Problem 94 - Thomas Calculus in SI Units

Question

Sketch a smooth connected curve \(y=f(x)\) with \(f(-2)=8, \quad f^{\prime}(2)=f^{\prime}(-2)=0\), \(f(0)=4, \quad f^{\prime}(x)<0 \quad\) for \(\quad|x|<2\), \(f(2)=0, \quad f^{\prime \prime}(x)<0 \quad\) for \(\quad x<0\), \(f^{\prime}(x)>0\) for \(|x|>2, \quad f^{\prime \prime}(x)>0\) for \(x>0\).

Step-by-Step Solution

Answer

Sketch a curve with inflection at \(x = 0\), peaks at \(x = -2\), and trough at \(x = 2\).

Step 1: Understanding Given Points

First, identify the key points provided: - At \(x = -2\), \(f(-2) = 8\) and the slope \(f^{\prime}(-2) = 0\), which indicates a horizontal tangent. - At \(x = 0\), \(f(0) = 4\). - At \(x = 2\), \(f(2) = 0\) with \(f^{\prime}(2) = 0\), indicating another horizontal tangent.

Step 2: Analyzing Derivative Information

Note the behavior of the first derivative: - \(f^{\prime}(x) < 0\) for \(|x| < 2\), meaning the function is decreasing between \(-2\) and \(2\). - \(f^{\prime}(x) > 0\) for \(|x| > 2\), meaning the function is increasing outside the interval from \(-2\) to \(2\).

Step 3: Analyzing Second Derivative Information

Assess the concavity using the second derivative: - \(f^{\prime\prime}(x) < 0\) for \(x < 0\), indicating the curve is concave down on this interval. - \(f^{\prime\prime}(x) > 0\) for \(x > 0\), indicating the curve is concave up on this interval.

Step 4: Sketching the Curve Overview

Combine the information: - The curve starts at \(x = -2\) with a horizontal tangent at \((x, y) = (-2, 8)\), decreases as it moves toward \(x = 0\), where it passes through \((0, 4)\). - Because \(f^{\prime}(x) < 0\) for \(-2 < x < 2\), it continues to decrease until \(x = 2\) where there's a horizontal tangent at \((2, 0)\). - The curve is concave down for \(x < 0\) and becomes concave up for \(x > 0\), affecting the shape between key points.

Step 5: Drawing the Curve

Start sketching: - Begin at \(x = -2\), point upwards with a concave down segment that decreases to \(x = 0\), then continue decreasing but switch to concave up as you approach \(x = 2\). - After crossing \(x = 2\), the curve turns upward, increasing sharply due to \(f^{\prime}(x) > 0\).

Key Concepts

Derivatives

In the context of calculus, derivatives provide us with vital information about the behavior of a function. Simply put, the derivative of a function at a given point indicates the slope of the tangent line at that point. If the slope is positive \(f^{\prime}(x) > 0\), it means the function is increasing at that interval. Conversely, if the slope is negative \(f^{\prime}(x) < 0\), the function is decreasing.
For the exercise, we note that \(f^{\prime}(x) < 0\) for \(|x| < 2\). This tells us that the function decreases in the interval from -2 to 2, depicting a downward slope. On the other hand, \(f^{\prime}(x) > 0\) for \(|x| > 2\), which indicates the function is increasing once the interval exceeds 2 in absolute value. Understanding derivatives like this helps us form a picture of how the function behaves across different segments.

Concavity

Concavity is a property of a function that tells us how the function curves. It is determined by the second derivative \(f^{\prime\prime}(x)\). When the second derivative is positive \(f^{\prime\prime}(x) > 0\), the graph of the function is concave up, resembling the shape of a "U". If the second derivative is negative \(f^{\prime\prime}(x) < 0\), the function is concave down, similar to an upside-down "U", or a hump.
In this exercise, for \(x < 0\), \(f^{\prime\prime}(x) < 0\) which indicates the function curves downwards in that region. For \(x > 0\), \(f^{\prime\prime}(x) > 0\), making the curve concave up. This switch in concavity affects how we draw the curve between key points and reveals changes in the turning direction of the curve.

Horizontal Tangent

A horizontal tangent occurs at a point on a function where the slope of the tangent line is zero. It is indicated when \(f^{\prime}(x) = 0\). At this point, the graph flattens out momentarily. In our task, we have horizontal tangents at two places: \(x = -2\) and \(x = 2\), with \(f^{\prime}(-2) = 0\) and \(f^{\prime}(2) = 0\). For \(x = -2\), it means the function reaches its maximum or minimum and flattens before possibly changing direction. Similarly, at \(x = 2\), since it is accompanied by the fact that \(x > 2\) has \(f^{\prime}(x) > 0\), the function possibly begins to increase after this flat point. Horizontal tangents often signify important stopping points that change the trajectory of a function.

Function Behavior

Function behavior refers to the combination of all insights from derivatives and concavity to describe how a function progresses over an interval. This involves the patterns of increases, decreases, and curvature direction within specific segments. In the exercise, the function starts at point \(x = -2\) with a value of 8 and an initial horizontal tangent, symbolizing a momentary halt in change. From \(x = -2\) to \(x = 0\), the function decreases, reaching another key value of 4. Concavity influences this portion by ensuring it's concave down. As \(x\) approaches 2, the function remains decreasing until it flattens at \(x = 2\), then begins to increase as \(x\) exceeds 2 due to positive slope indicators. Grasping these intertwining elements helps depict an accurate, comprehensive picture of the function and its graphical representation.