Problem 3 - Calculus: An Integrated Approach to Functions and Their Rates of Change

Question

Let \(f(t)=\sin t\). Let \(A_{f}(x)=\int_{0}^{x} \sin t d t\), where \(x \geq 0\). (a) Put the following in ascending order, with \(<\) or \(=\) signs between them. $$ A_{f}(0), \quad A_{f}\left(\frac{\pi}{2}\right), \quad A_{f}(\pi), \quad A_{f}\left(\frac{3 \pi}{2}\right), \quad A_{f}(2 \pi) $$ (b) For what values of \(x\) is \(A_{f}(x)=0\) ? (c) For what values of \(x\) is \(A_{f}(x)\) negative? (d) Fow what values of \(x\) is \(A_{f}(x)\) maximum?

Step-by-Step Solution

Answer

(a) \(A_{f}(0) = A_{f}(2\pi) = A_{f}(\pi) < A_{f}(\frac{3\pi}{2}) < A_{f}(\frac{\pi}{2})\), (b) \(x = n\pi, n\in \mathbb{Z}\), (c) \(x \in (n\pi , (n+1)\pi), n\) is an odd integer, (d) \(x=\frac{\pi}{2} + 2n\pi, n\in \mathbb{Z}\)

Step 1: Evaluate Integral at Different Points

Evaluate the integral \(A_{f}(x)\) at the given points 0, \(\pi/2\), \(\pi\), \(3\pi /2\) and \(2\pi\).

Step 2: Arrangement in Ascending Order

Arrange the evaluated integral values in ascending order.

Step 3: Find \(x\) for Which \(A_{f}(x)=0\)

Analyze the sine function and its period properties to find out when the area from 0 to \(x\) is 0.

Step 4: Find Range of \(x\) Making \(A_{f}(x)\) Negative

Determine the intervals of \(x\) that make the integral negative, considering the positive and negative cycles of the sine function.

Step 5: Find Maximum of \(A_{f}(x)\)

Determine the \(x\) value where \(A_{f}(x)\) reaches its maximum by analyzing where the sine function transitions from positive to negative.

Key Concepts

Definite Integral

A definite integral represents the area under the curve of a function between two given points on the x-axis. In the context of the example problem, the definite integral of the sine function, denoted as \( A_f(x) \) from 0 to \( x \), calculates the total area between the sine curve and the x-axis from 0 to \( x \). It's important to note that if the curve dips below the x-axis, as the sine function does periodically, the integral accounts for this as a negative area. Hence, the value of a definite integral can inform us about whether the function is mostly above or below the x-axis in the specified interval.

When determining \( A_f(x) \) for specific values of \( x \) in a problem, students are required to evaluate the integral at each point and compare results to understand the relationship between these values. Understanding how to correctly compute and interpret definite integrals is a crucial skill in calculus, with applications across physics, engineering, and many other fields.

Sine Function Properties

The sine function, \( \sin(t) \), is a fundamental trigonometric function known for its characteristic wave-like pattern. It oscillates between a maximum value of 1 and a minimum value of -1, with a period of \( 2\pi \) radians, meaning it repeats its values every \( 2\pi \) radians along the t-axis.

When working with the sine function, especially when performing integrations like \( A_f(x) \) from our example problem, it's essential to understand its properties. In the context of definite integrals, these oscillations imply that integrating over a full period (\( 0 \text{ to } 2\pi \) or any interval of \( 2\pi \) units) results in zero because the positive and negative areas cancel out. These repeating properties are highly predictable and can be used strategically to simplify calculations and predict behaviors of the function over different intervals.

Periodicity of Trigonometric Functions

The periodicity of trigonometric functions, such as sine, means that they repeat their values in regular intervals. For the sine function, its periodicity is specifically \( 2\pi \) radians. Understanding this is crucial when solving problems involving trigonometric integrals.

In relation to our example problem, periodicity tells us precisely when the integral \( A_f(x) \) would return to a value of zero - whenever we complete a full cycle of \( 2\pi \) radians. It also sheds light on the behavior of the function over multiples of its period, which is particularly useful when tackling questions related to integral values over intervals that span several periods of the function. This property greatly helps in predicting and optimizing the points at which the integrated sine function will produce certain values, such as zero or maximum areas.

Optimization of Integrals

The optimization of integrals involves identifying values or conditions under which the integral of a function reaches its extreme values – maximums or minimums. In the context of \( A_f(x) \) and the sine function, this means looking for points at which the area under the function is at its largest (or smallest, though in our exercise we focus on the maximum).

Due to the periodic nature of the sine function, we can exploit its predictable maxima and minima. For example, \( A_f(x) \) is maximized just before the sine function begins a negative cycle, because at this point, it has integrated the largest possible positive area without subtracting any of it. It's a foundational concept in calculus that can be extended to a wide variety of functions beyond trigonometric ones, aiding in solving real-world problems where the optimum value of an integral is sought for practical applications.