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• Limits and Continuity

• Derivatives Basics

• Applications of Derivatives

• Chain Rule

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Limits & Continuity

Learn limits and continuity — the building blocks of calculus. Understand how functions behave as they approach specific points, such as calculating the maximum load a structure can handle before breaking.

• Limits intro

• Estimating limits from graphs

• Estimating limits from tables

• Properties of limits

• Limits by direct substitution

• Limits using algebraic manipulation

• Formal definition of limits (epsilon-delta)

• Strategy in finding limits

• Squeeze theorem

• Types of discontinuities

• Continuity at a point

• Continuity over an interval

• Removing discontinuities

• Infinite limits

• Limits at infinity

• Intermediate value theorem

Derivatives Basics

Derivatives are a core component of calculus, representing the rates at which a function changes. We can help with topics like the Power Rule to help you understand how to calculate and apply derivatives, like calculating an object’s speed by measuring how its position changes over time.

• Derivatives and tangent lines

• Power rule

• Product and quotient rules

• Higher-order derivatives

• Applications of basic derivatives

Advanced Derivatives: Chain Rule & More

Master more advanced derivative concepts and techniques that are crucial for solving complex problems, such as determining how changing one variable affects another in real-world situations like physics, where the chain rule can help calculate the rate of change of velocity over time.

• Chain rule

• Implicit differentiation

• Inverse functions and derivatives

• Derivatives of trigonometric, exponential, and logarithmic functions

• L'Hopital's rule

• Related rates

Applications of Derivatives

Explore real-world applications of derivatives. Understand how things change and make predictions based on that change; for example, derivatives can help calculate how your car’s velocity changes at any moment — which is crucial in understanding acceleration.

• Understanding Rate of Change

• Tangent Lines and Slopes

• Optimization Problems

• Maxima and Minima

• Related Rates

• Curvature and Concavity

• Inflection Points

• Motion and Velocity

• Population Growth Models

• Economics and Marginal Analysis

• Exponential Growth and Decay

Analyzing Functions

In calculus, applications of derivatives help you analyze how things change, such as determining speed by calculating the rate of change in its position. Learn how to analyze functions as tools for solving optimization problems and understanding real-world scenarios involving rates of change.

• Understanding functions and their graphs

• Intercepts of functions (x-intercepts and y-intercepts)

• Intervals of increase and decrease

• Relative (local) extrema

• Absolute (global) extrema

• Concavity and points of inflection

• Behavior at infinity

• Symmetry in functions (even and odd functions)

• Piecewise functions

• Step functions and absolute value functions

• Rate of change and slope of tangent lines

• End behavior of functions

• Using the first derivative test

• Using the second derivative test

• Critical points and their significance

• Sketching graphs using derivative information

• Understanding asymptotes (vertical, horizontal, and oblique)

Integrals

Integrals represent the area under a curve. In this calculus subtopic, learn how to calculate total quantities like distance, volume, or accumulated change. 

• Introduction to Integrals

• Definite Integrals

• Indefinite Integrals

• Fundamental Theorem of Calculus

• Techniques of Integration

• Applications of Integrals in Real Life

• Area Between Curves

• Volume of Solids of Revolution

• Improper Integrals

• Numerical Integration Methods

• Integration of Trigonometric Functions

• Integration of Exponential and Logarithmic Functions

• Applications in Physics and Engineering

• Graphical Interpretation of Integrals

• Challenges and Common Mistakes in Integration

Applications of Integrals

Learn to calculate the area under curves to determine real-life scenarios like the total distance traveled during a road trip.

• Average Value of a Function

• Straight-Line Motion

• Non-motion Applications of Integrals

• Vertical Area Between Curves

• Horizontal Area Between Curves

• Curves Intersecting at More than Two Points

• Volume, Squares and Rectangles

• Volume, Triangles and Semicircles

• Volume, Disc Method

• Volume, Washer Method

Differential Equations

Differential equations are mathematical equations that relate a function to its derivatives, helping us understand how things change over time. You can use differential equations in many real-life applications, like studying population growth or the spread of diseases. 

• Introduction to Differential Equations

• First-Order Differential Equations

• Applications of First-Order Differential Equations

• Second-Order Differential Equations

• Systems of Differential Equations

• Numerical Methods for Differential Equations

• Laplace Transforms

• Graphical and Qualitative Analysis

• Project and Real-Life Applications