Problem 9 - Applied Calculus

Question

Table \(1.10\) gives values of a function \(w=f(t) .\) Is this function increasing or decreasing? Is the graph of this function concave up or concave down? $$ \begin{array}{l} \text { Table } 1.10\\\ \begin{array}{c|c|c|c|c|c|c|c} \hline t & 0 & 4 & 8 & 12 & 16 & 20 & 24 \\ \hline w & 100 & 58 & 32 & 24 & 20 & 18 & 17 \\ \hline \end{array} \end{array} $$

Step-by-Step Solution

Answer

The function is decreasing and concave up.

Step 1: Determine if the Function is Increasing or Decreasing

To check if the function is increasing or decreasing, compare the values of \( w \) at successive \( t \) values. Since each \( w \) value is less than the previous one (e.g., \( 100 \to 58 \to 32 \to 24 \to 20 \to 18 \to 17 \)), the function is decreasing.

Step 2: Calculate Differences to Analyze Concavity

To analyze concavity, calculate the differences between successive \( w \) values: \( 58-100 = -42 \), \( 32-58 = -26 \), \( 24-32 = -8 \), \( 20-24 = -4 \), \( 18-20 = -2 \), and \( 17-18 = -1 \).

Step 3: Determine the Pattern in Differences

The differences \( -42 \), \( -26 \), \( -8 \), \( -4 \), \( -2 \), and \( -1 \) are increasing (becoming less negative), indicating that the function's rate of decrease is slowing down, which suggests the graph is concave up.

Key Concepts

Function Analysis

Analyzing a function involves understanding its behavior and properties based on its values. Functions like those in Table 1.10 are numeric representations that can provide insights into trends such as whether they're increasing or decreasing. By examining specific values of a function over a domain, one can determine the pattern or trend present. Further analysis involves looking for patterns in changes between values to get a complete picture of the function's behavior. This is a foundational skill in calculus and is crucial for understanding advanced mathematical concepts. For non-continuous data provided in tables, comparisons between consecutive entries provide clarity on the rise and fall of values, which can then be extended to discussions on their rate of change and other properties.

Increasing and Decreasing Functions

To determine if a function is increasing or decreasing, we compare its successive values. For each value, if the next is larger, the function is increasing; if smaller, it's decreasing.Consider the sample function in Table 1.10. Each value of \(w\) decreases as \(t\) increases: \(100 \rightarrow 58 \rightarrow 32 \rightarrow 24 \rightarrow 20 \rightarrow 18 \rightarrow 17\). This consistent drop indicates a decreasing function.Key Points:

• An increasing function goes upwards in a graph as one moves from left to right.
• A decreasing function, like the one in Table 1.10, descends as it progresses.

Understanding whether a function is increasing or decreasing can help analyze the function's overall behavior within a given interval.

Concavity

Concavity tells us about the shape or curvature of the graph of a function and how it bends. A graph can be concave up or concave down, which directly relates to how the "slope" changes as you move along the curve.To analyze concavity, you often need to examine changes in the differences between the values of the function. In Step 2 of the solution for the problem, the differences were calculated: \(-42, -26, -8, -4, -2, -1\). These are differences between successive values of \(w\), showing changes in the function's rate of decline.Key Points:

• If successive differences are increasing, even if negative, it suggests the rate of decrease is slowing, leading to a concave-up shape (like a cup).
• If differences were decreasing (becoming more negative), it would indicate a concave-down shape (like a frown).

In the given exercise, because the differences are becoming less negative (slowing rate of decrease), we can infer the graph is concave up. Recognizing these patterns is a vital part of calculus, aiding in understanding the underlying shape of functions visually and algebraically.