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Master the Fundamental Theorem of Calculus | Key Concepts

NL.SO.3208C1

Finding limits from graphs

Finding limits algebraically - direct substitution

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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

NL.SO.3208C11

Derivative of trigonometric functions

Derivative of Trigonometric Functions

Derivative of Composite Trig Functions

Position velocity acceleration

Velocity of a Particle

Total Distance Traveled by a Particle

Comparative Analysis of Particle's Velocity and Acceleration

NL.SO.3208C12

Finding inverse trigonometric function from its graph

Evaluating inverse trigonometric functions

Finding inverse reciprocal trigonometric function from its graph

Inverse reciprocal trigonometric function: finding the exact value

Inverse Reciprocal Trigonometric Function: Finding the Exact Value

Derivative of inverse trigonometric functions

NL.SO.3208C13

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

NL.SO.3208C14

Antiderivatives

Derivatives and Antiderivatives

Derivative and Antiderivative of a Function

U-Substitution

U-Substitution: Integrating Radical Functions

Not-So-Obvious U-Substitution

NL.SO.3208C15

Areas between curves

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

NL.SO.3208C2

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

Continuity

Discontinuities of Rational Functions (denominator=0)

NL.SO.3208C3

Estimating Derivatives from a table

Slope and equation of tangent line

Definition of derivative

Definition of Derivative

Derivative using Definition

Vertical Tangent Line of a Function

Applications to definition of derivative

NL.SO.3208C4

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

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

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

NL.SO.3208C5

Implicit differentiation

NL.SO.3208C6

Critical number & maximum and minimum values

Critical Number & Maximum and Minimum Values

Curve sketching

NL.SO.3208C8

Related rates

NL.SO.3208C9

Optimization

NL.SO.3208RF1

Composite functions

Adding functions

Subtracting functions

Multiplying functions

Dividing functions

Operations of Functions – In a Nutshell

Operations with functions

Inequalities of combined functions

Inequalities of Combined Functions

NL.SO.3208RF2

What is a rational function?

Point of discontinuity

Slant asymptote

Determining the Equation of a Slant Asymptote Using Long Division

Determining the Equation of a Slant Asymptote Using Synthetic Division

Solving rational equations

Solving rational inequalities

Solving Rational Inequalities With Two Fractions

Vertical asymptote

Horizontal asymptote

Horizontal Asymptote

Graphs of rational functions

Mathematics 3208

Calculus Reasoning

Relations and Functions

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Intro

Ex. 1

Ex. 2

Intro to the Fundamental Theorem of Calculus with chain rule

Using antiderivatives to evaluate definite integrals

Applying the Fundamental Theorem of Calculus with a simple upper limit

Applying the fundamental theorem with chain rule to a complex integral

Evaluating a definite integral of x squared from -1 to 3

Identifying a discontinuity that prevents integration

Evaluating a definite integral of 5/x from 1 to e squared

Evaluating a definite integral of 4cos(5θ) from π/5 to π

Evaluating a definite integral of e to the x with ln bounds

In this lesson, we will learn about part 1 and part 2 of the Fundamental Theorem of Calculus. In part 1, we see that taking the derivative of an integral will just result in giving us the original function. However in some cases, we get the original function AND the derivative of the upper limit. Lastly in part 2, we will learn another way of evaluating the <a href='[[glossary.html|{"keyword":"Definite integral","term":"Definite integral"}]]'>definite integral</a>. To evaluate the definite integral, we must take the difference of the anti-derivative of the function at the upper limit, and the anti-derivative of the function at the lower limit. We will apply this theorem to many types of definite integrals such as polynomial integrals, trigonometric integrals, logarithmic integrals, and exponential integrals.

What is the fundamental theorem of calculus?