Problem 23 - Applied Calculus

Question

Values of \(x\) and \(g(x)\) are given in the table. For what value of \(x\) does \(g^{\prime}(x)\) appear to be closest to \(3 ?\) $$\begin{array}{c|c|c|c|c|c|c|c|c} \hline x & 2.7 & 3.2 & 3.7 & 4.2 & 4.7 & 5.2 & 5.7 & 6.2 \\ \hline g(x) & 3.4 & 4.4 & 5.0 & 5.4 & 6.0 & 7.4 & 9.0 & 11.0 \\ \hline \end{array}$$

Step-by-Step Solution

Answer

The value of \(x\) is 5.2.

Step 1: Understanding the Problem

The problem asks us to find the value of \(x\) for which the derivative \(g'(x)\) is closest to 3. This means that we need to estimate the rate of change or slope of the function \(g(x)\) at various points and see which one is closest to 3.

Step 2: Estimating Derivatives

To estimate \(g'(x)\), we calculate the difference quotient for each interval: \(g'(x) \approx \frac{g(x+h) - g(x)}{h}\), where \(h\) is the difference in \(x\) values. We use this formula to estimate the derivatives between each pair of consecutive \(x\) values.

Step 3: Calculate Difference Quotients

1. \(x = 2.7\); \(g'(2.7) \approx \frac{g(3.2) - g(2.7)}{3.2 - 2.7} = \frac{4.4 - 3.4}{0.5} = 2\)2. \(x = 3.2\); \(g'(3.2) \approx \frac{g(3.7) - g(3.2)}{3.7 - 3.2} = \frac{5.0 - 4.4}{0.5} = 1.2\)3. \(x = 3.7\); \(g'(3.7) \approx \frac{g(4.2) - g(3.7)}{4.2 - 3.7} = \frac{5.4 - 5.0}{0.5} = 0.8\)4. \(x = 4.2\); \(g'(4.2) \approx \frac{g(4.7) - g(4.2)}{4.7 - 4.2} = \frac{6.0 - 5.4}{0.5} = 1.2\)5. \(x = 4.7\); \(g'(4.7) \approx \frac{g(5.2) - g(4.7)}{5.2 - 4.7} = \frac{7.4 - 6.0}{0.5} = 2.8\)6. \(x = 5.2\); \(g'(5.2) \approx \frac{g(5.7) - g(5.2)}{5.7 - 5.2} = \frac{9.0 - 7.4}{0.5} = 3.2\)7. \(x = 5.7\); \(g'(5.7) \approx \frac{g(6.2) - g(5.7)}{6.2 - 5.7} = \frac{11.0 - 9.0}{0.5} = 4.0\)

Step 4: Find Closest Value to 3

Among the estimated derivatives, we compare their proximity to 3. The calculated derivative that is closest to 3 is \(g'(5.2) = 3.2\). So, \(x = 5.2\) gives the closest value to 3.

Key Concepts

Difference Quotient

When discussing derivatives, the difference quotient is a fundamental concept. It's a technique to estimate the rate of change of a function at a particular point. In simple terms, it calculates the average change in the function’s value over a small interval. Given by the formula

• \( g'(x) \approx \frac{g(x+h) - g(x)}{h} \)

it provides an approximation of the derivative when the exact solution is difficult or unnecessary to find, like in our problem above.
The main idea here is that by considering values closely spaced (small \( h \)), the difference quotient gives a good estimate of the true instantaneous rate of change at a point. The smaller the \( h \), the more accurate this estimation becomes, approaching the true derivative as \( h \) approaches zero.
The exercise solution walks us through this by calculating the difference quotient for a series of \( x \) values, effectively helping us "guess" the slopes to find the one closest to 3.

Rate of Change

Rate of change is another key element in understanding derivatives. It describes how a function’s output changes as its input changes. In mathematics, it’s synonymous with the derivative, which explains why understanding derivatives is crucial to grasping rate of change.
For the problem at hand, the rate of change at any given \( x \) value (specifically the derivative of the function, \( g(x) \)) is estimated using the difference quotient. This means that the slope calculated between two points adjacent on either side of \( x \) represents how quickly the values of \( g(x) \) are rising or falling at that spot.
This concept is vital in many fields, as it also interprets how one quantity varies in response to another. For example, understanding how quickly a car accelerates (speed changes as time progresses) can be analyzed as a rate of change.

Function Analysis

Function analysis involves understanding the behavior of functions and how they change. It’s an indispensable skill in solving calculus problems efficiently. In our specific example, analyzing where the function's derivative (or rate of change) is close to a particular value (in this case, 3) involves calculating and comparing the derived values of \( g(x) \).
During the analysis, we focus on how the function behaves over intervals, gives clues about its direction and concavity, and helps identify where certain criteria are met, like reaching a specific rate of change. Through function analysis, we can deduce relationships and inverse relationships, predict patterns, and solve complex problems progressively by understanding smaller components.
It's akin to how scientists use trends in data to make predictions. Here, we analyzed each segment's slope to find which most approximately represents the derivative closest to 3, highlighting the point \( x = 5.2 \) as our answer.