Problem 10 - Calculus: An Integrated Approach to Functions and Their Rates of Change

Question

(a) Find the second degree Taylor polynomial generated by sec \(x\) at \(x=0\). (b) Graph \(P_{2}(x)\) and sec \(x\) on the same set of axes.

Step-by-Step Solution

Answer

The second degree Taylor polynomial generated by sec \(x\) at \(x=0\) is \(1 + 1/2 x^2\). To graph \(P_{2}(x)\) and sec \(x\), mark the point of x = 0 and draw the graphs of sec \(x\) and the polynomial.

Step 1: Calculate the derivatives

Firstly, calculate the first and second derivative of sec x. The first derivative of sec x is \(sec(x)tan(x)\) and the second derivative is \(sec(x)(tan^2(x) + sec^2(x))\)

Step 2: Evaluate at x = 0

Next, evaluate the function and its derivatives at x = 0. At x = 0, sec(0) = 1, \(sec(0)tan(0) = 0\) and \(sec(0)(tan^2(0) + sec^2(0)) = 1\)

Step 3: Construct the Taylor polynomial

The second degree Taylor polynomial is given by \(f(0) + f'(0)(x - 0) + f''(0)(x - 0)^2/2 = 1 + 0 + 1/2 x^2 = 1 + 1/2 x^2\)

Step 4: Graph the function and the polynomial

For part (b), the function sec x and the Taylor polynomial can be graphed on the same set of axes. Mark the point x = 0 on the x-axis, and draw the graphs of sec x and the polynomial.

Key Concepts

Secant Function

The secant function, denoted as \( \sec x \), is one of the six fundamental trigonometric functions. It can be defined as the reciprocal of the cosine function. Thus, \( \sec x = \frac{1}{\cos x} \). This function has certain properties and values that are important in calculus, especially when finding derivatives for use in Taylor series.The function \( \sec x \) helps in studying angles and their relationships in triangles, often involved when working with periodic phenomena. For those learning calculus, understanding how \( \sec x \) behaves is critical for both derivative calculations and subsequent function approximation through Taylor Series.Key points to know about the secant function:

• The function \( \sec x \) is undefined wherever \( \cos x = 0 \). This leads to vertical asymptotes at odd multiples of \( \frac{\pi}{2} \).
• Its range is \(( -\infty, -1 ] \cup [ 1, \infty)\).
• When graphing, \( \sec x \) has a repeating pattern every \( 2\pi \), which means it's periodic like other trigonometric functions.

Second Derivative

The second derivative of a function provides important information about the concavity or curvature of the graph of the function. In calculus, the second derivative is fundamentally a measure of how the rate of change (expressed by the first derivative) itself changes.When specifically dealing with \( \sec x \), we first derive the first derivative which is \( \sec(x)\tan(x) \). The second derivative involves finding the derivative of this expression:\[ \frac{d}{dx}(\sec(x)\tan(x)) = \sec(x)(\tan^2(x) + \sec^2(x)) \]This derivative gives us the second term to use in forming the Taylor Series. Evaluating this at a point allows us to incorporate it into the polynomial. For \( x = 0 \):

• The first derivative evaluated at \( x = 0 \) is 0, as \( \tan(0) = 0 \).
• The second derivative evaluated at \( x = 0 \) becomes 1, since \( \sec(0) = 1 \) and \( \tan^2(0) + \sec^2(0) = 1 \).

Graphing Functions

Graphing functions serves as a crucial step for visual understanding of mathematical concepts. When we graph the second degree Taylor polynomial of \( \sec x \) alongside the actual function, it helps to see how well the approximation fits around the point of expansion.Here's how one approaches this process:

• Begin by marking the essential point of approximation, which is \( x = 0 \) in this case.
• The second degree Taylor polynomial derived is \( P_2(x) = 1 + \frac{1}{2}x^2 \). This polynomial is graphed along with the original \( \sec x \) function.
• At \( x = 0 \), both \( \sec x \) and the polynomial will meet since the Taylor polynomial is designed to reflect the value of the original function and its derivatives at \( x = 0 \).
• Pay attention to how the polynomial deviates as you move away from the point of approximation, showing the intervals where the approximation remains accurate.

Graphing these functions together provides a visual tool for understanding how Taylor series work, acting as a bridge between numerical calculations and conceptual insights.