Calculus BC – 10.12 Lagrange Error Bound

Summary

In this lesson, Mr. Bean explains the concept of the Lagrange error bound, which helps determine how far off an approximation is from the exact value of a function. Building on previous lessons about Taylor polynomials, he demonstrates how to calculate the error bound for approximations. Using examples like approximating cosine and exponential functions, he explains how the Lagrange error bound provides an upper limit on the error, helping students understand how accurate their approximations can be. Through these examples, students learn to apply error bounds to various problems and understand their significance in calculus.

Takeaways

• 😀 Taylor polynomials are approximations of actual functions, and the Lagrange error bound helps to quantify how much the approximation might differ from the actual function.
• 😀 The Lagrange error bound calculates the maximum possible error between the Taylor polynomial and the exact function.
• 😀 The remainder (or error) is the difference between the actual function and the Taylor polynomial.
• 😀 The Lagrange error bound formula involves calculating the maximum value of the (n+1)-th derivative of the function on a given interval.
• 😀 The function's (n+1)-th derivative is evaluated at some point z, which lies between the center of the approximation (c) and the point where the approximation is being evaluated (x).
• 😀 When approximating a function using a Taylor polynomial, we focus on finding the largest possible error, not the exact error itself.
• 😀 For sine or cosine functions, the maximum value of their derivatives on any given interval is 1, simplifying the calculation of the error bound.
• 😀 In the example of approximating cosine at 0.2, the error bound is calculated and results in an extremely small error of approximately 0.00002667.
• 😀 The error bound is useful for understanding how accurate our approximation is; for example, a Taylor polynomial of degree 4 gives a very precise approximation of cosine at 0.2.
• 😀 When approximating e^x using a third-degree Taylor polynomial, the Lagrange error bound helps determine the accuracy of the approximation of e at x=1, resulting in a small error of about 0.051615.
• 😀 The Lagrange error bound can also be used to determine the smallest order of a Taylor polynomial needed for the error to be less than a certain value, such as 1 in the given example, where the smallest order is a fifth-degree polynomial.

Q & A

What is the main topic of this lesson in calculus?

The main topic of the lesson is the Lagrange Error Bound, which is used to determine the maximum error between a Taylor polynomial approximation and the actual function.

What are Taylor polynomials, and why are they important in this lesson?

Taylor polynomials are an approximation of a function. In this lesson, they are used to estimate the value of a function at a given point, and the Lagrange Error Bound helps us understand how much the approximation might differ from the actual function.

What does the remainder in the context of Taylor polynomials represent?

The remainder represents the difference between the actual function and the Taylor polynomial approximation. It is essentially the error that occurs in the approximation.

How is the Lagrange Error Bound formula derived?

The Lagrange Error Bound formula is derived by considering the remainder of a Taylor polynomial approximation. It involves finding the maximum value of the n+1 derivative of the function over the interval of approximation.

What is the significance of the variable 'z' in the Lagrange Error Bound formula?

In the Lagrange Error Bound formula, 'z' is a variable representing some value between 'c' (the center of the approximation) and 'x' (the point where the approximation is being evaluated). The value of 'z' helps determine the maximum error.

In the example involving the cosine function, what does the Lagrange Error Bound tell us?

The Lagrange Error Bound tells us that the error between the fourth-degree Taylor polynomial approximation of cosine at x=0.2 is at most 0.00002667. This means the approximation is very close to the actual value, with only a very small possible error.

What is the maximum value of the fifth derivative of the cosine function used in the error bound calculation?

The maximum value of the fifth derivative of the cosine function is 1, since the sine and cosine functions always oscillate between -1 and 1, and the absolute value is considered in the error bound calculation.

In the second example with e^x, what is the approximation of e^1 using a third-degree Taylor polynomial?

The approximation of e^1 using a third-degree Taylor polynomial is 2.6667, which is the result of the sum 1 + 1 + 1^2/2! + 1^3/3!.

How does the Lagrange Error Bound work in this e^x example?

The Lagrange Error Bound for e^x tells us that the error in approximating e^1 with a third-degree Taylor polynomial is at most 0.0516, which is less than 1 and shows the error is relatively small.

In the third example, how do you determine the smallest order of the Taylor polynomial to achieve a given error bound?

The smallest order of the Taylor polynomial is determined by testing different degrees of Taylor polynomials and using the Lagrange Error Bound until the error is less than the given limit. In the example, a fifth-degree Taylor polynomial was required to achieve an error less than 1.