# Avon High School - AP Calculus BC - Topic 10.12 - Example 2

TLDRIn this engaging video, the presenter guides viewers through the process of using Taylor's inequality and the Lagrange error bound to approximate the value of e^x at x=1 with a third-degree polynomial. The step-by-step explanation involves calculating derivatives, evaluating them at zero for the Maclaurin polynomial, and then using the polynomial to estimate e. The video also demonstrates how to find the range of possible values for e^x at x=1 using the remainder term from Taylor's inequality, ultimately showing that the approximation falls within a range that closely matches the known value of e.

###### Takeaways

- 📘 The topic is the Lagrange Error Bound for approximating e^x at x=1 using a third-degree polynomial.
- 🌟 The process involves using Taylor's Inequality and not relying on memorization of the value of e.
- 🔍 The problem requires constructing the Taylor polynomial without it being explicitly given, starting from the original function e^x.
- 📈 The derivatives of e^x up to the third degree all evaluate to 1 at x=0, leading to a simple polynomial form.
- 🧮 The third-degree polynomial approximation for e^x at x=1 is 1 + (1/1!)x - 0^1 + (1/2!)x^2 - 0^2 + (1/3!)x^3.
- 🎯 The polynomial simplifies to 1 + x/2 + x^2/6, which when evaluated at x=1 gives 8/3 ≈ 2.67.
- 📊 Taylor's Inequality is used to find the range of possible values for e^x at x=1 by evaluating the remainder term.
- 🧬 The fourth derivative of e^x is e^x, which is constant and simplifies the process of finding the maximum value for the error bound.
- 🏹 A conservative estimate for the maximum value of the remainder term is used to ensure a safe error bound.
- 🤔 The final range of possible values for e^x at x=1, using the third-degree Maclaurin polynomial, is approximately between 2.61 and 2.73.
- 📚 The video emphasizes the importance of understanding the process and not just the final answer, and encourages further practice for better understanding.

###### Q & A

### What topic is being discussed in the video?

-The topic being discussed is the Lagrange Error Bound and its application to find the third-degree polynomial approximation for e^x at x equals one, centered at zero.

### Why is the topic of Lagrange Error Bound considered overwhelming for students?

-The topic is considered overwhelming because it involves understanding complex mathematical concepts, such as Taylor's inequality and polynomial approximations, which can be challenging for students who are new to these ideas.

### What is the significance of the third-degree polynomial in this context?

-The third-degree polynomial is significant as it is used to approximate the function e^x at x equals one. This approximation helps in understanding the behavior of the function without relying on memorized values.

### How many derivatives are taken to get to the third-degree polynomial?

-Three derivatives are taken to get to the third-degree polynomial, as it is a third-degree approximation.

### What is the value of the third-degree polynomial at x equals one?

-The value of the third-degree polynomial at x equals one is 8/3, which is approximately 2.67.

### What is Taylor's Inequality used for in this problem?

-Taylor's Inequality is used to find the range of possible values for e^x at x equals one by determining the maximum error that can occur using the third-degree polynomial approximation.

### What is the role of the fourth derivative in the Lagrange Error Bound calculation?

-The fourth derivative plays a crucial role in the Lagrange Error Bound calculation as it represents the maximum value of the function's rate of change within the interval, which is used to estimate the error of the approximation.

### How does the讲师 (instructor) handle the complexity of evaluating the fourth derivative at some point z between 0 and 1?

-The讲师 (instructor) chooses a conservative approach by evaluating the fourth derivative (which is e^x) at a point z slightly beyond 1 (using 3/4 as an example) to ensure a safe maximum value without needing a calculator.

### What is the final range of possible values for e^x at x equals one, according to the video?

-The final range of possible values for e^x at x equals one, as determined using the Lagrange Error Bound, is between 2.67 (the approximation) and a value slightly greater than 2.718, which is close to the actual value of e.

### How does the讲师 (instructor) encourage students to approach problems that seem overwhelming?

-The讲师 (instructor) encourages students to hang in there, to work through examples, and to check answers repeatedly. They emphasize that with practice, students will feel better about tackling complex topics.

### What is the讲师's (instructor's) advice for dealing with the complexity of the Lagrange Error Bound?

-The讲师 (instructor) advises students not to let the complexity of the Lagrange Error Bound bother them. They suggest that sometimes, you just can't do anything about it and should focus on working through the problem without getting caught up in the details that might not be solvable without a calculator.

###### Outlines

##### 📚 Introduction to Lagrange Error Bound

The video begins with an introduction to the concept of Lagrange Error Bound, emphasizing the importance of understanding the process rather than relying on memorization. The speaker encourages viewers to stick with the topic despite its complexity, assuring that familiarity will come with practice. The problem at hand involves finding a third-degree polynomial approximation for e^x at x=1, centered at zero, using Taylor's Inequality to determine the range of possible values for e^x at x=1. The speaker acknowledges the challenge but guides the audience through the steps of deriving the necessary polynomial and evaluating it at the given point.

##### 🔢 Derivatives and Polynomial Construction

The speaker proceeds to calculate the derivatives of the exponential function e^x, which are all equal to e^x itself, and evaluates them at x=0 to construct the Maclaurin polynomial. The third-degree polynomial approximation is derived and simplified to 1 + (1/1!)x - (1/2!)x^2 + (1/3!)x^3. The video then moves on to approximate e^1 using this polynomial, resulting in a value close to the known approximate value of e. The speaker introduces Taylor's Inequality to find the range of possible values for e^x at x=1, explaining the concept of the remainder term and its calculation using the fourth derivative of the function.

##### 📈 Lagrange Error Bound Application

The speaker applies the Lagrange Error Bound to determine the maximum possible error in the approximation of e^1. By evaluating the fourth derivative of e^x and finding its maximum value, the speaker calculates the bound of the error to be 1/8. The video then sets up an inequality to find the possible range of values for e^x at x=1, considering the error bound. The speaker solves this inequality to find the range of values that e^x at x=1 could fall within, which aligns with the known value of e. The video concludes with a reminder of the importance of understanding the process and the application of Taylor's Inequality and Lagrange Error Bound for error analysis in approximation problems.

###### Mindmap

###### Keywords

##### 💡Lagrange Error Bound

##### 💡Taylor's Inequality

##### 💡Maclaurin Polynomial

##### 💡Derivatives

##### 💡Polynomial Approximation

##### 💡e to the power of x (e^x)

##### 💡Taylor Series

##### 💡Approximation

##### 💡Error Analysis

##### 💡Factorial

##### 💡Calculus

###### Highlights

The topic is about using Taylor's Inequality to find the range of possible values for e^x at x=1 with a third-degree polynomial approximation centered at zero.

The process involves finding the third-degree polynomial approximation for e^x, which is an unusual problem as it essentially asks to determine e to the first power.

The Taylor polynomial is not given explicitly, so it must be constructed using derivatives of the function e^x.

The derivatives of e^x are all 1, making the construction of the Maclaurin polynomial straightforward.

The third-degree polynomial approximation is 1 + (1/1!)x - 0^1 + (1/2!)x^2 - 0^2 + (1/3!)x^3.

The polynomial simplifies to 1 + x/1! + x^2/2! + x^3/6, which approximates e^x at x=1 as 8/3 or 2.67, close to the actual value of e.

Taylor's Inequality is used to determine the range of possible values for e^x at x=1 by calculating the remainder term.

The remainder term is less than or equal to the absolute value of the maximum of the fourth derivative divided by 4! times (x-c)^(n+1), where z is between c and x.

The fourth derivative of e^x is simply e^x, which is constant and equal to 1 when evaluated at z.

A conservative approach is taken to estimate the maximum value of the remainder term by using a value of z slightly greater than 1.

The maximum bound of error is calculated to be 18/24 or 3/4, which is used to determine the possible range of values for e^x at x=1.

By solving the compound inequality, the possible range of values for e^x at x=1 is found to be between 23/24 and 41/24.

The actual value of e is approximately 2.71828, which falls within the calculated range using the third-degree Maclaurin polynomial.

The example demonstrates the practical application of Taylor's Inequality and the Lagrange error bound in approximating function values without a calculator.

The video is part of a series discussing Taylor's Inequality and its applications, with more examples and problems to be covered in class.

The use of Taylor's Inequality and the Lagrange error bound helps in understanding error analysis in approximations.

The video encourages viewers to practice and work through examples to build confidence in handling complex topics like Taylor's Inequality.

###### Transcripts

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