Problem 74 - Thomas Calculus

Question

In Exercises \(71-74,\) let \(F(x)=\int_{a}^{x} f(t) d t\) for the specified function \(f\) and interval \([a, b] .\) Use a CAS to perform the following steps and answer the questions posed. a. Plot the functions \(f\) and \(F\) together over \([a, b]\) b. Solve the equation \(F^{\prime}(x)=0 .\) What can you see to be true about the graphs of \(f\) and \(F\) at points where \(F^{\prime}(x)=0 ?\) Is your observation borne out by Part 1 of the Fundamental Theorem coupled with information provided by the first derivative? Explain your answer. c. Over what intervals (approximately) is the function \(F\) increasing and decreasing? What is true about \(f\) over those intervals? d. Calculate the derivative \(f^{\prime}\) and plot it together with \(F .\) What can you see to be true about the graph of \(F\) at points where \(f^{\prime}(x)=0 ?\) Is your observation borne out by Part 1 of the Fundamental Theorem? Explain your answer. $$ f(x)=x \cos \pi x, \quad[0,2 \pi] $$

Step-by-Step Solution

Answer

1. Use CAS to plot f and F. 2. Solve F'=0; observe horizontal tangents. 3. Use sign of f for intervals. 4. Differentiate f, plot f'. 5. Relate back to the Fundamental Theorem.

Step 1: Plot Functions

Use a CAS (Computer Algebra System) tool to plot the functions. Given: \( f(x) = x \cos(\pi x) \) over \([0, 2\pi]\).Calculate \( F(x) = \int_{0}^{x} t \cos(\pi t)\, dt \) using a CAS.Plot \( f(x) \) and the computed function \( F(x) \) over the interval \([0, 2\pi]\).By plotting, you should observe how the integral function \( F(x) \) behaves compared to \( f(x) \).

Step 2: Solve Derivative Equation

Find \( F'(x) \) by using the Fundamental Theorem of Calculus, which states \( F'(x) = f(x) \).Solve \( F'(x) = 0 \) for \( x \) by finding the roots of \( f(x) = 0 \) in the interval \([0, 2\pi]\).Observe that where \( F'(x) = 0 \), \( f(x) \) will intersect the x-axis. According to the Fundamental Theorem, at these points \( F(x) \) should have horizontal tangents.

Step 3: Determine Increasing/Decreasing Intervals

To find intervals where \( F(x) \) is increasing, determine where \( F'(x) > 0 \). Conversely, find where \( F'(x) < 0 \) for decreasing intervals.These intervals are where \( f(x) > 0 \) (increasing) or \( f(x) < 0 \) (decreasing) respectively, since \( F'(x) = f(x) \).Check these results by comparing the graph of \( f(x) \) identified in Step 1 with these intervals.

Step 4: Calculate Derivative of f

Find \( f'(x) \) by differentiating \( f(x) = x \cos(\pi x) \) using the product rule:\[ f'(x) = \cos(\pi x) - \pi x\sin(\pi x) \].Plot \( f'(x) \) along with the integral function \( F(x) \).Observe points where \( f'(x) = 0 \). These correspond to points where \( F(x) \) might have extrema. This works alongside the Fundamental Theorem, indicating critical points of \( F \).

Step 5: Fundamental Theorem Observations

Confirm that parts of the Fundamental Theorem of Calculus support your observations:1. \( F'(x) = f(x) \) gives the relationship between the graph intersections and tangents.2. Where \( f'(x) = 0 \), it indicates critical points of \( F(x) \), showing local maxima or minima in \( F \).Check your findings against expected behavior of derivative sign changes and graph intersections.

Key Concepts

Integral Function

In mathematics, an integral function is central to the Fundamental Theorem of Calculus. In this case, the integral function is represented as \( F(x) = \int_{a}^{x} f(t) \, dt \). This function essentially accumulates the area under the curve of \( f(t) \) from a starting point \( a \) to a variable endpoint \( x \).
The integral function \( F(x) \) provides valuable insight into the behavior of the original function \( f(x) \). For instance, by plotting the function \( f(x) = x \cos(\pi x) \) and its integral \( F(x) \), we can visualize how the area under \( f(x) \) is summed to create \( F(x) \). Observing these plots helps us understand the nature of \( F(x) \) as a smooth and continuous reflection of the changes in \( f(x) \).
By understanding integral functions, students can better grasp how calculus bridges the gap between geometry, algebra, and real-world applications.

Derivative

The derivative of a function illustrates the rate at which it changes, acting as a cornerstone of calculus. In the context of our exercise, the Fundamental Theorem of Calculus tells us that \( F'(x) = f(x) \).
This means that the derivative of the integral function \( F(x) \) is precisely the function \( f(x) \) itself. It implies that wherever the derivative \( F'(x) \) equates to zero, it's an indicator of a horizontal tangent in \( F(x) \).
By solving \( F'(x) = 0 \), we find points where \( f(x) = 0 \). These are critical because they highlight where \( F(x) \) transitions smoothly, potentially from increasing to decreasing, indicating a maximum or minimum point.

• Horizontal tangents suggest that at these points, \( F(x) \) might have flat spots or turning points.
• Through calculating derivatives, we can track changes in the slope of functions efficiently.

Critical Points

Critical points occur where the derivative of a function equals zero or is undefined, leading to potential maxima, minima, or points of inflection. To identify critical points in our function \( F(x) \), we solve \( F'(x) = 0 \).
In the exercise, finding \( F'(x) = f(x) = 0 \) gives us critical points of \( F(x) \), meaning points on \( f(x) \) where it intersects the x-axis. At these points, \( F(x) \) either stops increasing or decreasing and could potentially indicate extremas.

• Local maxima occur where \( F(x) \) moves from increasing to decreasing.
• Local minima occur where \( F(x) \) transitions from decreasing to increasing.

This understanding aids in predicting the behavior of functions during analysis and is a crucial skill in mathematics and scientific computing.

Increasing and Decreasing Intervals

Functions can be increasing or decreasing over specific intervals. Understanding these intervals provides invaluable insights into the behavior and properties of a function.
Given that \( F'(x) = f(x) \), the function \( F(x) \) is increasing wherever \( f(x) > 0 \) and decreasing wherever \( f(x) < 0 \). This is because the sign of \( F'(x) \) indicates the slope of \( F \); a positive slope means climbing, while a negative slope means falling.
By plotting \( f(x) \) and identifying where it is above or below the x-axis, we define where \( F(x) \) increases and decreases. Visualizing both plots together can effectively highlight:

• Segments of the graph where \( F(x) \) climbs upward (\( f(x) > 0 \)).
• Segments where \( F(x) \) descends (\( f(x) < 0 \)).

This relationship is critical for identifying and interpreting functional behavior in calculus and beyond.