Problem 57 - Essential Calculus

Question

(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 indicated 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 \(f^{\prime}\). $$ f(t)=t^{2} \sin t,[0,2 \pi] $$

Step-by-Step Solution

Answer

The derivative of the function is \(f^{\prime}(t) = 2t\sin(t) + t^{2}\cos(t)\). The critical numbers and intervals where the derivative is positive or negative should be determined from the solution of \(f^{\prime}(t) = 0\). The behavior of \(f\) increases when \(f^{\prime}\) is positive, and \(f\) decreases when \(f^{\prime}\) is negative. These observations can be noticed from the graphs.

Step 1: Derive the Function

To derive the given function \(f(t) = t^{2} \sin(t)\) within a computer algebra system, the product rule will be used. The derivative is \(f^{\prime}(t) = 2t\sin(t) + t^{2}\cos(t)\).

Step 2: Sketch the Functions

Next, the graphs of \(f(t)\) and \(f^{\prime}(t)\) should be sketched over the same set of coordinate axes. Note that the critical numbers (when the derivative is zero) will help in identifying the maximum and minimum values on the graph, which helps in sketching the curve.

Step 3: Find the Critical Numbers

To find the critical numbers of the function, set \(f^{\prime}(t)\) equal to zero and solve for \(t\). From \(f^{\prime}(t) = 2t\sin(t) + t^{2}\cos(t)\), the critical numbers are obtained by solving \(2t\sin(t) + t^{2}\cos(t) = 0\).

Step 4: Determine Intervals where \(f^{\prime}\) is Positive or Negative

To determine where the derivative of the function is positive or negative, make a sign test chart. Start by marking the critical numbers on the line. Then test numbers from each interval on the derivative equation to see whether the derivative is positive or negative.

Step 5: Analyze Comparison between \(f\) and \(f^{\prime}\)

Finally, compare the behavior of the function \(f\) and its derivative \(\fcrep\). When the derivative is positive, \(f\) should be increasing. When the derivative is negative, \(f\) should be decreasing. This can be analyzed from the graphs of \(f\) and \(f^{\prime}\) that were sketched in step 2.

Key Concepts

Critical Numbers in Calculus

Critical numbers play a crucial role in calculus. They are points in the domain of a function where the derivative is either zero or undefined. To find these critical numbers for a function like

• Start by differentiating the function to get its first derivative.
• Set the derivative equal to zero.
• Solve for the variable to find the critical numbers.

In the original exercise for the function \(f(t) = t^{2} \sin(t)\), the derivative was found to be \(f^{\prime}(t) = 2t\sin(t) + t^{2}\cos(t)\). Setting this equal to zero gives you an equation to solve for \(t\), which determines the critical numbers. These points help in understanding the graph's behavior, pinpointing where the function may have relative maxima, minima, or saddle points.
Finding critical numbers is vital as they indicate where a function changes its increasing or decreasing trend, aiding in sketching its graph accurately.

Using a Computer Algebra System for Differentiation

A Computer Algebra System (CAS) streamlines and enhances the differentiation process by computing all necessary mathematical steps nearly instantaneously. In mathematical problems and exercises, particularly those involving derivatives, using a CAS like Mathematica, Maple, or MATLAB proves beneficial. It helps in ensuring:

• Accuracy: Reduces chances of manual error.
• Efficiency: Quickly computes complex derivatives.
• Insight: Visualizes results, such as derivative functions.

For example, differentiating the function \(f(t) = t^{2} \sin(t)\), manually involves applying the product rule. A CAS simplifies this, producing the derivative \(f^{\prime}(t) = 2t\sin(t) + t^{2}\cos(t)\) accurately and promptly.
By utilizing a CAS, students can focus more on understanding concepts rather than becoming bogged down by intricate calculations. It also assists in corroborating manually derived solutions, providing an additional layer of verification for mathematical results.

Interval Analysis in Calculus

Interval analysis is a valuable process to examine the behavior of functions, especially concerning their derivatives. After determining the critical numbers of a function, the next step is to analyze intervals formed by these critical points.The original problem involves analyzing where the derivative \(f^{\prime}(t) = 2t\sin(t) + t^{2}\cos(t)\) is positive or negative. Interval analysis helps accomplish this through these steps:

• Establish the critical numbers as boundaries for the intervals.
• Choose test points within each interval to substitute into the derivative equation.
• Observe if the derivative's sign is positive or negative in each segment.

This positive or negative indication informs you whether the function is increasing or decreasing in that interval.
For instance, pinpointing intervals where the derivative is positive indicates that the function \(f(t)\) is increasing. Conversely, negative intervals show where \(f(t)\) is decreasing. This technique is crucial for fully understanding the shape and behavior of the graph of a function over specific ranges.