Problem 40 - Calculus Early Transcendentals

Question

The function f whose graph is shown below has values as given in the accompanying table. $$\begin{array}{|c|c|c|c|c|c|}\hline x & {-1} & {0} & {1} & {2} & {3} \\\ \hline f(x) & {1.56} & {0.58} & {2.12} & {2.34} & {2.2} \\\ \hline\end{array}$$ Use data from the table to approximate the derivative values. (a) \(f^{\prime}(0.5)\) (b) \(f^{\prime}(2.5)\)

Step-by-Step Solution

Answer

(a) 1.54; (b) -0.14

Step 1: Understand Derivative Approximation

We approximate the derivative at a point by using the average rate of change, which is calculated from function values close to the given point. For a point \( x = a \), we use nearby points \( x = a - h \) and \( x = a + h \). The derivative \( f'(a) \) is approximated by \( \frac{f(a + h) - f(a - h)}{2h} \).

Step 2: Approximate f'(0.5)

To approximate \( f'(0.5) \), choose \( x = 0 \) and \( x = 1 \) as the closest points. Use:\[ f'(0.5) \approx \frac{f(1) - f(0)}{1 - 0} = \frac{2.12 - 0.58}{1} = 1.54 \]

Step 3: Approximate f'(2.5)

To approximate \( f'(2.5) \), choose \( x = 2 \) and \( x = 3 \) as the closest points. Use:\[ f'(2.5) \approx \frac{f(3) - f(2)}{3 - 2} = \frac{2.2 - 2.34}{1} = -0.14 \]

Key Concepts

Understanding the Average Rate of Change

The average rate of change is a fundamental concept in calculus that helps us understand how a function behaves as we move from one point to another. Essentially, it measures how much the function's output (or value) changes, on average, for each unit increase in the input or the independent variable. Imagine you are driving a car and you want to calculate your average speed over a journey. You would look at how many miles you traveled divided by the number of hours it took. In a similar way, the average rate of change of a function uses changes in function values and changes in input values.

For a function \( f(x) \), the average rate of change over an interval \([a, b]\) is given by the formula:

• \( \frac{f(b)-f(a)}{b-a} \)

This expression gives the slope of the line that connects the function values at points \( a \) and \( b \), also known as the secant line. This concept sets the foundation for understanding derivatives, which gives us the rate of change at an exact point.

An Introduction to Calculus

Calculus is a branch of mathematics that focuses on how things change. At its core, calculus deals with two fundamental concepts:

• Derivatives: They tell us about the rate of change of a quantity.
• Integrals: These help us find the total accumulation of quantities.

Derivatives provide a deeper understanding of how a function changes at any given point. While the average rate of change looks at changes over intervals, calculus lets us zoom in to find the instantaneous rate of change at a specific point. This is one of the primary reasons calculus is so powerful—it allows us to deal with dynamic and continuous changes effectively.

Calculus is widely used across various fields such as physics, engineering, economics, and even biology, because it offers tools to model and solve complex problems involving change.

Delving into Function Derivatives

Function derivatives are a way of representing the rate at which a function is changing at any given point. Unlike the average rate of change, which looks at the change over a range of values, the derivative finds the instant rate of change—like capturing a single frame in a video.

In mathematical terms, for a function \( f(x) \), the derivative at point \( a \), denoted by \( f'(a) \), can be approximated by looking at points very close to \( a \). The formula used for this approximation is:

• \( f'(a) \approx \frac{f(a + h) - f(a - h)}{2h} \), where \( h \) is a small number

This approximation gives us an idea of how the function is behaving right at point \( a \). When \( h \) approaches zero, we get an accurate measure of the derivative.

Derivatives are crucial because they help in optimizing functions, predicting future behavior, and understanding the dynamics of systems that vary continuously with their inputs.