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❶Lesson 6 - Optimizing Simple Systems.

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Unlimited access to all video lessons Lesson Transcripts Tech support. See all other plans. First Name Name is required. Last Name Name is required. Phone number is required. Phone number is invalid. Have a Coupon Code? You have not applied your coupon. Card Number Have a Coupon Code? Card number is required. Credit card number invalid. Please correct or use a different card. This card has been declined. Please use a different card.

Prepaid cards not accepted. Expiration is not a valid, future date. Year Expiration Year is required. Zip Code Zip code is required. Secure Server tell me more. Lesson 1 - What is a Function: Basics and Key Terms.

Lesson 2 - Graphing Basic Functions. Lesson 4 - Understanding and Graphing the Inverse Function. Lesson 5 - Polynomial Functions: Lesson 6 - Polynomial Functions: Lesson 8 - Slopes and Tangents on a Graph.

Lesson 10 - Horizontal and Vertical Asymptotes. Lesson 11 - Implicit Functions. Lesson 1 - Continuity in a Function. Lesson 2 - Discontinuities in Functions and Graphs. Lesson 3 - Regions of Continuity in a Function. Lesson 4 - Intermediate Value Theorem: Lesson 5 - Intermediate Value Theorem: Lesson 3 - Finding Distance with the Pythagorean Theorem.

Lesson 4 - Trigonometry: Lesson 5 - Trigonometry and the Pythagorean Theorem. Lesson 1 - Using a Graph to Define Limits. Lesson 2 - Understanding Limits: Lesson 3 - One-Sided Limits and Continuity. Lesson 4 - How to Determine the Limits of Functions. Lesson 5 - Understanding the Properties of Limits. Lesson 6 - Squeeze Theorem: Lesson 7 - Graphs and Limits: Defining Asymptotes and Infinity. Lesson 1 - Velocity and the Rate of Change. Lesson 2 - Slopes and Rate of Change.

Lesson 3 - What is the Mean Value Theorem? Lesson 5 - Derivatives: Lesson 6 - Derivatives: Lesson 1 - Using Limits to Calculate the Derivative. Lesson 2 - The Linear Properties of a Derivative. Lesson 3 - Calculating Derivatives of Trigonometric Functions. Lesson 4 - Calculating Derivatives of Polynomial Equations. Lesson 5 - Calculating Derivatives of Exponential Equations.

Lesson 7 - Differentiating Factored Polynomials: Product Rule and Expansion. Lesson 10 - Calculating Higher Order Derivatives. Lesson 1 - Graphing the Derivative from Any Function. Lesson 2 - Non Differentiable Graphs of Derivatives. Lesson 5 - Concavity and Inflection Points on Graphs.

Lesson 7 - Data Mining: Function Properties from Derivatives. Lesson 8 - Data Mining: Identifying Functions From Derivative Graphs. Lesson 1 - Linearization of Functions. Lesson 5 - Optimization and Differentiation. Lesson 6 - Optimizing Simple Systems. Lesson 7 - Optimizing Complex Systems.

Lesson 1 - Summation Notation and Mathematical Series. Lesson 4 - What is the Trapezoid Rule? Lesson 6 - Definite Integrals: Lesson 8 - Linear Properties of Definite Integrals. Lesson 9 - Average Value Theorem. Lesson 10 - The Fundamental Theorem of Calculus. Lesson 11 - Indefinite Integrals as Anti Derivatives.

Lesson 1 - Calculating Integrals of Simple Shapes. Lesson 2 - Anti-Derivatives: Calculating Indefinite Integrals of Polynomials. Lesson 6 - Substitution Techniques for Difficult Integrals. The function is not continuous at this point. This kind of discontinuity is called a removable discontinuity. Removable discontinuities are those where there is a hole in the graph as there is in this case.

A function is continuous on an interval if we can draw the graph from start to finish without ever once picking up our pencil. The graph in the last example has only two discontinuities since there are only two places where we would have to pick up our pencil in sketching it.

Rational functions are continuous everywhere except where we have division by zero. So all that we need to is determine where the denominator is zero. With this fact we can now do limits like the following example. Below is a graph of a continuous function that illustrates the Intermediate Value Theorem. Also, as the figure shows the function may take on the value at more than one place. It only says that it exists. These are important ideas to remember about the Intermediate Value Theorem.

A nice use of the Intermediate Value Theorem is to prove the existence of roots of equations as the following example shows. For the sake of completeness here is a graph showing the root that we just proved existed.

Note that we used a computer program to actually find the root and that the Intermediate Value Theorem did not tell us what this value was.

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AP Calculus‎ > ‎AP Calculus Limits and Continuity‎ > ‎ AP Calculus Limits and Continuity Homework. Showing 19 items Assignment Day Solutions Assignment Number Video Solutions; Sort Intro to Limits Homework.

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In this section we will introduce the concept of continuity and how it relates to limits. We will also see the Intermediate Value Theorem in this section and how it can be used to determine if functions have solutions in a given interval. Reach infinity within a few seconds! Limits are the most fundamental ingredient of calculus. Learn how they are defined, how they are found (even under extreme conditions!), and how they relate to continuous functions.

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