Problem 13 - Applied Calculus

Question

Use the values given for each function. (a) Does the derivative of the function appear to be positive or negative over the given interval? Explain. (b) Does the second derivative of the function appear to be positive or negative over the given interval? Explain. $$\begin{array}{c|c|c|c|c|c|c}\hline t & 0 & 1 & 2 & 3 & 4 & 5 \\\\\hline s(t) & 12 & 14 & 17 & 20 & 31 & 55 \\\\\hline\end{array}$$

Step-by-Step Solution

Answer

(a) Positive; function values increase. (b) Positive; rate of increase accelerates.

Step 1: Analyze First Derivative

To determine if the first derivative is positive or negative, we need to calculate the differences between consecutive values of the function, \(s(t)\). These differences represent the changes in the function over time. These are \(14-12=2\), \(17-14=3\), \(20-17=3\), \(31-20=11\), and \(55-31=24\). Since all changes in \(s(t)\) are positive, the first derivative is positive over the interval \([0, 5]\).

Step 2: Analyze Second Derivative

To determine if the second derivative is positive or negative, calculate the differences between successive rate changes or first derivatives. In this case: \(3-2=1\), \(3-3=0\), \(11-3=8\), \(24-11=13\). Since the second derivative changes from 0 in the second calculation to all positive values thereafter, the second derivative appears to be positive over most of the interval.

Key Concepts

Derivative Analysis

In calculus, derivative analysis is the study of how a function changes with respect to one of its variables. Derivatives help us understand the behavior of functions by providing insights into the rate of change at any given point. This analysis is crucial for understanding the trend and concavity of the function, which respectively pertain to the first and second derivatives.

When analyzing a table of values for a function, you look at the differences between consecutive outputs to get an idea of how a function behaves. A positive change indicates that the function is increasing, while a negative change indicates a decrease. This basic principle forms the foundation of derivative analysis as it applies to understanding real-world scenarios, such as velocity analysis in physics. Keeping these concepts in mind helps demystify subjects that rely heavily on rate of change and curvature, making them critical in mathematical applications across various fields.

First Derivative

The first derivative of a function, often denoted as \(f'(x)\), provides information about the rate of change of the function. In simpler terms, it tells us whether the function is increasing or decreasing at any point. Calculating the first derivative is essential for determining the slope or gradient of a function.

In the context of the given exercise, to find whether the first derivative is positive or negative, we look at the differences between the function's values at successive points. In mathematical terms, for the function \(s(t)\), we compute the differences as follows:

• \(14-12=2\)
• \(17-14=3\)
• \(20-17=3\)
• \(31-20=11\)
• \(55-31=24\)

Each of these differences is positive, indicating that the first derivative over the interval \([0, 5]\) is positive. This implies that the function is consistently increasing throughout the interval. Such information is vital for applications like determining the velocity in a physics problem, where a positive first derivative means an increasing speed.

Second Derivative

The second derivative, often represented as \(f''(x)\), provides crucial information about a function's concavity and the rate of change of the first derivative. Simply put, it tells us how the function's rate of increase or decrease is itself increasing or decreasing. This is useful for determining whether the graph of the function is curving upwards (concave up) or downwards (concave down).

In the given problem, to investigate whether the second derivative is positive or negative, we derive the differences between consecutive first derivatives:

• \(3-2=1\)
• \(3-3=0\)
• \(11-3=8\)
• \(24-11=13\)

The initial value of zero transitions into positive values, indicating a generally positive second derivative over most of the interval. This suggests that the acceleration or the rate of growth of the function is itself increasing, and the curve represented by the function \(s(t)\) is bending upwards. Understanding second derivatives allows deeper insights into the nature of a function's growth, often used in economics to predict trends, or in animation for motion analysis.