Problem 73 - 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. $$ f(x)=\sin 2 x \cos \frac{x}{3}, \quad[0,2 \pi] $$

Step-by-Step Solution

Answer

Plot and analyze functions; \( F'(x)=0 \) where \( f(x)=0 \); \( F \) increases/decreases with \( f \); \( f'(x)=0 \) marks changes in \( f \).

Step 1: Plotting Functions f and F

We start by defining the function \( f(x) = \sin(2x) \cos\left(\frac{x}{3}\right) \). Next, we define \( F(x) = \int_0^x f(t) \, dt \). Using a computational tool, plot both functions over the interval \([0, 2\pi]\). Observe that \( f(x) \) oscillates due to the sine and cosine components, and \( F(x) \), being the integral of \( f \), reflects the cumulative area under the curve of \( f \).

Step 2: Solving for F'(x)=0

Using the Fundamental Theorem of Calculus, we know \( F'(x) = f(x) \). To solve \( F'(x) = 0 \), we find the roots of \( f(x) = 0 \). These are the values of \( x \) where the graph of \( f(x) \) crosses the x-axis. Observe that at these points, the graph of \( F(x) \) has horizontal tangent lines (local maxima, minima, or inflection points).

Step 3: Analysis of Increasing/Decreasing Intervals

Determine where \( F(x) \) is increasing and decreasing by examining \( f(x) \). \( F(x) \) is increasing where \( f(x) > 0 \) and decreasing where \( f(x) < 0 \). Use the plots to identify these intervals, understanding that \( F(x) \) increases as the area under the curve of \( f(x) \) is added positively, and decreases as area is subtracted (when \( f(x) \) is negative).

Step 4: Calculating and Plotting f'(x)

Calculate the derivative \( f'(x) \) using the product rule: \[ f'(x) = 2 \cos(2x) \cos\left(\frac{x}{3}\right) - \frac{1}{3} \sin(2x) \sin\left(\frac{x}{3}\right) \]. Plot \( f'(x) \) along with \( F(x) \). Observe that where \( f'(x) = 0 \), the slope of \( f(x) \) changes, indicating potential points of inflection or critical points for \( f(x) \), which can affect the concavity of \( F(x) \) based on the accumulation from \( f \).

Key Concepts

Function Plotting

Function plotting is an essential tool to visually understand how a function behaves across its domain. In our original exercise, we are given the function

• \( f(x) = \sin(2x) \cos\left(\frac{x}{3}\right) \)

which is defined over the interval \([0, 2\pi]\). This particular function exhibits oscillatory behavior due to the sine and cosine components. To better analyze this, we plot both the function \( f(x) \) and its associated integral function \( F(x) \), which is defined as:

• \( F(x) = \int_0^x f(t) \, dt \)

Plotting these functions allows us to clearly see how the oscillations in \( f(x) \) translate into the cumulative behavior shown by \( F(x) \), as it reflects the area under the curve of \( f(x) \). When you observe the plot, \( F(x) \) will typically have smoother transitions compared to the often rapid oscillations shown by \( f(x) \), giving insight into the nature of accumulation over the interval.

Derivative Analysis

Derivative analysis involves understanding the rate of change of functions. Here, we primarily focus on the relationship between \( f(x) \) and its integral \( F(x) \), where \( F'(x) = f(x) \) as derived from the Fundamental Theorem of Calculus. To solve for \( F'(x) = 0 \), we identify the roots of \( f(x) = 0 \). These roots represent the points where the integral function \( F(x) \) has a horizontal tangent, indicating it is neither increasing nor decreasing.

What we observe is that at these points, where \( f(x) \) crosses the x-axis, \( F(x) \) experiences local extrema—either a peak or a trough. Therefore, the sign of \( f(x) \) around these roots can tell us about the intervals over which \( F(x) \) increases (\( f(x) > 0 \)) or decreases (\( f(x) < 0 \)). This derivative analysis is crucial for identifying important characteristics of \( F(x) \) and understanding its tendency to form peaks or troughs.

Integral Functions

Integral functions like \( F(x) = \int_0^x f(t) \, dt \) are powerful in understanding how functions accumulate over a given interval. In our exercise, \( F(x) \) embodies the cumulative area under the curve of \( f(x) \). This function provides insight into how the small sections of \( f(x) \) unite to form the bigger picture.

When \( f(x) \) is positive over an interval, the integral accumulates positively, leading to an increasing \( F(x) \). Conversely, when \( f(x) \) is negative, \( F(x) \) decreases as its integral reflects the negative accumulation. Calculating integrals like \( F(x) \) simplifies understanding the relationship between immediate function values and the broader trends they cause over their domain, returning a deeper look at the overall picture of the function's behavior.

Oscillatory Functions

Oscillatory functions, such as \( f(x) = \sin(2x) \cos\left(\frac{x}{3}\right) \), have patterns of repeating behavior, moving between extremes in a regular interval. This behavior is often related to trigonometric components that can produce wave-like plots.

When working with oscillatory functions, it's important to pay attention to their amplitude and period. The amplitude is determined by the coefficient in front of the trigonometric function and reflects how high and low the function goes. The period, on the other hand, specifies how often the cycle repeats over the x-axis.

Understanding these functions is key for interpreting their effects on other calculations, like integrals and derivatives. Recognizing their periodic properties provides a reliable structure for analyzing other advanced mathematical functions and applications, making them a fundamental aspect of function analysis.