Problem 47 - Calculus Early Transcendentals: Pearson New International Edition

Question

The graph of \(y=f(x)\) depends on a parameter c. Using a CAS, investigate how the extremum and inflection points depend on the value of \(c .\) Identify the values of \(c\) at which the basic shape of the curve changes. $$ f(x)=c+\sin c x $$

Step-by-Step Solution

Answer

Extremum and inflection points become more frequent as \(c\) increases. The curve's basic shape changes with \(c\).

Step 1: Define the Function

Consider the function in question: \( f(x) = c + \sin(cx) \). Here, \( c \) is the parameter that affects both the amplitude and the period of the sine function.

Step 2: Determine the First Derivative

To find the extremum points, calculate the first derivative of \( f(x) \) with respect to \( x \). Use the derivative formula: \( f'(x) = c \cos(cx) \).

Step 3: Solve for Critical Points

Set the first derivative equal to zero to find critical points: \( c \cos(cx) = 0 \). Solving this equation gives the critical points at \( x = \frac{\pi}{2c}, \frac{3\pi}{2c}, \frac{5\pi}{2c}, \ldots \) where \( \cos(cx) = 0 \).

Step 4: Determine the Second Derivative

Compute the second derivative to examine inflection points: \( f''(x) = -c^2 \sin(cx) \).

Step 5: Solve for Inflection Points

Set the second derivative equal to zero to find inflection points: \( -c^2 \sin(cx) = 0 \). Solving results in inflection points at \( x = \frac{n\pi}{c} \) for integer \( n \), where \( \sin(cx) = 0 \).

Step 6: Investigate How \(c\) Affects Extrema and Inflections

The parameter \( c \) influences the spacing of the extrema and inflection points. Specifically, as \( c \) increases, these points occur more frequently within a fixed interval of \( x \). Conversely, decreasing \( c \) stretches them further apart.

Step 7: Identify Changes in Curve Shape

The basic shape of the curve changes when the number of extrema and inflection points in a given interval changes, which happens due to variations in \( c \). Key values to investigate are where \( c \) results in full coverage of periods and amplitudes in the interval being considered.

Key Concepts

Critical Points

Critical points occur where the derivative of a function is zero or undefined. For our function, \( f(x) = c + \sin(cx) \), we determined the first derivative as \( f'(x) = c \cos(cx) \). To find critical points, we set this derivative equal to zero: \( c \cos(cx) = 0 \). This immediately tells us that the function's slope is zero at these points.

The critical points are found at \( x = \frac{\pi}{2c}, \frac{3\pi}{2c}, \frac{5\pi}{2c}, \ldots \), where \( \cos(cx) = 0 \). Here are some key insights about these points:

• Critical points indicate locations where the function reaches either a local maximum or minimum.
• The spacing of critical points is inversely proportional to the parameter \( c \). This means as \( c \) increases, the critical points become closer together.
• These points give valuable information about the overall shape and turning points of the curve in any interval.

Understanding critical points helps in predicting the peaks and valleys of the wave-like graph of the sine function.

Inflection Points

Inflection points are places where the curvature of a graph changes from concave up to concave down or vice versa. To find these in the function \( f(x) = c + \sin(cx) \), we use the second derivative: \( f''(x) = -c^2 \sin(cx) \).

Setting this second derivative equal to zero will locate the inflection points: \( -c^2 \sin(cx) = 0 \). Solving this, we find inflection points at \( x = \frac{n\pi}{c} \) for any integer \( n \). Important aspects of inflection points include:

• Inflection points reveal changes in the curvature of the graph, showing where it switches from bending upward to downward.
• The parameter \( c \) dictates the frequency of these points within a given interval of \( x \). As \( c \) grows, inflection points appear more often.
• Locations of inflection points enrich our understanding of the function's waveform and help depict the rhythm of sine variations across different values of \( c \).

Inflection points can be crucial for understanding the overall "shape shift" in the graph's appearance.

Parameter c Effect

The parameter \( c \) in the function \( f(x) = c + \sin(cx) \) has a profound impact on the graph's appearance. It affects both the horizontal spacing of extrema and inflection points as well as the frequency of oscillations.

To comprehend the parameter's effect, let's examine some of its influences:

• As \( c \) increases, the function exhibits more frequent peaks, troughs, and curvature changes within any fixed interval. This means \( c \) plays a role in determining the \'density\' of the wave\'s oscillations.
• On the other hand, if \( c \) is decreased, the same features become more stretched out along the \( x \)-axis, creating broader waves.
• The changes in parameter \( c \) also impact the visual 'shape' and rhythm of the graph seriously, as higher \( c \) produces a dense and more intricate pattern, suitable for modeling quickly fluctuating phenomena.

The parameter \( c \) is crucial when examining dynamic systems or phenomena that rely on periodic functions, allowing us to adjust the wave properties effectively.