Problem 65 - Essential Calculus Early Transcendentals

Question

Describe how the graph of \(f\) varies as \(c\) varies. Graph several members of the family to illustrate the trends that you discover. In particular, you should investigate how maximum and minimum points and inflection points move when \(c\) changes. You should also identify any transitional values of \(c\) at which the basic shape of the curve changes. $$f(x)=c x+\sin x$$

Step-by-Step Solution

Answer

As \( c \) increases, slope changes and critical points shift on the sine curve. Transitional values are at \( |c| = 1 \), altering curve shape.

Step 1: Understand the Function

Consider the function given by \( f(x) = c x + \sin x \), where \( c \) is a parameter that affects the shape of the graph. This function is a combination of a linear function \( c x \) and a sinusoidal function \( \sin x \).

Step 2: Analyze the Linear Component

The linear component \( c x \) suggests that as \( c \) varies, the slope of the line changes: increasing \( c \) makes the slope steeper, while decreasing \( c \) makes it gentler. If \( c = 0 \), the function is purely sinusoidal.

Step 3: Graphical Representation for Different Values of c

Graph the function for several values of \( c \), such as \( c = -2, -1, 0, 1, 2 \). Observe that for negative \( c \), the graph slants downwards, and for positive \( c \), it slants upwards. At \( c = 0 \), the graph is purely a sine wave.

Step 4: Identify Critical Points

To find the critical points, differentiate \( f(x) \): \( f'(x) = c + \cos x \). Set the derivative to zero: \( c + \cos x = 0 \). Solving gives \( \cos x = -c \). Critical points occur where \( x \) satisfies this equation.

Step 5: Study the Inflection Points

Second derivative \( f''(x) = -\sin x \). Inflection points occur where \( f''(x) = 0 \). Hence, \( -\sin x = 0 \), implying \( x = n\pi \) (where \( n \) is an integer). The inflection points remain locations of the sine wave.

Step 6: Investigate Transitional Values of c

For \( |c|\leq 1 \), the term \( \cos x = -c \) is valid, meaning critical points can exist within [-1, 1] for \( \cos x \). As \( |c| \) increases beyond 1, no real solution exists for \( \cos x = -c \). Here, the shape changes as the graph becomes a monotonic increase or decrease, dependent on the sign of \( c \).

Step 7: Conclude Observations

Observe that increasing \( c \) shifts the minimum and maximum points left or right (essentially moving along the sine curve), while the general slope of the graph also becomes steeper in the positive (negative) direction for positive (negative) \( c \). Inflection points remain unchanged but move across the graph according to the slope.

Key Concepts

Critical Points

Critical points of a function are where the first derivative is zero or undefined. For our function, these points are found by setting the derivative to zero:

• Start by differentiating the function: \( f'(x) = c + \cos x \).
• Set this derivative equal to zero: \( c + \cos x = 0 \), which simplifies to \( \cos x = -c \).

These critical points indicate the x-values where the slope of the tangent to the graph is horizontal. This can signal either a local maximum, local minimum or a point of inflection under certain conditions. The parameter \( c \) significantly influences these points:

• As \( c \) changes, the critical points shift along the x-axis.
• For \( |c| \leq 1 \), real critical points exist because \(-c\) lies within the range of the cosine function, which is \([-1, 1]\).
• When \( |c| > 1 \), no real solutions exist for \( \cos x = -c \), and thus no critical points occur.

Understanding how these points move as \( c \) varies helps identify the behavior and trends in the graph, like peaks and valleys.

Inflection Points

Inflection points are where the function changes concavity. To find these for the function, observe where the second derivative equals zero:

• The second derivative is \( f''(x) = -\sin x \).
• Set \( f''(x) = 0 \), resulting in \(-\sin x = 0\).
• This implies \( x = n\pi \) for integers \( n \), where the sine function is zero.

These points mark where the curve goes from being concave upwards to downwards or vice versa. In this function, the inflection points coincide with the zeros of the sine component.

The role of \( c \) does not directly affect the positions of inflection points, as they remain stability marks of the sine wave, iterating with each cycle. However, the way they manifest on the graph becomes apparent as the overall slope changes, making them a clear map of how the function "waves."

Transitional Values

Transitional values of \( c \) are specific points where the fundamental shape of the graph changes. This happens when the behavior of critical points shifts or when they cease to exist:

• For \( |c| \leq 1 \), there are real critical points on the graph, indicating portions where the function has local maxima or minima.
• When \( |c| = 1 \), critical points just reach the threshold of existing, leading to special transitional behaviors.
• Once \( |c| > 1 \), the graph becomes monotonous, as the absence of real critical points means no peaks or valleys, changing the graph's shape significantly.

Understanding these transitional values is crucial. They signal a shift from oscillatory behavior to uniform direction, either increasingly positive or negative, ensuring the role of \( c \) is deeply grasped in shaping the graph.

Parametric Variations

Parametric variations involve how adjusting the parameter \( c \) alters the function's behavior. Particularly:

• \( c \) represents a linear component in the function, impacting the overall slope: positive \( c \) results in an upward-slanting slope, while negative \( c \) leads to a downward slant.
• At \( c = 0 \), the function reduces purely to \( \sin x \), a standard sine wave without any linear modification.
• Graphically, modifying \( c \) scales the inclination but does not affect periodic components. It instead shifts them, either amplifying or diminishing the wave's context.

Through varying \( c \), one can visualize how the sine component meshes with the linear nature, creating diverse graph shapes and patterns.