Problem 30 - Calculus

Question

Let \(W(h)\) be an invertible function which tells how many gallons of water an oak tree of height \(h\) feet uses on a hot summer day. Give practical interpretations for each of the following quantities or statements. (a) \(\quad W(50)\) (b) \(W^{-1}(40)\) (c) \(\quad W^{\prime}(5)=3\)

Step-by-Step Solution

Answer

(a) Water used by 50-ft tree, (b) Height for 40 gallons, (c) Rate of water increase at 5 ft.

Step 1: Interpretation of W(50)

The function notation \(W(50)\) refers to the function \(W\) evaluated at \(h = 50\). This implies that at a height of 50 feet, the oak tree consumes \(W(50)\) gallons of water. Therefore, \(W(50)\) represents the total gallons of water that an oak tree of 50 feet tall uses on a hot summer day.

Step 2: Interpretation of W^(-1)(40)

The expression \(W^{-1}(40)\) indicates the inverse function \(W\) evaluated at 40. The inverse function reverses the original function's input and output. Therefore, \(W^{-1}(40)\) means the height, \(h\), of an oak tree which uses 40 gallons of water on a hot summer day.

Step 3: Interpretation of W'(5) = 3

\(W'(5) = 3\) conveys that the derivative of the function \(W\) at \(h = 5\) is 3. This means that when the tree is 5 feet tall, its water usage increases by 3 gallons for every additional foot in height. Thus, the rate of change of water consumption with respect to the tree's height at 5 feet is 3 gallons per foot.

Key Concepts

Function Notation

Function notation is a way of representing functions that makes it easier to understand and communicate their behavior. When we see something like \( W(h) \), it indicates a function named \( W \), with \( h \) acting as the variable or input. This notation tells us that the function \( W \) outputs a specific value based on the input \( h \). For example, in the context of the oak tree problem, \( W(50) \) means we are evaluating the function at a tree height of 50 feet.

This notation is crucial because it helps us distinguish different functions and their specific evaluations. It also aids in comparing values by clearly specifying which input each value corresponds to. Thinking in terms of function notation empowers students to approach problems methodically, seeing \( W(h) \) as a dynamic tool rather than just a static number.

Derivative Interpretation

The derivative represents how a function changes as its input changes. In simpler terms, it tells us how steep a curve is at any given point. When we encounter a derivative notation, such as \( W'(5) = 3 \), it indicates the rate at which the function \( W \) is changing at the point where \( h = 5 \). This can be visualized as the slope of the tangent line to the curve at \( h = 5 \).

• \( W'(5) = 3 \) suggests that for every one-foot increase in the tree's height, the water usage increases by 3 gallons.
• It provides a snapshot of the function's behavior at a very specific point, allowing us to make predictions about small changes in inputs.
• Understanding derivatives is crucial for interpreting real-world problems where variables don't remain constant.

The concept of derivatives helps us discuss and solve problems about growth, change, and trends effectively, making it a fundamental tool in both calculus and applied mathematics.

Rate of Change

The rate of change is an essential concept that describes how one quantity changes in relation to another. It is straightforward to visualize it as the speed of change between the variables involved. In the exercise, \( W'(5) = 3 \) gives the rate of change of the oak tree's water usage with respect to its height at 5 feet.

Key insights about the rate of change include:

• It highlights the responsiveness of the dependent variable (water usage) to changes in the independent variable (tree height).
• The rate often ties directly to real-world applications, indicative of things like speed, cost efficiency, or ecological impacts.
• By understanding the rate, we can predict the direction and magnitude of changes in complex systems.

This concept is not just about numbers; it's about understanding dynamics. Learning about rate of change arms students with tools to anticipate and adjust to changes in various fields such as biology, economics, and engineering.