Problem 6 - Calculus: An Integrated Approach to Functions and Their Rates of Change

Question

Do the following. (a) Compute the fourth degree Taylor polynomial for \(f(x)\) at \(x=0 .\) (b) On the same set of axes, graph \(f(x), P_{1}(x), P_{2}(x), P_{3}(x)\), and \(P_{4}(x)\). (c) Use \(P_{1}(x), P_{2}(x), P_{3}(x)\), and \(P_{4}(x)\) to approximate \(f(0.1)\) and \(f(0.3) .\) Compare these approximations to those given by a calculator. $$ f(x)=2 x^{4}-3 x^{2}+x-1 $$

Step-by-Step Solution

Answer

The fourth degree Taylor polynomial for the function \(f(x)=2x^4-3x^2+x-1\) at x=0 is \(P_4(x) = -1 + x + \frac {2x^4}{3}\). The polynomial approximation values for \(f(0.1)\) and \(f(0.3)\) are -0.89997 and -0.602 respectively, which are very close to the original function values at these points (-0.8999 and -0.598).

Step 1: Computation of the Taylor Polynomial

Start by finding the derivatives of \(f(x)\) till the fourth degree. This is done as follow:\[f(x) = 2x^4 - 3x^2 + x - 1\]\[f'(x) = 8x^3 - 6x + 1\]\[f''(x) = 24x^2 - 6\]\[f'''(x) = 48x\]\[f''''(x) = 48\]The nth degree Taylor polynomial \(P_4(x)\) of \(f(x)\) around 0 is given by:\[P_4(x) = f(0) + f'(0)x + \frac {f''(0)x^2}{2!} + \frac {f'''(0)x^3}{3!} + \frac {f''''(0)x^4}{4!}\]Substituting the respective values, we get\[P_4(x) = -1 + x + 0x^2 - \frac {0x^3}{3!} + \frac {48x^4}{4!} = -1 + x + \frac {2x^4}{3}\]

Step 2: Graphing the function and its Polynomial approximations

With the original function and its Taylor Polynomial approximations at hand, graph these on the same axis. This will result in a clear depiction of how each polynomial approximation estimates the original function. It cannot be practically shown in textual format in this step. You should use scientific graphing calculators or computer software to get the graphical representation.

Step 3: Using the Polynomials to approximate function values

Now use the Taylor Polynomial approximations to approximate the value of function at \(x=0.1\) and \(x=0.3\). Calculate this for \(P_1(x)\), \(P_2(x)\), \(P_3(x)\) and \(P_4(x)\).\[P_1(0.1) = -1 + (0.1) = -0.9\]\[P_2(0.1) = P_1(0.1) = -0.9\]\[P_3(0.1) = P_2(0.1) = -0.9\]\[P_4(0.1) = -1 + (0.1) + \frac {2*0.1^4}{3} = -0.89997\]Similarly, \[P_1(0.3) = -1 + (0.3) = -0.7\]\[P_2(0.3) = P_1(0.3) = -0.7\]\[P_3(0.3) = P_2(0.3) = -0.7\]\[P_4(0.3) = -1 + (0.3) + \frac {2*0.3^4}{3} = -0.602\]Using a calculator, \[f(0.1) = 2*0.1^4 - 3*0.1^2 + 0.1 - 1 = -0.8999\]\[f(0.3) = 2*0.3^4 - 3*0.3^2 + 0.3 - 1 = -0.598\]

Step 4: Comparison of the Approximation Results

Observe and compare these approximation results with those calculated directly from the calculator. The results show that \(P_4(x)\) approximates the function quite well. The higher the degree of the polynomial, the more accurately it approximates the function.

Key Concepts

Function Approximation

Function approximation is a powerful mathematical concept used to estimate complex functions with simpler ones. In this context, the Taylor polynomial is used to approximate a given function by polynomials of increasing degrees.

These polynomials enable us to predict the function's behavior around a specific point. For our example, the Taylor polynomial is an approximation of the function\( f(x) = 2x^4 - 3x^2 + x - 1 \)\.

• The approximation improves as we consider polynomials of higher degrees (i.e., more terms).
• For instance, \( P_1(x) \), a first-degree polynomial, offers a basic approximation, while \( P_4(x) \) is much closer to the true function.

The fundamental idea is that by using these approximations, we can simplify calculations and gain insights into the function's behavior, especially around the point of approximation.

Derivative

Derivatives play a crucial role in constructing Taylor polynomials. They measure how a function changes as its input changes. In our example, we calculate derivatives up to the fourth degree.

Each derivative provides vital information:

• \( f'(x) \) represents the slope of the tangent at any given point of the original function \( f(x) \).
• \( f''(x) \) involves concavity, showing how the slope changes.
• Higher-order derivatives, like \( f'''(x) \) and \( f''''(x) \), provide even more detail about the function's curvature and minute changes.

By substituting these derivatives into the polynomial formula, we obtain a Taylor polynomial that closely approximates the original function near the expansion point.

Polynomial Graphing

Graphing the function alongside its polynomial approximations is vital for visually understanding their behavior.

This step involves plotting both the original function \( f(x) \) and its Taylor polynomials such as \( P_1(x), P_2(x), \) etc., on the same set of axes. Here's why it's useful:

• Visual comparison helps us see how closely the polynomials follow the original curve.
• As we increase the degree of the polynomial from \(P_1(x)\) to \(P_4(x)\), the approximations become tighter around the expansion point \(x=0\).
• Graphing software or graphing calculators allow for a clear depiction, illustrating how the fourth-degree polynomial practically hugs the graph of \( f(x) \) while simpler polynomials show deviations.

This graphical method provides a highly intuitive grasp of how polynomial approximations work and their effectiveness.

Taylor Series Expansion

The Taylor series expansion is at the heart of creating Taylor polynomials.

This method expands a function into an infinite sum of terms calculated from the function's derivatives at a specific point. In our exercise, we focus on a degree-four approximation:

• Expand the function \( f(x) \) into its Taylor polynomial at \(x = 0\).
• Each term in this expansion uses derivatives and factorials to weigh the influence of each potential component, such as \( f'(0)x \) and \( \frac{f''''(0)x^4}{4!} \).
• This method shows how both low and high-order derivatives contribute to a more precise approximation as they account for more subtleties of the original function's behavior.

Therefore, the Taylor series expansion not only facilitates efficient approximation of functions but also enhances our understanding of the function's intrinsic behaviors around the point of expansion.