Problem 45 - Thomas Calculus

Question

Sketch the graph of a continuous function \(y=g(x)\) such that a. \(g(2)=2, \quad 02,\) and \(g^{\prime}(x) \rightarrow-1^{+}\) as \(x \rightarrow 2^{+}\) b. \(g(2)=2, \quad g^{\prime}<0\) for \(x<2, \quad g^{\prime}(x) \rightarrow-\infty\) as \(\quad x \rightarrow 2^{-}, g^{\prime}>0\) for \(x>2,\) and \(g^{\prime}(x) \rightarrow \infty\) as \(x \rightarrow 2^{+}\)

Step-by-Step Solution

Answer

Sketch a gentle transition at x=2 for (a); draw sharp slopes near x=2 for (b).

Step 1: Analyze the function's behavior for x < 2 (Part a)

For part (a), we need a function where the derivative is between 0 and 1 for values of \(x\) less than 2. This indicates that the graph is increasing but with a slope less steep than 1. Additionally, as \(x\) approaches 2 from the left, \(g'(x)\) approaches 1, so the slope becomes exactly 1 as it gets very close to \(x = 2\). A possible curve is a gentle upward sloping line or curve that flattens to a slope of 1 near \(x = 2\).

Step 2: Analyze the function's behavior for x > 2 (Part a)

For \(x > 2\) in part (a), the derivative \(g'(x)\) is between -1 and 0, indicating a gently decreasing function. As \(x\) approaches 2 from the right, the slope approaches -1, meaning it becomes steeper in the negative direction near \(x = 2\). This could look like a gradual decline that steepens to a slope of -1 near \(x = 2\).

Step 3: Analyze the function's behavior for x < 2 (Part b)

In part (b), the derivative \(g'(x)\) must be negative for \(x < 2\), which means the function is decreasing on this interval. As \(x\) approaches 2 from the left, the derivative goes to \(-\infty\), indicating a very steep negative slope approaching \(x = 2\). We might consider a curve that becomes nearly vertical at \(x = 2\) from the left.

Step 4: Analyze the function's behavior for x > 2 (Part b)

For \(x > 2\) in part (b), \(g'(x)\) is positive, so the function is increasing. As \(x\) approaches 2 from the right, \(g'(x)\) approaches \(\infty\). This suggests a function that is steeply sloping upwards and becomes nearly vertical at \(x = 2\) from the right side. The graph may abruptly shoot upwards as \(x\) approaches 2 from the right.

Step 5: Sketch the Graphs

For part (a), sketch a curve that is gently increasing to \(x = 2\), leveling to a slope of 1, then gently decreasing past \(x = 2\) with a slope flattening to -1. For part (b), the graph should sharply decline to \(x = 2\) and sharply rise after \(x = 2\), near vertical at 2, creating a sharp V-turn with a negative slope transitioning to a positive slope.

Key Concepts

Continuous Functions

Continuous functions play a fundamental role in graph sketching, especially when dealing with derivatives and the overall behavior of the graph. A continuous function has no breaks, jumps, or holes in its graph. This means if you can draw it without lifting your pencil off the paper, it’s continuous. In the exercise, the function \(y = g(x)\) is continuous, so you will generally notice smooth transitions as you analyze its behavior.

For a function like \(g(x)\), understanding continuity helps predict how the graph behaves at points like \(x = 2\), ensuring there are no surprises like jumps or mini gaps in the slope. Keeping continuity in mind allows you to focus on the changes in the derivative and how these affect graph direction.

Derivatives

Derivatives are critical when analyzing the rate of change of a function. They tell us the slope at any given point on a graph, thus, indicating how the graph is moving. A positive derivative means the function is increasing, while a negative one means it is decreasing.

In this exercise, the derivative \(g'(x)\) is the key player. For instance, when \(x < 2\) in Part (a), \(0 < g'(x) < 1\), indicating that the graph increases but is not steep. As \(x\) approaches 2, \(g'(x)\) heads towards 1, showing that the function's slope is close to leveling out. Understanding the behavior of derivatives in these small intervals allows for precise and educated sketches of the graph's paths.

For Part (b), different derivative behaviors determine whether the graph sharply dips and then rises near \(x = 2\). With \(g'(x)\) heading towards negative or positive infinity, the steepness becomes nearly vertical as \(x\) closes in on 2.

Graph Behavior Analysis

Graph behavior analysis helps you understand how and why a graph changes direction at various points. It involves looking at derivatives and continuity to paint a picture of the graph's overall shape.

For this exercise, analyzing the behavior at \(x = 2\) is crucial. It's where the most significant changes occur in both parts (a) and (b). For example, in part (a), the function transitions from a subtly climbing slope to a gently declining one starting steep at \(x = 2\). Meanwhile, in part (b), the function exhibits sharp changes – a steep drop before climbing again.

Observing these characteristics helps in predicting graph appearance, which can significantly aid learners in understanding how subtle changes in derivatives affect the graph's path.

Slope Interpretation

Understanding the slope is key to successfully interpreting graph changes. The slope, given by the derivative \(g'(x)\), helps determine if the graph is ascending or descending. At various points, the interpretation of its magnitude and sign can tell a much deeper story.

In Part (a), where \(g'(x)\) hovers between 0 to 1 for \(x < 2\), means increased but restrained slope and smooth ascent. As \(x\) approaches 2, nearing a slope of 1 means becoming almost flat. On the flipside, post-\(x = 2\), a derivative between -1 and 0 hints a gentle descent. Hence, knowing just these values makes interpreting the graph’s turns intuitive.

Similarly, slope behavior in Part (b) tells the powerful shifts with nearly infinite negative or positive slopes at \(x = 2\). Such information makes sketching much more straightforward, as you can better visualize how steep or mild these changes look.