation of Exponential Functions

Differentiation: Chain Rule with Logarithmic Functions

Differentiation of Logarithmic Function

Product rule

Quotient rule

Differentiation using Quotient Rule

Differentiation using Chain Rule and Quotient Rule

Derivative of inverse trigonometric functions

Definition of derivative  

Definition of Derivative

Definition of derivative

Derivative using Definition

Vertical Tangent Line of a Function

Applications to definition of derivative

Power rule

Power Rule

Application of the Power Rule

Power rule application

Applying Power Rule

Power Rule with Negative Exponents

Power rule with rational exponents

Power rule and rational exponents

Derivative of trigonometric functions  

Derivative of Trigonometric Functions

Derivative of Composite Trig Functions

Derivative of exponential functions

Derivative of Exponential Functions

Differentiation of Composite Functions

Derivative of power and exponential functions

Derivative of logarithmic functions

Derivative of log with arbitrary base

Derivative of Natural Logarithm Functions

Derivative of Logarithmic Functions

Implicit differentiation

Higher order derivatives

Higher Order Derivatives

First and second derivatives

First and Second Derivatives

Second derivatives with implicit differentiation

Implicit Differentiation

Higher Order Derivatives with Repeating Patterns

Derivative Applications

Rolle&apos;s theorem

Quadratic approximation

Marginal profit, and maximizing profit & average profit

Elasticity of demand

Mean value theorem 

Mean Value Theorem

Mean value theorem

Mean Value Theorem Application

Critical number & maximum and minimum values

Critical Number & Maximum and Minimum Values

Linear approximation

Use of Linear Approximations in Estimating Square Roots

Linearization of Radical Functions

Linearization of Polynomial Functions

Linearization of Rational Functions

Linearization of Exponential Functions

Linearization of Logarithmic Functions

Linear Approximation of Trigonometric Functions

Optimization

Position velocity acceleration

Velocity of a Particle

Total Distance Traveled by a Particle

Comparative Analysis of Particle's Velocity and Acceleration

Curve sketching

Related rates

l'Hospital's rule

l'Hopital's rule

l'Hospital's Rule

Evaluating the limit using L'Hospital's Rule

l"Hospital"s rule

Demand, revenue, cost & profit

Marginal revenue, and maximizing revenue & average revenue

Marginal cost, and minimizing cost & average cost

Integrals

Antiderivatives

Derivatives and Antiderivatives

Derivative and Antiderivative of a Function

Riemann sum

Riemann Sum

Evaluating integrals with a Riemann sum

Evaluating integrals with a Riemann Sum

Riemann Sum with Trapezoid Approximation

Trapezoidal Approximation of Riemann Sums

Fundamental theorem of calculus

Fundamental Theorem of Calculus Part II

Fundamental Theorem of Calculus Part I

Fundamental Theorem of Calculus

Definite integral

Definite Integral

Integration Techniques

Nuerical integration

Numerical integration

Numerical Integration

U-Substitution

U-Substitution: Integrating Radical Functions

Not-So-Obvious U-Substitution

Trigonometric substitution

Trigonometric Substitution

Integration of rational functions by partial fractions

Improper integrals

Improper Integrals

Integration by parts

Integration using trigonometric identities

Integration Applications

Consumer and producer surplus

Continuous money flow

Areas between curves

Volumes of solids of revolution - Disc method

Volumes of solids with known cross-sections

Volumes of solid with known cross-sections

Volumes of solids of revolution - Shell method

Volumes of solid of revolution - Shell method

Arc length

Average value of a function

Order and solutions to differential equations

Separable equations

Separable Equations

Modeling with differential equations

Sequence and Series

Introduction to sequences

Monotonic and bounded sequences

Introduction to infinite series

Convergence and divergence of normal infinite series  

Convergence & divergence of geometric series 

Divergence of Geometric Series

Convergence & divergence of telescoping series 

math

<h2>What is calculus?</h2>
<p>Calculus math is the branch of mathematics involved with studying the rate of change, motion, and limits. As a joint discovery in Mathematics, Sir Isaac Newton and Gottfried Newbitz are credited as the Fathers of calculus for their early work in developing the fundamental concepts, theories and calculus rules we study today. Recent findings in the past decade however, suggest that this honour must now be split 4-fold to include Madhava and Nilakantha on the throne. Sir Isaac Newton was responsible for establishing the relationship between the two key topics in calculus: Differentials and Integrals. Gottfried Newbitz work was more focused on the rate of change in graphs alongside developing the mathematical notation for calculus that we use today.  Research has now confirmed that Madhava and Nilakantha were responsible for discovering infinite series in calculus, two centuries before Sir Isaac Newton and Gottfried Newbitz were born.</p>
<p>If you&apos;re just getting started with basic calculus or intro to calculus, differential calculus is the first order of business. Here you will learn how to calculate the slope of curves (e.g. parabolas), learn to use and apply the different derivative rules, and identify rates of change in velocity and acceleration.</p>
<p>In <a href="[[region_textbook.html|{"textbook_key":"calculus"}]]">Calculus 1</a>, <a href="[[region_textbook.html|{"textbook_key":"business-calculus"}]]">Business Calculus</a>, or any of your <a href="[[region_textbook.html|{"textbook_key":"ap-calculus-ab"}]]">AP Calculus classes</a>, you will also take a closer look at the concepts of limits and infinity. On the surface, these concepts appear to be at odds with one another. In the world of calculus however, this is far from true. Finding the limits of any graph requires you to look at what is happening at the microscopic level, dissecting the graph into infinitely small pieces and examining what happens as points on a graph get infinitesimally close together.  These foundational notions links differentials and integrals together.</p>
<p>Integral Calculus falls into the realm of advanced calculus. Topics covered here you&apos;ll likely see in <a href="[[region_textbook.html|{"textbook_key":"ap-calculus-ab"}]]">AP Calculus AB</a>, but more so in <a href="[[region_textbook.html|{"textbook_key":"ap-calculus-bc"}]]">AP Calculus BC</a> and <a href="[[region_textbook.html|{"textbook_key":"calculus2"}]]">Calculus 2</a>. Integral calculus is where you learn new methods of calculating area, volume and arc lengths of objects, learn integration techniques, and apply calculus to sequences and series. Science, Technology, Math, and Engineering (STEM) majors may be required to study <a href="[[region_textbook.html|{"textbook_key":"differential-equations"}]]">differential equations</a>, a higher level of specialized topics related to differential calculus. Here you will examine first and second order differential equations alongside Laplace transformations.</p>
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<h2>Is calculus hard?</h2>
<p>Calculus has a notorious reputation for being difficult. Many who fear math have poised this perception of calculus. However, there are many who excel and enjoy calculus, some more so than the likes of <a href="[[canon_textbook.html|{"textbook_key":"algebra-help"}]]">Algebra</a> and <a href="[[canon_textbook.html|{"textbook_key":"trigonometry-help"}]]">Trigonometry</a>. The reality here however is that calculus is and can be hard - but it can be dealt with.</p>
<p>If you are just beginning with <a href="[[region_textbook.html|{"textbook_key":"pre-calculus"}]]">Precalculus for dummies</a> or even Calculus for dummies, there&apos;s no need to fret. Take your time to learn calculus essentials, familiarize yourself with calculus equations and follow along calculus examples to learn new techniques and problem solving strategies. Laying the groundwork here is important and will help you advance to more difficult calculus topics with ease. Remember that concepts are sequential and compounded on one another, so skip back if you feel like you need a quick refresher on common antiderivatives, more limits practice or a <a href="[[region_textbook.html|{"textbook_key":"calculus"}]]">complete Calculus 1 review</a>.</p>
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<h2>How to learn calculus?</h2>
<p>When it comes to calculus, or any math subject, students are quick to ask "how to do calculus" instead of seeking "How to learn calculus".  In this modern day and age, to learn calculus online is quickly becoming the norm, whether it is for College Calculus or Calculus for beginners. With the advancement in technology and increase in accessibility of online resources, StudyPug aims to provide a superior service for those looking to learn calculus online, a calculus refresher, or access online calculus review material.</p>
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<p>Looking for something more to chew on like our resident StudyPugs? Take a gander at our articles below each video lesson. These articles are your personal online notes where we break down calculus formulas, provide more calculus review, go over examples and frequently asked questions.</p>
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P Series: Exploring Convergence and Divergence in Infinite Sums

What You'll Learn

Recognize the standard form of a p-series: sum of 1/n^p from n=1 to infinity

Apply the p-series test: convergence when p > 1, divergence when p 1

Transform series algebraically to reveal p-series form

Adjust series starting indices by adding and subtracting terms

Determine convergence or divergence based on parameter values

What You'll Practice

Testing basic p-series for convergence using the exponent p

Reindexing series that start at values other than n=1

Simplifying complex rational expressions into p-series form

Analyzing series with parameters to find convergence intervals

Why This Matters

Before You Start — Make Sure You Can:

This Unit Includes

4 Video lessons

Practice exercises

Learning resources

Skills

P-Series

Convergence Tests

Divergence Tests

Series Manipulation

Infinite Series

Algebraic Simplification

Parameter Analysis