Problem 19 - Calculus

Question

Find the second-degree Taylor polynomial for \(f(x)=\) \(4 x^{2}-7 x+2\) about \(x=0 .\) What do you notice?

Step-by-Step Solution

Answer

The second-degree Taylor polynomial is the same as the original function: \(P_2(x) = f(x)\).

Step 1: Evaluate Function and Derivatives at Center

First, we need to evaluate the function and its derivatives at the center, which is given as \(x=0\). The value of the function at \(x=0\) is \(f(0) = 4(0)^2 - 7(0) + 2 = 2\). Next, we calculate the first and second derivatives at \(x=0\):1. \(f'(x) = 8x - 7\), so \(f'(0) = 8(0) - 7 = -7\).2. \(f''(x) = 8\), so \(f''(0) = 8\).

Step 2: Construct the Taylor Polynomial

Now, we use the function and its derivatives' values to construct the Taylor polynomial up to the second degree about \(x=0\). The formula for the Taylor polynomial is:\[ P_2(x) = f(0) + f'(0)x + \frac{f''(0)}{2}x^2 \]Substituting the values we computed earlier:\[ P_2(x) = 2 - 7x + \frac{8}{2}x^2 \]This simplifies to:\[ P_2(x) = 2 - 7x + 4x^2 \]

Step 3: Compare Polynomial and Original Function

Compare the resulting Taylor polynomial to the original function \( f(x) = 4x^2 - 7x + 2 \). Notice that \( P_2(x) \) exactly matches \( f(x) \). This indicates that for this particular quadratic function, the second degree Taylor polynomial is the function itself.

Key Concepts

Second Degree Polynomial

A second-degree polynomial is a mathematical expression of the form \( ax^2 + bx + c \), where \( a \), \( b \), and \( c \) are constants, and \( a eq 0 \). In other words, the highest exponent of the variable \( x \) is 2. This type of polynomial is significant because it represents the simplest form of a curve that has the potential to change direction (i.e., a parabola).

Key characteristics of a second-degree polynomial include:

• The graph of the polynomial is a parabola.
• The vertex form of a parabola is useful for understanding its shape and location.
• The polynomial can have zero, one, or two real roots, depending on the discriminant \( b^2 - 4ac \).

Understanding second-degree polynomials is essential in learning about more complex polynomial behavior and analyzing various mathematical phenomena.

Derivatives at a Point

Derivatives measure how a function changes as its input changes. When evaluating derivatives at a specific point, we are looking at the slope of the tangent line at that point. For a given function \( f(x) \), its derivative \( f'(x) \) tells us how \( f(x) \) changes per unit change in \( x \).

To find derivatives at a specific point:

• First derive the general expression of the derivative.
• Substitute the point's \( x \)-value into the derivative to find its slope at that point.

For example, in the provided solution, the first derivative of \( f(x) = 4x^2 - 7x + 2 \) is \( f'(x) = 8x - 7 \). Calculating \( f'(0) = -7 \) gives us the slope at \( x = 0 \). Similarly, the second derivative, \( f''(x) = 8 \), remains constant, indicating that the curvature of the parabola is concave upwards.

Understanding how derivatives behave at a point is crucial because:

• It helps in building Taylor polynomials, approximating functions near that point.
• It provides insight into the function's behavior, such as increasing or decreasing trends and curvature.

Quadratic Functions

Quadratic functions are a special type of polynomial function, specifically those of the form \( f(x) = ax^2 + bx + c \) where \( a eq 0 \). These functions produce the familiar "U" shaped graphs known as parabolas. Parabolas can open upwards or downwards depending on the sign of the leading coefficient \( a \).

Some important aspects to consider when working with quadratic functions are:

• The vertex, which is the maximum or minimum point of the parabola. It's a crucial feature for graph interpretation.
• The axis of symmetry, a vertical line passing through the vertex dividing the parabola into two mirror-image halves.
• The roots or zeros of the function, which are the \( x \)-values where \( f(x) = 0 \), important for understanding intersections with the x-axis.

When comparing a Taylor polynomial to the original quadratic function, as shown in the exercise, you may observe that sometimes the two are identical. This happens when the Taylor polynomial perfectly captures the behavior of quadratic functions, as they naturally match up to a second-degree approximation.

Quadratic functions serve as fundamental building blocks in algebra and calculus, providing insights into curve behavior and aiding in solving real-world problems.