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) \(a=\) \(-0.1 .\) [Use separate coordinate systems 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

(b) The \(\sin x\) part produces oscillations. (c) \(a\) affects amplitude growth/decay. (d) For \(a = -0.1\), \(\lim_{x \to \infty} f(x) = 0\).

Step 1: Graph the Function

Use a graphing calculator to plot the graph of the function \(f(x) = e^{ax} \sin x\) for each value of \(a = -0.1, -0.01, 0, 0.01, 0.1\). Observe the impact on the graphs as \(a\) changes. Notice the oscillations in the graph result from the \(\sin x\) component, as it is a periodic function with oscillations between -1 and 1.

Step 2: Identify Oscillation Source

In the function \(f(x) = e^{ax} \sin x\), the oscillations in the graph are produced by the \(\sin x\) part. This is because \(\sin x\) is periodic and oscillates between -1 and 1, causing the overall function to fluctuate in value.

Step 3: Analyze the Effect of 'a'

The parameter \(a\) affects the amplitude of \(f(x) = e^{ax} \sin x\). If \(a > 0\), the function grows exponentially as \(x\) increases, amplifying the oscillations. If \(a < 0\), the function decays exponentially, causing oscillations to diminish as \(x\) increases. When \(a = 0\), \(f(x)\) becomes a constant amplitude sinusoidal wave.

Step 4: Graph Functions Together

Graph \(f(x) = e^{ax} \sin x\), \(g(x) = -e^{ax}\), and \(h(x) = e^{ax}\) on the same axes for \(a = 0.1\) and \(a = -0.1\). Observe how \(f(x)\) is bounded within \(-e^{ax}\) and \(e^{ax}\), due to the oscillatory behavior of \(\sin x\) between -1 and 1.

Step 5: Analyze Inequality Bound

The inequality \(-e^{ax} \leq f(x) \leq e^{ax}\) holds because \(-1 \leq \sin x \leq 1\). Thus, \(f(x)\) is bounded by \(-e^{ax}\) and \(e^{ax}\). For \(\lim_{x \to \infty} f(x)\) to exist, both \(-e^{ax}\) and \(e^{ax}\) must tend to zero, which occurs when \(a < 0\). The limit is then 0.

Key Concepts

Oscillation in Functions

Oscillation in functions occurs when a function periodically varies in value between two extremes, typically in a wave-like pattern. In the function \( f(x) = e^{ax} \sin x \), the oscillations are directly influenced by the \( \sin x \) part. The sine function is well known for its periodic oscillations, fluctuating between -1 and 1.

This means that regardless of the \( e^{ax} \) factor, \( f(x) \)’s core oscillatory behavior is governed by \( \sin x \):

• When \( \sin x \) reaches 1, \( f(x) \) hits a local positive peak.
• When \( \sin x \) is -1, \( f(x) \) hits a local negative trough.

In any graph of \( f(x) \), the peaks and valleys will align with the natural periodic oscillation of the sine function.
The exponential part \( e^{ax} \) modifies the amplitude of these oscillations, but not the period or frequency.

Exponential Growth and Decay

Exponential growth and decay are pivotal concepts describing how the amplitude of a function changes over time. In the context of \( f(x) = e^{ax} \sin x \), whether the function grows or decays depends on the sign of the value \( a \).

When \( a > 0 \), \( e^{ax} \) represents exponential growth. This means:

• The oscillations in the function intensify with increasing \( x \).
• The amplitude of the sine oscillations expands, potentially leading to larger positive and negative values.

Conversely, if \( a < 0 \), \( e^{ax} \) conveys exponential decay, and:

• The oscillations in the sine function diminish as \( x \) becomes larger.
• Ultimately, the influence of \( e^{ax} \) shrinks the amplitude, converging towards zero.

With \( a = 0 \), the function \( f(x) = \sin x \) becomes a pure oscillatory function with a constant amplitude.

Bounded Functions

A bounded function is one that stays within a fixed range for its entire domain. For \( f(x) = e^{ax} \sin x \), bounding typically results from interaction between \( e^{ax} \) and \( \sin x \).

The sine function \( \sin x \) inherently oscillates between -1 and 1, contributing a natural binding element. Thus, the inequalities \( -e^{ax} \leq f(x) \leq e^{ax} \) are formed due to:

• \( e^{ax} \sin x \) maxes out when \( \sin x = 1 \), resulting in \( e^{ax} \).
• Conversely, it minimizes when \( \sin x = -1 \), producing \( -e^{ax} \).

For the limit \( \lim_{x \to \infty} f(x) \) to exist, the bounding functions \( -e^{ax} \) and \( e^{ax} \) must both approach zero. This solely occurs if \( a < 0 \), indicating exponential decay,
which eventually nullifies the effects of oscillation, leaving a limit of zero.