Problem 50 - Calculus of a Single Variable

Question

Applying the First Derivative Test In Exercises \(41-48\) , consider the function on In Exercises \(49-54,\) (a) use a computer algebra system to differentiate the function, (b) sketch the graphs of \(f\) and \(f^{\prime}\) on the same set of coordinate axes over the given interval, (c) find the critical numbers of \(f\) in the open interval, and (d) find the interval(s) on which \(f^{\prime}\) is positive and the interval(s) on which it is negative. Compare the behavior of \(f\) and the sign of \(f^{\prime}\) $$ f(x)=10\left(5-\sqrt{x^{2}-3 x+16}\right), \quad[0,5] $$

Step-by-Step Solution

Answer

Use the steps to find the derivative of the function, find the critical numbers, sketch the graphs of the function and its derivative, and identify where the function is increasing or decreasing. The critical points and signs of the derivative help to illustrate how the value of the function is changing over the interval [0, 5].

Step 1: Find the derivative

Use a computer algebra system to find the derivative of \(f(x)=10(5-\sqrt{x^{2}-3x+16})\). The derivative, also known as \(f^{\prime}(x)\), gives the function's rate of change at any given point.

Step 2: Find critical numbers

Find where \(f^{\prime}(x)\) equals 0 or is undefined in the interval (0, 5). These points are known as critical numbers and give potential maximums, minimums, or inflection points for the function.

Step 3: Sketch the graphs

Use the original function \(f(x)\) and its derivative \(f^{\prime}(x)\) to sketch their respective graphs on the same set of axes over the given interval [0, 5]. This shows how the function's rate of change relates to its values.

Step 4: Analyze the derivative

Identify intervals on which \(f^{\prime}(x)\) is positive and intervals on which it is negative. A positive derivative indicates that the function is increasing, and a negative derivative indicates that it is decreasing. Compare the behavior of \(f(x)\) with the sign of \(f^{\prime}(x)\) on the graph.

Key Concepts

Critical Numbers

Critical numbers are essential to understanding the behavior of a function. They occur where the derivative of the function, noted as \( f'(x) \), is either zero or undefined. These numbers can indicate where a function might have maximum or minimum values, or where the function's graph has a point of inflection, affecting how the graph curves.

When solving a problem involving critical numbers, you start by finding the derivative of the function. In this exercise, the derivative was found using a computer algebra system, which revealed how the function changes over time. The next task was to set \( f'(x) = 0 \) or determine where it is undefined within the specified interval.

This step is crucial because it helps to identify the potential turning points of a function graph. Once you find these critical numbers, you can further examine the function's increasing or decreasing nature.

Derivative

The derivative, denoted as \( f'(x) \), is fundamental in calculus for determining the rate at which a function is changing at any given point. It essentially measures how fast or slow a function's value is climbing or falling as its input varies.

For the function in this exercise \( f(x) = 10(5-\sqrt{x^{2}-3x+16}) \), finding the derivative means calculating \( f'(x) \). This involves using rules of differentiation, and technological tools can simplify this process by performing complex algebraic manipulations quickly.

Once the derivative is determined, it can be analyzed to understand better the dynamics of the function \( f(x) \), such as identifying critical numbers and the function's behavior over specific intervals.

Increasing and Decreasing Intervals

Determining where a function increases or decreases is one of the primary applications of the first derivative test. By analyzing the sign of the derivative \( f'(x) \), we can tell whether the function \( f(x) \) is climbing or descending at any point.

- **Positive Derivative:** If \( f'(x) > 0 \), then the function is increasing on that interval. This implies that as you move along the x-axis, the values of \( f(x) \) are getting larger.- **Negative Derivative:** If \( f'(x) < 0 \), then the function is decreasing in that interval. Here, the values of \( f(x) \) reduce as you proceed along the x-axis.
By identifying these intervals using the computed derivative, we can effectively map out where the function's graph rises and falls, offering a richer understanding of its overall behavior. Comparing these intervals with the function graph allows us to visually confirm these increasing and decreasing trends.