Problem 8 - Applied Calculus

Question

During a flood, the water level in a river first rose faster and faster, then rose more and more slowly until it reached its highest point, then went back down to its preflood level. Consider water depth as a function of time. (a) Is the time of highest water level a critical point or an inflection point of this function? (b) Is the time when the water first hegan to rise more slowly a critical point or an inflection point?

Step-by-Step Solution

Answer

(a) Critical point; (b) Inflection point.

Step 1: Understanding Critical and Inflection Points

A critical point of a function occurs where its derivative is zero or undefined, which usually corresponds to a maximum or minimum of the function. An inflection point is where the second derivative of the function changes sign, indicating a change in the concavity of the function.

Step 2: Analyzing the Highest Water Level

When the water level reaches its highest point, the rate of change of the height is zero, indicating a local maximum. Thus, this point is characterized by a zero first derivative (i.e., a critical point) of the function representing the water level.

Step 3: Analyzing Changing Conformity in Water Rise

As the water begins to rise more slowly, there is a change from increasing at an increasing rate to increasing at a decreasing rate. This change in concavity indicates that it is an inflection point because the second derivative changes sign.

Key Concepts

Critical Point

In calculus, a critical point of a function provides insightful information regarding the behavior of the function. It occurs when the derivative (the rate of change) of a function is either zero or undefined. This typically happens at points where a function reaches its highest or lowest values, known as maxima or minima, respectively. For example, in the case of the water level during a flood, the maximum level corresponds to a critical point because the derivative at this point is zero. Here, the water stops rising and is at its peak before it begins to descend. Understanding critical points is key in determining key features of any graph, including where the direction of the function may change.

Inflection Point

An inflection point is found where a graph changes its concavity, moving from being concave up to concave down, or vice versa. This change of concavity is identified through the second derivative of the function. An inflection point occurs precisely where the second derivative changes sign. In the flood scenario, when the water starts rising more slowly, an inflection point is encountered. At this point, the rate at which the water level increases begins to decelerate, shifting from a steep rise to a gentler slope. Recognizing inflection points in graphs helps us understand nuanced aspects of the function's behavior, such as changes in acceleration.

Derivative

The derivative of a function is a core concept in calculus. It measures how a function changes as its input changes, essentially providing the slope of the tangent to the curve at any point. Derivatives are vital for identifying critical points because they reach zero or an undefined value precisely at these points. For example, understanding how fast the water level is rising or falling during a flood involves examining the derivative of the water depth function. Calculating the derivative helps in identifying where the water reaches its highest point, which would be a place where the increase in water level stops momentarily. Mastery of derivatives provides essential insights into all sorts of changing quantities.

Second Derivative

The second derivative of a function offers deeper insights into the curve's behavior, revealing information about the concavity and inflection points. The first derivative tells us where and how quickly the function is rising or falling, while the second derivative indicates how this rise or fall is accelerating or decelerating. Specifically, a change in the sign of the second derivative indicates an inflection point. In the context of the flood, when the water level transition shifts from rising faster to rising slower, the second derivative changes from positive to negative. Understanding the second derivative is essential for getting a more detailed picture of a function's geometry, particularly how it "curves" in different intervals.