Problem 17 - Calculus Single Variable

Question

In each of Exercises 13-22, a function \(f,\) an order \(N, a\) base point \(c,\) and a point \(x_{0}\) are given. The interval with endpoints \(c\) and \(x_{0}\) is denoted by \(J\) a. Calculate the approximation \(T_{N}\left(x_{0}\right)\) of \(f\left(x_{0}\right)\). b. By showing that \(M=\max _{t \in J}\left|f^{(N+1)}(t)\right|\) is the greater of \(\left|f^{(N+1)}(c)\right|\) and \(\left|f^{(N+1)}\left(x_{0}\right)\right|,\) and by using inequality (8.8.7) of Theorem 2 , find an upper bound for the absolute error \(\left|R_{N}\left(x_{0}\right)\right|=\left|f\left(x_{0}\right)-T_{N}\left(x_{0}\right)\right|\) that results from the approximation of part a. $$ f(x)=x^{2}+e^{1+x} \quad N=2 \quad J=\left[x_{0}, c\right]=[-1.2,-1] $$

Step-by-Step Solution

Answer

The approximation \( T_2(-1.2) \) is 2.26 with an error bound of approximately 0.00133.

Step 1: Calculate Derivatives

The first task is to compute the derivatives required for the Taylor polynomial approximation. Given the function \( f(x) = x^2 + e^{1+x} \):1. Calculate \( f'(x) = 2x + e^{1+x} \).2. Calculate \( f''(x) = 2 + e^{1+x} \).3. Calculate \( f'''(x) = e^{1+x} \).These derivatives will be used in later steps.

Step 2: Evaluate Derivatives at Base Point

Evaluate each of the derivatives at the base point \( c = -1 \):1. \( f(-1) = (-1)^2 + e^{1-1} = 1 + 1 = 2 \).2. \( f'(-1) = 2(-1) + e^{1-1} = -2 + 1 = -1 \).3. \( f''(-1) = 2 + e^{1-1} = 2 + 1 = 3 \).

Step 3: Construct Taylor Polynomial \( T_2(x) \)

Using the derivatives evaluated at \( c = -1 \), construct the second-order Taylor polynomial:\[ T_2(x) = f(-1) + f'(-1)(x + 1) + \frac{f''(-1)}{2!}(x + 1)^2 \]\[ = 2 + (-1)(x + 1) + \frac{3}{2}(x + 1)^2 \]This polynomial will be used to approximate \( f(x_0) \) where \( x_0 = -1.2 \).

Step 4: Calculate \( T_2(x_0) \)

Substitute \( x_0 = -1.2 \) into \( T_2(x) \):\[ T_2(-1.2) = 2 + (-1)(-1.2 + 1) + \frac{3}{2}(-1.2 + 1)^2 \]\[ = 2 + 0.2 + \frac{3}{2}(0.2)^2 \]\[ = 2 + 0.2 + \frac{3}{2}(0.04) \]\[ = 2.2 + 0.06 = 2.26 \].This evaluates the approximation of \( f(x_0) \).

Step 5: Find Maximum for Error Bound

Determine the maximum value of the third derivative over the interval \([-1.2, -1]\):Given \( f'''(x) = e^{1+x} \), evaluate at the endpoints:1. \( f'''(-1.2) = e^{-0.2} \approx 0.8187 \).2. \( f'''(-1) = e^{0} = 1 \).Max value \( M = 1 \) (since it's greater than \( 0.8187 \)).

Step 6: Calculate Error Bound

Use Taylor's remainder theorem (Inequality 8.8.7) for \( N+1 = 3 \):The bound for the error is given by:\[ \left| R_2(x_0) \right| = \frac{M |x_0 + 1|^3}{3!} \]\[ = \frac{1 \cdot (0.2)^3}{6} \]\[ = \frac{0.008}{6} \approx 0.00133 \].Thus, the error bound is approximately \( 0.00133 \).

Key Concepts

Understanding Derivatives

Derivatives play a crucial role in the construction of Taylor polynomials. In essence, a derivative represents how a function changes as its input changes. For a Taylor polynomial, we need to determine the rate of change of a function at a specific point, known as the base point.
In the given problem, we have the function \( f(x) = x^2 + e^{1+x} \). To find the Taylor polynomial, we must calculate the derivatives of this function. The first derivative, \( f'(x) \), is found by using basic differentiation rules: if \( f(x) = Ax^n + B e^x \), then \( f'(x) = Anx^{n-1} + Be^x \).
For our function, this yields \( f'(x) = 2x + e^{1+x} \). Continuing, the second derivative \( f''(x) = 2 + e^{1+x} \) and the third derivative \( f'''(x) = e^{1+x} \) are computed by differentiating again.
The derivatives provide the coefficients in the Taylor polynomial expansion, showing the rates of change at the base point. Evaluating these at \( c = -1 \), we get specific numbers that will be used to approximate the function's value at another point.

Calculating Error Bound

When approximating functions using Taylor polynomials, it's crucial to consider how accurate the approximation is. The concept of an "error bound" helps us gauge this. The error bound provides a limit to how off our approximation can be from the true function value. This is especially important when we're only using a few terms from an infinite series.
In the given exercise, after constructing the Taylor polynomial \( T_2(x) \) for \( f(x) \), we focus on finding an upper bound for \( |R_N(x_0)| = |f(x_0) - T_N(x_0)| \). The theorem states:

• Identify \( M = \max_{t \in J} |f^{(N+1)}(t)| \), where \( J \) is the interval from the base point to \( x_0 \).
• Evaluate the error bound as \( \frac{M |x_0 + 1|^3}{3!} \).

For the problem, we determined \( f'''(x) = e^{1+x} \), which needed evaluation at the endpoints \(-1.2\) and \(-1\), giving us a maximum \( M = 1 \). Calculating the expression results in an error bound of approximately \( 0.00133 \), signaling that our approximation is extremely close to the actual value.

Introduction to Taylor's Remainder Theorem

Taylor's remainder theorem provides the mathematical foundation to determine how precise a Taylor polynomial approximation is for a function. This theorem, sometimes called Lagrange's form of the remainder, deals with the portion of the function that is not captured by the polynomial.
Taylor's remainder theorem asserts that the error \( R_N(x) \) between the function \( f(x) \) and its Taylor polynomial \( T_N(x) \) is given by:
\[ R_N(x) = \frac{f^{(N+1)}(z)(x-c)^{N+1}}{(N+1)!} \text{ for some } z \text{ in the interval } (c, x_0). \]
This formula shows how the remainder depends on higher-order derivatives of the function. In practice, finding \( f^{(N+1)}(z) \) exactly can be complex, so we often use a maximum value over the interval \( J \), as demonstrated in the original exercise.
The simplification using bounds leads to a manageable calculation that predicts the error margin. Thanks to Taylor's remainder theorem, mathematicians and students can ascertain that their polynomial approximations are not only easier to compute but also reliable in terms of accuracy.