Problem 37 - College Algebra with Corequisite Support

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

Use derivative and graphing to identify critical points and intervals of increase/decrease.

Step 1: Rewrite the function

First, recognize that the function can be rewritten in a different form to help analyze it. The function \(g(t) = t \sqrt{t + 3}\) is equivalent to \(g(t) = t (t + 3)^{1/2}\). This form helps when taking derivatives.

Step 2: Find the first derivative

To identify where the function is increasing or decreasing, calculate the first derivative \(g'(t)\). Use the product rule and chain rule: \[g'(t) = \frac{d}{dt}[t] \cdot (t + 3)^{1/2} + t \cdot \frac{1}{2}(t+3)^{-1/2}\cdot 1 \]. Simplifying gives \(g'(t) = (t + 3)^{1/2} + \frac{t}{2\sqrt{t+3}} \). Further simplification may be needed for analysis.

Step 3: Set the first derivative to zero

To find critical points, set the first derivative \(g'(t)\) equal to zero and solve for \(t\). Determine where \((t+3)^{1/2} + \frac{t}{2\sqrt{t+3}} = 0\). This equation will help locate local extrema.

Step 4: Test intervals around critical points

Using a sign chart or another method, test the intervals around the critical points found in Step 3 to determine where \(g(t)\) is increasing or decreasing. Check the sign of \(g'(t)\) in each interval. If \(g'(t) > 0\), the function is increasing; if \(g'(t) < 0\), it is decreasing.

Step 5: Use a graphing utility

Plot \(g(t) = t \sqrt{t+3}\) using a graphing calculator or software to estimate the local extrema and verify your solutions from the derivative calculations. Observe where the function changes direction to identify local maxima and minima.

Step 6: Compile results

Document the intervals of increase and decrease, and estimate the coordinates of the local extrema based on the graph. Specify both the intervals and the turning points determined from the derivative and graphing analysis.

Key Concepts

Local Extrema

Local extrema refer to the points on a graph where a function reaches a local maximum or minimum value. At these points, the function changes direction, making them critical for analyzing the behavior of a graph. In the case of the function \(g(t) = t \sqrt{t+3}\), understanding the local extrema helps us understand where the function peaks or dips within specific intervals.

To find local extrema:

• Identify critical points by setting the first derivative to zero.
• Evaluate these points using additional tests or a graphing utility to determine their nature (maximum or minimum).
• Observe the graph for turning points where the function shifts from increasing to decreasing and vice versa.

Understanding local extrema can provide valuable insights into real-world applications, such as finding optimal values within certain contexts.

First Derivative

The first derivative of a function, often denoted as \(f'(x)\), describes the rate of change or the slope of the function. For \(g(t) = t \sqrt{t+3}\), the first derivative, \(g'(t)\), shows how \(g(t)\) changes with respect to \(t\).

Using calculus, we calculate the first derivative to explore critical points and intervals. This helps decide whether those intervals reflect increasing or decreasing trends. For instance:

• Use the product rule and chain rule to derive \(g'(t)\).
• Simplify the expression to conduct further analysis.

Knowing how to compute and interpret the first derivative is essential for examining the behavior and attributes of any function.

Increasing and Decreasing Intervals

Increasing and decreasing intervals show us where a function continuously rises or falls. These intervals are determined by analyzing the first derivative. For the function \(g(t) = t \sqrt{t+3}\), identifying such intervals aids in understanding how the function behaves globally.

When \(g'(t) > 0\), the function is increasing, while \(g'(t) < 0\) indicates a decreasing function. This is determined by:

• Finding the sign of \(g'(t)\) in different intervals.
• Using critical points as boundaries for these intervals.

Graphically, these intervals can often be observed directly, making them easier to estimate with a graphing utility. Having a clear view of these intervals allows us to predict the function's behavior across its entire domain.

Graphing Utility

Graphing utilities are powerful tools used to visualize mathematical functions, making concepts like local extrema and intervals much clearer. By plotting \(g(t) = t \sqrt{t+3}\) on a graphing calculator or software:

• Estimate local maxima and minima by observing where the graph changes direction.
• Visually confirm calculated intervals of increase and decrease.
• Detect any unexpected behavior that might not be obvious from analytical methods alone.

Using technology in mathematics enhances our understanding and gives us a reliable picture of the function's behavior.

Critical Points

Critical points are values of \(t\) where the first derivative \(g'(t)\) is either zero or undefined, and they can potentially indicate where the local extrema occur. For the function \(g(t) = t \sqrt{t+3}\):

- Set \(g'(t) = 0\) to solve for these points.- Determine if each critical point is a maximum, minimum, or saddle point by analyzing surrounding intervals or using the second derivative test.
Recognition of critical points is pivotal when analyzing and graphing a function, as they form the basis for many subsequent analyses, such as determining behavior in different intervals.