Problem 1 - Calculus: An Integrated Approach to Functions and Their Rates of Change

Question

Water is being poured into a bucket at a steady rate. \(h(t)\) gives the height of water at time \(t\). Let \(t_{*}\) be the time when the bucket is half full. What can you say about the signs of \(h^{\prime}\left(t_{*}\right)\) and \(h^{\prime \prime}\left(t_{*}\right) .\) Explain your reasoning precisely in plain English.

Step-by-Step Solution

Answer

The sign of \(h'(t_*)\) is positive, indicating that the water level in the bucket is rising at time \(t_*\). The sign of \(h''(t_*)\) is zero since the rate of change is constant, indicating that there is no acceleration or deceleration in the water rise at time \(t_*\).

Step 1: Sign of \(h'(t_*)\)

Since the water is poured into the bucket at a steady rate, the height of the water increases linearly over time, implying that the first derivative \(h'(t)\) is positive for all \(t\). This means the function is increasing. Therefore, \(h'(t_*) > 0\). In the context of this problem, the positive \(h'(t_*)\) means the water level in the bucket is rising at time \(t_*\).

Step 2: Sign of \(h''(t_*)\)

Since \(h(t)\) is a function that increases linearly with time, \(h(t)\) is a function of the form \(y=mx+n\), where \(m\) and \(n\) are constants. The second derivative of such a function is zero (i.e., \(h''(t) = 0\) for all \(t\)), because the slope of the tangent line at each point is constant and does not change. Therefore, \(h''(t_*) = 0\). In the context of this problem, \(h''(t_*) = 0\) means that the rate at which the water level is rising is constant at time \(t_*\) and does not show any acceleration or deceleration.

Key Concepts

Derivatives and Their Applications

Calculus is a branch of mathematics that deals with rates of change and the accumulation of quantities. One of its fundamental concepts is the derivative. The derivative of a function at a point is the rate at which the function value is changing at that point. It is an essential tool in various fields such as physics, economics, and engineering.

For instance, if you're looking at the height of water in a bucket, represented by a function such as h(t), the derivative, denoted by h'(t), tells us the rate at which the water's height is changing with respect to time. If the water is poured at a steady rate, the height increases at a constant rate, indicating that h'(t) is positive. A positive derivative at any time t means the function is increasing at that point.

In practical terms, derivatives allow us to model and predict the behavior of moving objects, changing markets, and growth rates of populations, making them extremely useful for making informed decisions in real-world scenarios.

Related Rates

Related rates are a particular type of problem in calculus that involve finding the rate at which one quantity changes in relation to another. These problems utilize the chain rule to connect the rates of change of multiple variables that depend on a common variable, such as time.

Take the earlier example where water is being poured into a bucket. If we not only had to figure out the rate at which the height of the water rises, but also needed to know how the volume of water is changing over time, we would be dealing with related rates. Since both the height and volume change with respect to time, by knowing the rate of one, calculus allows us to deduce the rate of the other.

In real-world application, related rates can help in understanding relationships in moving objects, such as the distance between two cars traveling at different speeds, or the rate at which the shadow of a building grows as the sun sets.

Linear Functions

In the world of mathematics, a linear function is one of the simplest forms to understand and graph. It is a function that has a constant rate of change, represented typically in the form y = mx + n, where m is the slope and n is the y-intercept. This linearity implies a direct proportionality between variables: as one variable changes, the second changes at a constant rate defined by the slope, which is the derivative of the function.

In our example, since the height of the water in the bucket changes linearly with time when poured at a steady rate, we are describing a linear function. Thus, the derivative h'(t) is the slope and remains constant for any value of t, which is why the second derivative, or the slope of the slope, is zero. This zero second derivative reflects no curvature or 'bend' in the graph of the function; it’s a straight line.

Linear functions are foundational in various disciplines. Economists use them to predict demand or cost over time, and scientists use them to describe constant-speed motion. Understanding the simplicity yet versatility of linear functions helps in grasifying concepts of more complex functions.