Problem 24 - Calculus: Early Transcendentals

Question

Sketch the graph of a function \( g \) for which \( g(0) = g(2) = g(4) = 0 \), \( g'(1) = g'(3) = 0 \), \( g'(0) = g'(4) = 1 \), \( g'(2) = -1 \), \( \displaystyle \lim_{x \to \infty} g(x) = \infty \), and \( \displaystyle \lim_{x \to -\infty} g(x) = -\infty \).

Step-by-Step Solution

Answer

The graph has roots at 0, 2, and 4, critical points at 1 and 3, and rises towards infinity as \( x \to \infty \).

Step 1: Identify Critical Points

The critical points occur where the derivative is equal to zero. Here, these points are at \( x = 1 \) and \( x = 3 \). At these points, the function may have local maxima, minima, or inflection points.

Step 2: Determine Behavior at Critical Points

Given \( g'(1) = 0 \) and \( g'(3) = 0 \), check the derivative on either side of these points to determine the nature of these points. For simplification, assume behavior similar to standard test functions at these points (e.g., using the second derivative test, if possible, or analyzing sign changes).

Step 3: Analyze Given Values and Behavior

The function \( g(x) \) is zero at \( g(0) = 0 \), \( g(2) = 0 \), and \( g(4) = 0 \), suggesting roots or intercepts at these points. Between these, the sign of the derivative will indicate whether the graph increases or decreases.

Step 4: Examine End Behavior

\( \lim_{x \to \infty} g(x) = \infty \) and \( \lim_{x \to -\infty} g(x) = -\infty \) suggest a positive, unbounded slope as \( x \to \infty \) and a negative, unbounded slope as \( x \to -\infty \). This will guide the graph's general direction.

Step 5: Construct Skeleton of Graph

Start by plotting the zeros at \( x = 0, 2, 4 \), then sketch the curve passing smoothly through these points while adhering to the derivative information: increasing at \( x = 0 \), level at \( x = 1 \), decreasing at \( x = 2 \), level at \( x = 3 \), and increasing again at \( x = 4 \).

Step 6: Adjust Graph for Slope Directions

Adjust the graph ensuring \( g'(0) = 1 \), meaning the initial point at \( x = 0 \) has a positive gentle slope. Then match \( g'(4) = 1 \) and \( g'(2) = -1 \) by sketching appropriate tangents or curves, keeping the sign and magnitude of slopes consistent.

Key Concepts

Critical Points

Critical points are key features of a function's graph where the slope of the tangent (i.e., the derivative) is zero. These occur at points where the function can switch direction, potentially having a local maximum, minimum, or an inflection point. In this context, at both \( x = 1 \) and \( x = 3 \), the derivative \( g'(x) \) is zero.

• This means at \( x = 1 \) and \( x = 3 \), the function could either peak (local maximum) or dip (local minimum), or it could change its curvature (inflection point).
• To distinguish between these possibilities, further analysis is required, typically involving the second derivative or examining the sign changes around these points.

Understanding critical points helps in predicting the framework of your graph – where it might rise or fall, providing foundational insights necessary for graph sketching.

End Behavior

End behavior refers to how the function behaves as it approaches infinity or negative infinity. In the given exercise:

• As \( x \to \infty \), \( g(x) \to \infty \). This tells us that as \( x \) grows larger, the graph continues to rise indefinitely.
• As \( x \to -\infty \), \( g(x) \to -\infty \). Here, this indicates the graph falls without bound as \( x \) moves to the negative extremes.

These behaviors shape the outermost parts of the graph. Knowing them is crucial because it defines the overarching trend of the function. It ensures your graph stretches in the correct directions as \( x \) moves far from the origin.

Derivative Analysis

Analyzing the derivative gives us detailed insights into the function's behavior. Here, we focus on the function \( g \) and its first derivative \( g'(x) \).

• Where \( g'(x) > 0 \), the function \( g(x) \) is increasing.
• Where \( g'(x) < 0 \), the function is decreasing.
• The slope at \( x = 0 \) and \( x = 4 \) equals \( 1 \), indicating the function increases gently from these points.
• At \( x = 2 \), \( g'(x) = -1 \), suggesting a decreasing graph at this point.

By tracking where the derivative is positive or negative, we can construct how the graph looks between the critical points and intercepts, determining the form and direction of each segment.

Roots and Intercepts

Roots and intercepts are the points where the graph crosses the x-axis. For function \( g(x) \), the roots are given as \( g(0) = 0 \), \( g(2) = 0 \), and \( g(4) = 0 \).

• These points are crucial as they mark where \( g(x) \) changes sign, transitioning from negative to positive or vice versa.
• Every zero or root should align with the sketch to where the function intersects the x-axis.
• Between these intercepts, analyzing the derivative informs whether this section of the graph should rise or fall before hitting the next intercept.

Recognizing these intercepts is vital. They provide anchor points for the sketch and shape how sections of the function transition smoothly over the x-axis.