Question

Conjecture Consider the function \(f(x)=\cos x\) and its Maclaurin polynomials \(P_{2}, P_{4},\) and \(P_{6}\) (see Example 5 ). (a) Use a graphing utility to graph \(f\) and the indicated polynomial approximations. (b) Evaluate and compare the values of \(f^{(n)}(0)\) and \(P_{n}^{(n)}(0)\) for \(n=2,4,\) and 6 (c) Use the results in part (b) to make a conjecture about \(f^{(n)}(0)\) and \(P_{n}^{(n)}(0)\)

Short answer

The conjecture might be that \(f^{(n)}(0)\) and \(P_{n}^{(n)}(0)\) are equal for \(n = 2, 4, 6\), hinting that the Maclaurin polynomial of order \(n\) for the cosine function has the same nth derivative at 0 as the original function.

Step-by-step solution

Graphing the function and approximations

Use a graphing tool to plot the function \(f(x) = \cos x\) and its Maclaurin polynomials \(P_{2}, P_{4},\) and \(P_{6}\). Notice how the approximations become more accurate as \(n\) increases.

Evaluating the derivatives

To work out \(f^{(n)}(0)\) and \(P_{n}^{(n)}(0)\), you'll need to differentiate the cosine function and its polynomials two, four, and six times, respectively. Evaluate each resulting derivative at \(x = 0\). Be mindful that the derivative of cosine is \(-\sin x\), and subsequently alternates in sign with each repeated differentiation.

Making a conjecture

Look at the values obtained in step 2. Do you notice any patterns? Perhaps there's a relationship between \(n\) and the value of \(f^{(n)}(0)\) or \(P_{n}^{(n)}(0)\). Whatever your observation is, formulate it into a conjecture. Common conjectures in exercises like this often revolve around the symmetry present in the cosine function and its derivatives, as well as patterns that emerge from the Maclaurin series expansion.

Key concepts

Cosine Function

The cosine function, represented by \( \cos x \), is one of the fundamental components of trigonometry. It is an even periodic function that varies between -1 and 1, and it tells us the x-coordinate of a point on the unit circle at a given angle. When graphing \( \cos x \) versus \( x \) on a Cartesian plane, we get a smooth, wavelike curve that is known as the cosine wave.

The behavior and properties of the cosine function are essential to understand when dealing with trigonometric applications, such as in physics for waveform analysis, in engineering for signal processing, and in pure mathematics when working with series expansions like Maclaurin and Taylor series.

The Maclaurin series for \( \cos x \) is an infinite sum that approximates the cosine function using polynomials. The more terms we include from the Maclaurin series, the closer the polynomial will be to the actual cosine curve within a certain interval around the point \( x=0 \). Students often work with finite Maclaurin polynomials, such as \( P_{2}, P_{4}, \) and \( P_{6} \), to grasp the behavior of \( \cos x \) near the origin.

Graphing Polynomials

When we talk about graphing polynomials, we're looking at translating algebraic expressions into visual curves on a graph. Polynomial functions of degree \( n \) will have at most \( n \) turning points, and their end behaviors will depend on the leading term's coefficient and exponent.

For the case of Maclaurin polynomials for the cosine function, these are even-degree polynomials that offer a snapshot approximation of \( \cos x \) near the point \( x = 0 \). By graphing \( P_{2}, P_{4}, \) and \( P_{6} \) alongside \( \cos x \), students can visually assess the accuracy of the approximation.

Seeing the Pattern

As more terms from the Maclaurin series are included, we observe that the polynomial curve becomes increasingly similar to the \cos x curve over a broader range. In mathematical terms, these polynomials 'converge' to the cosine function. This is a powerful illustration of how a complex function like \( \cos x \) can be represented by simpler polynomial expressions in certain regions, making it a cornerstone of understanding in calculus and numerical analysis.

Derivatives of Cosine

The concept of a derivative is at the heart of differential calculus and concerns the rate at which a function's value changes. For the cosine function, the derivative is particularly interesting because it demonstrates periodic behavior with alternating signs.

The first derivative of \( \cos x \) is \( -\sin x \), which reflects a 90-degree phase shift and a sign change from the original function. If we continue to differentiate, we will see a pattern: after each differentiation, the output function shifts and changes sign, following a cycle: \( -\sin x \), \( -\cos x \), \( \sin x \) and back to \( \cos x \) after the fourth derivative.

Maclaurin Polynomials and Derivatives

Each \( P_{n} \) for \( n=0,2,4,\ldots \) in the Maclaurin polynomials of the cosine function will exhibit the alternating pattern of derivatives with each corresponding degree. When \( n \) is odd, the derived function disappears at \( x = 0 \) because all sine terms vanish, leaving us with zero. For even \( n \) values, the derivatives at \( x = 0 \) will alternate between 1 and -1, aligning with the cosine function's own derivatives at that point. By examining these patterns, students develop a deeper understanding of the symbiotic relationship between a function and its derivatives and polynomials.