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Master Riemann Sum: Unlock the Power of Area Approximation

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

math

intro

Ex. 1

Finding area under x squared from 1 to 3 using a graphing calculator

Ex. 2

E. 3

Ex. 4

Estimating area under x squared using trapezoid approximation

Understanding area under a curve and the idea behind Riemann sums

Understanding sigma notation and summation formulas

Understanding Riemann sum formulas and right, left, and middle methods

Expressing integrals as Riemann sums and computing exact area under x squared

Right Riemann sum for x squared on [1, 3] with 4 rectangles

Left Riemann sum for f(x)=x² on [1,3] with 4 subintervals

Computing a middle Riemann sum for x squared on [1, 3] with 4 subintervals

Right Riemann sum of x² - 5x + 3 on [2,5] with 6 subintervals

Evaluating a definite integral using the Riemann sum with infinite subintervals

In this lesson, we will learn how to approximate the area under the curve using rectangles.  First we notice that finding the area under the curve is easy if the function is a straight line. However, what if it is a curve? We will actually have to approximate curves using a method called "Riemann Sum". This method involves finding the length of each sub-interval (delta x), and finding the points of interest, finding the y values of each point of interest, and then use the find the area of each rectangle to sum them up. There are 3 methods in using the Riemann Sum. First is the "Right Riemann Sum", second is the "Left Riemann Sum", and third is the "Middle Riemann Sum". Lastly, we will look at the idea of infinite sub-intervals (which leads to integrals) to exactly calculate the area under the curve.

What is a Riemann sum?