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Master the Fundamental Theorem of Calculus | Key Concepts

AP Calculus BC: Master Advanced Concepts & Skills

Limits & Continuity

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

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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  

Squeeze theorem

Quadratic Function with Integer Vertex

l'Hospital's rule

l'Hopital's rule

l'Hospital's Rule

Evaluating the limit using L'Hospital's Rule

l"Hospital"s rule

Limit laws

Evaluating Limits of Functions

limit laws

Evaluating Limits with specific limits given

Derivatives

Estimating derivatives from a table

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

Slope and equation of tangent line

Mean value theorem

Mean Value Theorem

Mean Value Theorem Application

Critical number & maximum and minimum values

Critical Number & Maximum and Minimum Values

Optimization

Rectilinear motion: Derivative

Velocity of a Particle

Position velocity acceleration

Total Distance Traveled by a Particle

Comparative Analysis of Particle's Velocity and Acceleration

Curve sketching

Related rates

Tangents and concavity of parametric equations

Tangent and concavity of parametric equations

Tangents of polar curves

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

Improper integrals

Improper Integrals

Definite integral

Definite Integral

Integration techniques

U-Substitution

U-Substitution: Integrating Radical Functions

Not-So-Obvious U-Substitution

Trigonometric substitution

Trigonometric Substitution

Integration of rational functions by partial fractions

Integration by parts

Integration using trigonometric identities

Integral Applications

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

Average value of a function

Arc length and surface area of parametric equations

Arc Length of a Parametric Curve

Area of polar curves

Arc length of polar curves

Introduction to differential equations

Order and solutions to differential equations

Separable equations

Separable Equations

Slope fields

Understanding Slope Fields

Euler's method

Euler's Method

Euler"s Method

Euler"s method

Sequence and Series

Introduction to sequences

Introduction to infinite series

Convergence and divergence of normal infinite series  

Convergence and divergence of geometric series 

Convergence & divergence of geometric series

Divergence of Geometric Series

Divergence of harmonic series

Divergence of Harmonic Series

P Series

Convergence and Divergence of P Series

Alternating series test

Alternating Series Test

Convergence of the Alternating Series Test

Divergence test

Divergence Test

Comparison and limit comparison test 

Comparison & Limit Comparison Test

Comparison & limit comparison test

Integral test

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AP Calculus BC

math

https://dmn92m25mtw4z.cloudfront.net/img/textbook/textbook-APcalculusBC.png

Fundamental Theorem of Calculus: Bridging Differentiation and Integration

What You'll Learn

Recognize that differentiation and integration are inverse operations that cancel each other out

Apply Part 1 to evaluate derivatives of integrals with variable upper limits using the chain rule

Use Part 2 to evaluate definite integrals by finding antiderivatives and computing endpoint differences

Calculate definite integrals of polynomial, trigonometric, logarithmic, and exponential functions

Identify when integrals cannot be evaluated due to discontinuities in the interval

What You'll Practice

Finding derivatives of integrals with variable upper limits like x and x³

Evaluating definite integrals using antiderivatives at upper and lower limits

Working with trigonometric integrals involving sine and cosine functions

Integrating exponential and logarithmic functions with e and ln

Checking for discontinuities before evaluating definite integrals

Why This Matters

This Unit Includes

9 Video lessons

Practice exercises

Learning resources

Skills

Definite Integrals

Antiderivatives

Chain Rule

Integration Techniques

Trigonometric Functions

Exponential Functions

Logarithmic Functions

Calculus