Problem 21 - Calculus for Biology and Medicine

Question

(a) Use a graphing calculator to sketch the graph of $$f(x)=e^{a x} \sin x, \quad x \geq 0$$ for \(a=-0.1,-0.01,0,0.01\), and 0.1. (b) Which part of the function \(f(x)\) produces the oscillations that you see in the graphs sketched in (a)? (c) Describe in words the effect that the value of \(a\) has on the shape of the graph of \(f(x)\). (d) Graph \(f(x)=e^{a x} \sin x, g(x)=-e^{a x}\), and \(h(x)=e^{a x}\) together in one coordinate system for (i) \(a=0.1\) and (ii) \(a=-0.1\). [Make separate graphs for (i) and (ii).] Explain what you see in each case. Show that $$-e^{a x} \leq e^{a x} \sin x \leq e^{a x}$$ Use this pair of inequalities to determine the values of \(a\) for which \(\lim _{x \rightarrow \infty} f(x)\) exists, and find the limiting value.

Step-by-Step Solution

Answer

The oscillations in the graph are due to \( \sin x \). The value of \( a \) affects the amplitude, with \( a > 0 \) increasing and \( a < 0 \) decreasing it over time. For \( a < 0 \), \( \lim_{x \to \infty} f(x) = 0 \).

Step 1: Sketch the Graphs

Use a graphing calculator or software to graph the function \( f(x) = e^{a x} \sin x \) for each given value of \( a = -0.1, -0.01, 0, 0.01, 0.1 \). You should observe that as \( x \) increases, the graph with positive \( a \) values will show increasing oscillations, while negative \( a \) values will show decreasing oscillations.

Step 2: Identify the Source of Oscillations

The oscillations seen in the graph are produced by the \( \sin x \) term, since the sine function is inherently oscillatory in nature (periodic and waves-like characteristic). The exponential function \( e^{a x} \) modifies the amplitude of these oscillations but does not produce them.

Step 3: Effect of Parameter 'a' on Graph Shape

For positive \( a \), the amplitude of the oscillations increases exponentially as \( x \) increases, causing the graph's envelopes to widen outward. For negative \( a \), the amplitude decreases exponentially, causing the oscillations to diminish as \( x \) increases, which results in a narrowing inward. For \( a = 0 \), the amplitude remains constant, as the envelope of the sine function is not affected by any exponential decay or growth.

Step 4: Graph Functions Together

Graph \( f(x) = e^{a x} \sin x \), \( g(x) = - e^{a x} \), and \( h(x) = e^{a x} \) on the same coordinate system for \( a = 0.1 \) in one set and \( a = -0.1 \) in another. For \( a = 0.1 \), observe that \( f(x) \) oscillates between the \( h(x) \) and \( g(x) \) curves, widening as \( x \) increases. For \( a = -0.1 \), \( f(x) \) is squeezed towards the x-axis between these curves as \( x \) increases.

Step 5: Verify and Use Inequalities

The inequalities are given by \( -e^{a x} \leq e^{a x} \sin x \leq e^{a x} \). To use them, consider the behavior of \( e^{a x} \). If \( a < 0 \), \( e^{a x} \to 0 \) as \( x \to \infty \), so \( f(x) \) is bounded and \( \lim_{x \to \infty} f(x) = 0 \). If \( a \geq 0 \), \( e^{a x} \to \infty \), making \( \lim_{x \to \infty} f(x) \) does not exist, as \( f(x) \)'s amplitude grows unbounded.

Key Concepts

Exponential functions

Exponential functions are prevalent in various mathematical areas, particularly in calculus. The function form is \(e^{a x}\), where \(e\) is a mathematical constant approximately equal to 2.71828. The key feature of these functions is the rate of growth or decay. This rate is determined by the exponent's coefficient, \(a\):

• If \(a > 0\), the function describes exponential growth as \(x\) increases. The larger the value of \(a\), the faster the growth.
• If \(a < 0\), it represents exponential decay, meaning the function decreases as \(x\) increases.
• If \(a = 0\), the expression simplifies to a constant equal to 1.

In the exercise, \(e^{a x}\) interacts with the trigonometric component \(\sin x\). The combination affects the overall function's amplitude as it modulates the sine wave, either amplifying in positive cases or damping in negative cases.

Trigonometric functions

Trigonometric functions like \(\sin x\) are periodic, meaning they repeat their values in regular intervals or cycles. This periodic behavior stems from their geometric origins in the unit circle. The sine function specifically has a host of properties essential for its role in combined functions:

• It is oscillatory, with values ranging from -1 to 1, producing wave-like graphs.
• The period of \(\sin x\) is \(2\pi\), indicating how often the wave pattern repeats.
• The combination with exponential functions results in changes to the amplitude but not the frequency of oscillations.

In the exercise, the sine function is responsible for the oscillations observed on the graph. The multiplication by \(e^{a x}\) alters the amplitude but does not affect these intrinsic wave properties.

Graphing calculator

Using a graphing calculator is a powerful way to visually understand complex functions like \(f(x) = e^{ax} \sin x\). Here are some tips on using these tools effectively:

• Input the function accurately, paying attention to operator placement and parentheses to ensure correct calculation.
• Adjust viewing windows to capture the essential parts of the graph, especially when dealing with exponential growth or decay that can quickly exceed common view limits.
• Look for features such as oscillations, intercepts, and bounding envelopes, which can reveal insights into function behavior.
• Observe the changes in the graph as you vary parameters like \(a\) to study their effects.

In our context, comparing graphs for different \(a\) values shows how the exponential factor affects oscillation shapes, either increasing them for positive \(a\) or damping them for negative \(a\).

Limits

Calculus often uses limits to understand behavior as functions approach specific points or infinity. This is particularly important when analyzing long-term behavior in functions:

• For the function \(f(x) = e^{ax} \sin x\), the limit as \(x\) approaches infinity depends heavily on the value of \(a\).
• If \(a < 0\), the function’s exponential decay drives it towards zero, so \(\lim_{x \to \infty} f(x) = 0\).
• If \(a \geq 0\), \(e^{ax}\) causes the amplitude to grow unbounded, so the limit does not exist.

Using these concepts helps establish the boundaries for the function \(f(x)\) as well as any asymptotic behavior, providing valuable insights into its overall behavior in the real number space.