Problem 8

Question

The zeros of $f$ are the zeros of $\sin z,$ that is, $n \pi, n=0,\pm 1,\pm 2, \ldots .$ From $f^{\prime}(z)=2 \sin z \cos z$ we see $f^{\prime}(n \pi)=0$ From $f^{\prime \prime}(z)=2\left(-\sin ^{2} z+\cos ^{2} z\right)$ we see $f^{\prime \prime}(n \pi) \neq 0 .$ This indicates that each zero is of order two.

Step-by-Step Solution

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Answer

Each zero at $ n\pi $ is a double zero of the function.

1Step 1: Identify the zeros of the function

The zeros of the function are the points where the function equals zero. The given function has zeros at all integer multiples of $ \pi $, i.e., $ n\pi $, where $ n $ is an integer ($ n = 0, \pm 1, \pm 2, \ldots $). These are the points where $ \sin z = 0 $.

2Step 2: First derivative analysis

The first derivative of the function, given as $ f'(z) = 2\sin z \cos z $, can be rewritten using the double angle formula: $ f'(z) = \sin(2z) $. At the zeros $ n\pi $, since $ \sin(n\pi) = 0 $, it results in $ f'(n\pi) = 0 $. This indicates stationary points at these zeros.

3Step 3: Second derivative analysis

The second derivative is given by $ f''(z) = 2(-\sin^2 z + \cos^2 z) $, which simplifies to $ 2 \cos(2z) $. At the points $ n\pi $, $ \cos(2n\pi) = 1 $ because $ 2n\pi $ is a multiple of $ 2\pi $. Therefore, $ f''(n\pi) = 2 \cos(2n\pi) = 2 eq 0 $. This non-zero value demonstrates that the zeros of $ f $ are not just simple zeros because the second derivative does not vanish at these points.

4Step 4: Conclusion on the order of the zeros

Since $ f'(n\pi) = 0 $ and $ f''(n\pi) eq 0 $, the zeros at $ n\pi $ are of order two. This means that each zero is a double zero, affirming that the function returns to the x-axis at these points with a multiplicity of two.

Key Concepts

Understanding Sinusoidal FunctionsExploring Derivative AnalysisUnderstanding the Order of Zeros

Understanding Sinusoidal Functions

Sinusoidal functions are a type of periodic function that describe oscillating or wave-like behaviors. The most common examples are the sine and cosine functions. In our exercise, the function has zeros related to the sine function, specifically where the sine equals zero. These zeros occur at integer multiples of $ \pi $.

Definition: A sinusoidal function like $ \sin(z) $ takes values along a continuous cyclical path. Its graph oscillates between -1 and 1.
Zeros: When $ \sin(z) = 0 $, the result is $ n\pi $, where $ n $ is any integer. These are key points where the wave "touches" the x-axis.
Amplitude and Period: The amplitude is the peak value of the wave, while the period is the distance between repeating waves, typically $ 2\pi $ for sine.

Sinusoidal functions model many real-world phenomena, from the motion of pendulums to sound waves.

Exploring Derivative Analysis

Analyzing derivatives helps us understand the behavior of functions, especially around their zeros and critical points. The first derivative $ f'(z) $ indicates where slopes of the function are zero, revealing stationary points. These occur where the sine function is zero, such as at integer multiples of $ \pi $. Thus, the derivative $ f'(n\pi) = 0 $.

First Derivative: The expression $ f'(z) = 2 \sin(z) \cos(z) $ simplifies using the double angle formula to $ \sin(2z) $. At zeros $ n\pi $, possibilities of local minima or maxima arise because the derivative equals zero.
Second Derivative: The second derivative $ f''(z) = 2 \cos(2z) $ provides insights about the concavity of the function. For zeros at $ n\pi $, we find $ f''(n\pi) eq 0 $, confirming the presence of a curve not simply touching the x-axis, but with some force behind the cross.

Derivatives provide much-needed insights into the nature and shape of function graphs.

Understanding the Order of Zeros

In mathematical analysis, the order of zeros explains how a function behaves around its roots. When we look at our function, the zeros occur at $ n\pi $, but it's not just about crossing the axis—it's how they do it. The first derivative $ f'(n\pi) = 0 $ reveals the zeros' locations, but the second derivative $ f''(n\pi) eq 0 $ indicates deeper characteristics.

Zero Order: A simple zero would reflect the function merely touching the axis at a single point.
Higher Order: Our given function has zeros of order two, signifying that the function returns to the axis at these roots with a hug-like or flat portion, rather than sharply.
Visual Insight: If you imagine the graph, a double zero might resemble a small "hill" or "valley" touching the x-axis momentarily, before continuing onwards.

Understanding zero orders is crucial for mastering the differentiability and continuity concepts in calculus.

Problem 8

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