Problem 47 - Calculus Early Transcendentals: Pearson New International Edition

Question

In Problems 43-47, the graph of \(y=f(x)\) depends on a parameter \(c\). Using a \(C A S\), 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

The shape of the graph and positions of extrema and inflection depend on parameter \( c \), changing spacing as \( c \) varies.

Step 1: Understand the Function

The function given is \( y = f(x) = c + \sin(cx) \). Here, \( c \) is a parameter that affects the shape of the graph. Our goal is to analyze how extremum (maxima and minima) and inflection points change with respect to \( c \).

Step 2: Find Critical Points

To find the critical points, we need the first derivative of the function, \( f'(x) = c \cos(cx) \). Set \( f'(x) = 0 \) to find critical points: \( c \cos(cx) = 0 \).The cosine function is zero at \( x = \frac{(2n+1)\pi}{2c} \), where \( n \) is an integer.

Step 3: Determine the Extremum Points

To verify whether these critical points are maxima or minima, check the second derivative, \( f''(x) = -c^2 \sin(cx) \). Depending on the sign of \( f''(x) \), determine if the critical points are maxima or minima.

Step 4: Find Inflection Points

Inflection points occur when the second derivative changes sign. Set \( f''(x) = 0 \) which gives \( -c^2 \sin(cx) = 0 \).The sine function is zero at \( x = \frac{n\pi}{c} \), where \( n \) is an integer.

Step 5: Analyze the Effect of Parameter c

As \( c \) changes, the spacing and number of extrema and inflection points change. Specifically, as \( c \to 0 \), the effects of these points spread apart, and as \( c \to \infty \), the features become more densely packed.

Step 6: Identify Basic Shape Changes

Identify values of \( c \) where the qualitative shape of the graph changes, such as when it transitions from a steep waveform to more gentle oscillations. This can happen at specific sign changes or where the magnitude of \( c \) alters the frequency of occurrences of these points.

Key Concepts

Critical Points

In calculus, identifying critical points is crucial for understanding the behavior of functions. These points occur where the derivative of a function is zero or undefined, indicating potential maxima, minima, or points of inflection.
For the function given in the exercise, \( y = f(x) = c + \sin(cx) \), the first derivative \( f'(x) = c \cos(cx) \) serves as our tool for finding these critical points.

• Set \( f'(x) = 0 \) to uncover where these points are: \( c \cos(cx) = 0 \).
• The angle for which \( \cos \) is zero is expressed as \( x = \frac{(2n+1)\pi}{2c} \), with \( n \) representing any integer. These are the critical points.

This step reveals potential points of change in direction on the curve, essential for analyzing extremum positions.

Extremum

Extremum points, encompassing both maxima and minima, are where a function reaches its highest or lowest values within a given interval. After determining the critical points using the first derivative, the second derivative helps assess their nature.
For the function \( y = c + \sin(cx) \), the second derivative is \( f''(x) = -c^2 \sin(cx) \). You can deduce the nature of extremum by examining this derivative at the critical points.

• If \( f''(x) < 0 \), the critical point is a local maximum, indicating the function is curving downwards.
• If \( f''(x) > 0 \), the critical point is a local minimum, with the curve turning upwards.

By providing this insight, extremum points illustrate where the function peaks or bottoms out, crucial for understanding the curve's overall profile.

Inflection Points

Inflection points are locations on the graph where the curvature changes sign, shifting from concave up to concave down, or vice versa. At these points, the second derivative is zero.
For our function, \( f''(x) = -c^2 \sin(cx) \) is the second derivative. To find inflection points, we need to set the equation to zero: \( -c^2 \sin(cx) = 0 \).

• The sine function is zero at angles \( x = \frac{n\pi}{c} \), indicating potential points of inflection.

Identifying these points helps track where the function's curvature transitions, allowing for a deeper understanding of the graph's dynamic nature.

Parameter Analysis

Parameter analysis examines how varying parameters influence a function's behavior. For the function \( y = c + \sin(cx) \), the parameter \( c \) significantly impacts the function's critical points, extremum, and inflection points.

• As \( c \to 0 \), the critical and inflection points appear further apart, spreading the graph's features along the axis.
• Conversely, as \( c \to \infty \), these points become densely packed, making the graph oscillate rapidly.

Identifying specific values of \( c \) where the graph's shape changes drastically is vital. Such analysis provides a comprehensive view of how the frequency and disposition of key points shift, which affects the curve's characteristics and visual representation.