Problem 5 - Calculus: An Integrated Approach to Functions and Their Rates of Change

Question

Consider the function \(f(x)=e^{-0.3 x} \sin x\) (a) For what values of \(x\) does \(f(x)\) have its local maxima and local minima? (b) Is \(f(x)\) a periodic function? (c) Sketch the graph of \(f(x)=e^{-0.3 x} \sin x\). (d) What is the maximum value of for \(e^{-0.3 x} \sin x\) for \(x \geq 0 ?\) At what \(x\) -value is this maximum attained? Your answers must be exact, not numerical approximations from a calculator. Give justification that this value is indeed the maximum.

Step-by-Step Solution

Answer

The function \(f(x)\) has local maxima and minima as determined by solving \(f'(x) = 0\) and applying the second derivative test. The locations of these will be determined by the equation \( \tan x = \frac{10}{3}\). The function is not periodic. The graph of the function shows a decaying sine wave with extrema at the locations determined in step 1. The maximum value of \(f(x)\) for \(x \geq 0\) will be found by evaluating the function at the critical points and endpoint \(x=0\), and comparing these values.

Step 1 Title: Local Maxima and Minima

For part (a), it is necessary to find the first derivative \(f'(x)\) which can be achieved using the product rule:\[f'(x)=e^{-0.3x}(-0.3\sin x) + e^{-0.3x}\cos x\]and then set \( f'(x) = 0 \) to find critical points. Calculate the value of \(x\) for which \(e^{-0.3x}\cos x = 0.3e^{-0.3x}\sin x\), and solve the resulting equation, \( \tan x = \frac{1}{0.3} = \frac{10}{3} \). Find all solutions in the domain of the function using the property of the tangent function, and then evaluate \(f''(x)\) to apply the second derivative test to determine whether these points are local maxima or minima.

Step 2 Title: Periodicity

For part (b), determine whether \(f(x)\) is periodic. The sine function \(\sin x\) is periodic with period \(2\pi\), and \(e^{-0.3x}\) is not periodic. Therefore, the overall function \(f(x)\) is not periodic as it does not repeat its values at regular intervals.

Step 3 Title: Graphical Representation

For part (c), sketch the function by noting its general shape due to sine, its amplitude governed by \(e^{-0.3x}\), and the locations of any critical points (from step 1) as well as other relevant points. The amplitude declines slowly (due to the exponential factor), and the wave shape comes from the sine function. Pay special attention to how the function shape changes near the extrema points.

Step 4 Title: Finding the Maximum Value

For part (d), the maximum value for \(f(x)\) for \(x\geq 0\) can be found by checking the function values at all critical points and endpoints (in this case, \(x=0\)). The maximum occurs at the highest of these values. Justify this by showing that other points are either less (for other critical points) or cannot exist (for values beyond the domain).

Key Concepts

Local Maxima and Minima

The concept of local maxima and minima refers to the highest or lowest points in a small region of a graph. To find these in the function \(f(x) = e^{-0.3 x} \sin x\), we need to look at **critical points**, which occur where the first derivative of the function \(f'(x)\) is zero. Using the product rule, the derivative is

• \(f'(x) = e^{-0.3x}(-0.3\sin x) + e^{-0.3x}\cos x\).

Setting \(f'(x) = 0\), we find the critical points by solving the equation \(\tan x = \frac{1}{0.3} = \frac{10}{3}\).
The second derivative test helps to classify these critical points. If \(f''(x) > 0\) at the critical point, the function has a local minimum; if \(f''(x) < 0\), it has a local maximum. These points provide key insights into the behavior of the function around certain \(x\)-values.

Periodicity

Periodicity in functions refers to the repeating pattern at regular intervals. For \(f(x) = e^{-0.3x} \sin x\), we need to consider the component functions:

• \(\sin x\) is periodic with a period of \(2\pi\), meaning it repeats every \(2\pi\) units.
• The exponential component \(e^{-0.3x}\) is not periodic; it continuously decreases as \(x\) increases.

Since \(f(x)\) combines both periodic and non-periodic functions, it cannot be periodic overall. A periodic function requires every element to repeat precisely at fixed intervals, which is not the case here due to the exponential decay.

Graphical Representation

The graphical representation of the function \(f(x) = e^{-0.3 x} \sin x\) offers a visual understanding of its behavior. Initially, we observe that the function has a wave-like appearance due to the sine component. However, as \(x\) becomes larger, the amplitude of the wave diminishes because of the \(e^{-0.3x}\) factor.

• Critical points, found from the derivative, represent the "peaks" and "valleys" of the sine wave.
• The function's amplitude decreases over time, resulting in shorter "waves."

This decrease in amplitude makes it look like the waves are "flattening" as \(x\) grows, a distinct trait impacted by the exponential term which acts to envelope the periodic function.

Exponential Function

Exponential functions are of the form \(e^{cx}\) and are known for their rapid increase or decrease. In \(f(x) = e^{-0.3x} \sin x\), the exponential part \(e^{-0.3x}\) influences the function's overall behavior significantly:

• The multiplier \(e^{-0.3x}\) causes the graph's wave peaks (and troughs) to decrease over time.
• This results in each subsequent crest of the sine wave being lower than the previous, showcasing exponential decay.

Thus, while the sine component creates a regular up-and-down motion, the exponential part dampens these oscillations progressively as \(x\) increases.