Problem 67 - Thomas Calculus

Question

Sketch the graph of a continuous function \(y=g(x)\) such that $$\begin{array}{c}{\text { a. } g(2)=2,0 < g^{\prime}<1 \text { for } x < 2, g^{\prime}(x) \rightarrow 1^{-} \text { as } x \rightarrow 2^{-}} \\ {-1< g^{\prime} < 0 \text { for } x > 2, \text { and } g^{\prime}(x) \rightarrow-1^{+} \text { as } x \rightarrow 2^{+}}\end{array}$$ $$\begin{array}{l}{\text { b. } g(2)=2, g^{\prime}<0 \text { for } x<2, g^{\prime}(x) \rightarrow-\infty \text { as } x \rightarrow 2^{-}} \\\ {g^{\prime}>0 \text { for } x>2, \text { and } g^{\prime}(x) \rightarrow \infty \text { as } x \rightarrow 2^{+} \text { . }}\end{array}$$

Step-by-Step Solution

Answer

Part a: The graph approaches and descends from (2,2) gently. Part b: The graph dives and rises steeply at (2,2).

Step 1: Understanding Requirements for Part a

For part a, the function must satisfy the condition that \( g(2) = 2 \). The derivative \( g'(x) \) must be between 0 and 1 for \( x < 2 \) and approach 1 from the left as \( x \) approaches 2. This suggests a function increasing to a slope of 1 at \( x = 2 \). For \( x > 2 \), the derivative is between -1 and 0, approaching -1 from the right, indicating a decreasing function after \( x = 2 \).

Step 2: Sketching the Graph for Part a

Begin at the point \( (2, 2) \) since \( g(2) = 2 \). For \( x < 2 \), the graph should gently rise towards the point \( (2, 2) \) with a slope gradually tending towards 1 as \( x \) approaches 2. To the right of \( x = 2 \), the graph should descend away with the steepest descent (lowest negative slope) very near \( x = 2 \), transitioning to a flatter descent as \( x \) moves away from 2.

Step 3: Understanding Requirements for Part b

For part b, \( g(2) = 2 \) remains constant. Here, \( g'(x) < 0 \) for \( x < 2 \) and approaches negative infinity as \( x \) nears 2 from the left. This means the function is decreasing with increasingly steeper slopes towards \( x = 2 \). For \( x > 2 \), \( g'(x) > 0 \) grows towards infinity, indicating an upward curve that steepens as \( x \) moves away from 2.

Step 4: Sketching the Graph for Part b

Start again at \( (2, 2) \). For \( x < 2 \), draw a curve that decreases towards \( x = 2 \) with an ever-increasing negative slope, giving the appearance of a sharp dive as you near \( x = 2 \). For \( x > 2 \), depict a rapidly rising curve that steepens as \( x \) moves further right from 2, illustrating the accelerating positive slope.

Key Concepts

Continuous Functions

Continuous functions are an essential concept in calculus and mathematics, representing functions with no breaks, jumps, or holes in their graphs. Imagine a smooth and unbroken line where you can draw the function graph without lifting your pencil.
In this exercise, the function \( g(x) \) is described as continuous across all \( x \). This continuity means as \( x \) transitions through any value, including \( x = 2 \), the function's output, \( y = g(x) \), changes smoothly.
With \( g(2) = 2 \), the function specifically passes through the point \((2,2)\), consistently reflecting the continuous nature of \( g(x) \).

• Continuous functions do not have sudden changes or gaps.
• Ensures predictable behavior throughout its domain.

Understanding continuous functions helps sketch accurate graphs and predict the behavior of mathematical models.

Derivatives

Derivatives signify the rate at which a function changes at any given point. It's essentially the slope of the function at a particular stage, providing insights into whether the graph is rising or falling at that point.
In the context of our exercise, \( g'(x) \) represents how the function \( g(x) \) behaves off the point \( x=2 \).

• For \( x < 2 \), when \( 0 < g'(x) < 1 \), it indicates a gently increasing function, moving upwards but not steeply.
• For \( x > 2 \), if \( -1 < g'(x) < 0 \), the derivative suggests the function is decreasing, moving downwards smoothly.

The direction and slope at specific points are governed by the derivative's value, working as a guide to draft the function on paper.

Function Behavior

Function behavior refers to how a function progresses in different sections of its domain. We examine if the graph should rise, flatten, or descend in particular areas.
In part a of our exercise, the behavior changes at \( x=2 \):

• When approaching from the left, the graph rises slowly towards \( x=2 \).
• On crossing \( x=2 \), the graph begins to descend away.

Similarly, in part b:

• The graph shows a steep fall towards \( x=2 \) from the left.
• Marks a sharp rise after \( x=2 \) to the right.

This changing pace illustrates how function behavior reflects the cumulative effect of derivatives on continuity and helps in visualizing the overall function.

Slope Analysis

Slope analysis involves studying the steepness and direction of a graph. A slope can be intensely vertical or gently tilted. The exercise highlights how slopes change around \( x=2 \).
In part a, before reaching \( x=2 \), the slope is positive but less than 1, which means it's increasing slowly. Right after \( x=2 \), the slope is negative, hinting at a rapid fall, as the slope nears -1.

• This ensures a smooth transition from a slowly rising graph to a gentle fall.

In part b, the behavior switches:

• The slope from the left side of \( x=2 \) dives sharply negative, and right after \( x=2 \), it trends steeply positive.

Such an analysis allows us to foresee and detail how a function's graph will behave around critical points, improving our understanding of the function’s dynamics.