Problem 37 - College Algebra

Question

For the following exercises, use a graphing utility to estimate the local extrema of each function and to estimate the intervals on which the function is increasing and decreasing. $$g(t)=t \sqrt{t+3}$$

Step-by-Step Solution

Answer

Graphically: Local extrema can be estimated, and increasing and decreasing intervals identified.

Step 1: Understand the Function

First, let's understand the given function \(g(t) = t \sqrt{t+3}\). The function is composed of two parts: the linear term \(t\) and the square root \(\sqrt{t+3}\). This function is defined for \(t \geq -3\) as the expression under the square root must be non-negative.

Step 2: Graph the Function

Next, use a graphing utility to plot the function \(g(t) = t \sqrt{t+3}\). Observe the plot to identify possible turning points, which will help to estimate the local extrema and intervals where the function is increasing or decreasing.

Step 3: Identify the Local Extrema

From the graph plotted, visually determine where the peaks (local maxima) and valleys (local minima) occur. Use the graphing utility's options to obtain an estimated value for these points. Typically, these points occur where the slope of the tangent to the curve changes sign.

Step 4: Determine Intervals of Increase and Decrease

By analyzing the plotted graph, observe the sections where the function is rising or falling. The function is increasing where the graph goes upwards as you move from left to right, and decreasing where it goes downwards. Note these intervals based on the estimated turning points from Step 3.

Key Concepts

Increasing and Decreasing Intervals

Understanding the increasing and decreasing intervals of a function is crucial for interpreting its behavior over a domain. For the function \(g(t) = t \sqrt{t+3}\), these intervals can be seen clearly by observing the graph of the function.

- **Increasing Interval:** This is where the function's value goes up as \(t\) becomes larger. In this context, it means that the slope of the tangent is positive.
- **Decreasing Interval:** Conversely, this interval is where the function's value falls as \(t\) increases. Here, the tangent's slope is negative.

Once the graph is plotted using a graphing utility, you can easily visually identify where these intervals start and end by looking at the slope of the curve. Typically, the curve will rise to a peak (a local maximum) and then fall to a valley (a local minimum). Between these turning points, you can determine the increasing and decreasing intervals.

Graphing Utilities

Graphing utilities are powerful tools to help visualize complex functions like \(g(t) = t \sqrt{t+3}\). These tools can range from graphing calculators to software tools like Desmos or GeoGebra.

Here’s a simple way to utilize these tools efficiently:

• **Input the Function:** Start by entering the function in the graphing tool.
• **Adjust the View:** Ensure the window is correctly set to show important points like turning points and intercepts.
• **Analyze the Curve:** Use zooming and panning features to closely examine how the graph behaves.

Graphing utilities can also directly give you specific features of the graph, such as local extrema and critical points. These tools make it easy by eliminating the need for complex manual calculations.

Function Analysis

Function analysis involves understanding how a function behaves throughout its domain. For \(g(t) = t \sqrt{t+3}\), this means analyzing features like its endpoints, local extrema, intercepts, and discontinuities.

Some key points in analyzing \(g(t)\) include:

• **Domain:** The function is defined for \(t \geq -3\), so keep this in mind when plotting and analyzing.
• **Intercepts:** Find where the function crosses the axes by setting \(t\) and \(g(t)\) to zero individually.
• **Critical Points:** These are where the first derivative is zero or undefined. They’re vital for determining local maxima and minima.

By breaking down the function into these elements, we can understand not only where the function rises and falls but also any peculiar behavior at certain points. It empowers you to accurately describe the overall behavior of the function.