Problem 13 - Calculus: An Integrated Approach to Functions and Their Rates of Change

Question

Let \(h(x)=\int_{0}^{x} \sin \left(t^{2}\right) d t\). (a) Graph \(\sin \left(t^{2}\right)\) on the domain \([-3,3]\). (b) On \((0, \infty)\), where is \(h(x)\) positive? Negative? (c) Is \(h(x)\) even, odd, or neither? Explain. (d) On \([0,3]\) where is \(h(x)\) increasing? Decreasing? Give exact answers. (e) What is the absolute maximum value of \(\sin \left(t^{2}\right)\) ? What is the absolute minimum value of \(\sin \left(t^{2}\right) ?\) (f) Where on \([0,3]\) does \(h\) attain its maximum and minimum values? Will your answers change if the domain is \((0, \infty)\) ? If the domain is \((-\infty, \infty)\) ? (g) Numerically approximate the maximum value of \(h\) on \([0,3]\).

Step-by-Step Solution

Answer

Graph of \(\sin(t^2)\) is a wave between -1 and 1. \(h(x)\) is positive where \(\sin(t^2)\) is greater than 0 and vice versa. \(h(x)\) cannot be classified as even or odd. \(h(x)\) increases where \(\sin(t^2)\) is positive and decreases where \(\sin(t^2)\) is negative. The absolute maximum and minimum values of \(\sin(t^2)\) are 1 and -1 respectively and maximum and minimum values of \(h(x)\) requires numerical approximation.

Step 1: Graph of \( \sin(t^2) \)

Use any Mathematics software or graphing calculator to plot the graph of \( \sin(t^2) \). You will notice that it is a wave that oscillates between -1 and 1.

Step 2: Positive and Negative Intervals of \( h(x) \)

As the integrand, \( \sin(t^2) \), oscillates between -1 and 1, it would produce a positive area under the curve when \( \sin(t^2) \) is greater than 0 and a negative area when less than 0. Note that \( h(x) \) is only defined for \( x >= 0 \).

Step 3: Nature of the function: Even, Odd or Neither

An even function satisfies \( f(-x) = f(x) \) and an odd function satisfies \( f(-x) = -f(x) \). The function \( h(x) \) is neither even nor odd because it is not defined for negative values of x.

Step 4: Increasing and Decreasing Intervals of \( h(x) \)

When the rate of change of a function is positive, the function is increasing. When the rate of change is negative, the function is decreasing. So we could infer that \( h(x) \) is increasing when \( \sin(t^2) \) is positive and decreasing when \( \sin(t^2) \) is negative.

Step 5: Absolute Maximum and Minimum Values of \( \sin(t^2) \)

For any \( t \), \( -1 \leq \sin(t^2) \leq 1 \). So, the absolute maximum value is 1, and the absolute minimum value is -1.

Step 6: Maximum and Minimum Values of \( h(x) \)

Since \( h(x) \) is a continuous function on the closed interval [0,3], by the Extreme Value Theorem, \( h(x) \) attains both maximum and minimum values. However, it's not immediately clear where those values occur. We would need to do some numerical analysis or approximations using Calculus or computer system.

Step 7: Numerical Approximation

We can numerically estimate the maximum value of \( h(x) \) on [0,3] using numerical methods or using any Mathematics Software.

Key Concepts

Graphing Functions

Understanding the graphical representation of functions is essential in calculus to visualize behavior and characteristics such as continuity, limits, and extrema. For instance, graphing the function \( \sin(t^2) \) on the domain [-3,3] involves plotting its values at various points within that interval. This sine function is unique because it oscillates more quickly as the value of \( t \) grows due to the squaring operation. A graphing calculator or software will show that this function produces a wave-like pattern that varies between -1 and 1.

When graphing functions, it's crucial to pay attention to the scale and the axes to ensure that the behavior of the function is accurately captured, particularly in regions where the function's rate of change increases, as with \( \sin(t^2) \). This graph lays the foundation for understanding how the integral of the function behaves, which emphasizes the integral role graphing plays in calculus.

Integration Concepts

Understanding integration is a cornerstone of calculus, especially when it comes to determining the area under a curve. In our exercise, the function \(h(x) = \int_{0}^{x} \sin(t^2) dt\) represents the accumulated area under the curve of the function \( \sin(t^2) \) from 0 to \(x\). As such, when the value of \( \sin(t^2) \) is positive, \(h(x)\) is accumulating positive area, making \(h(x)\) positive. Conversely, when \( \sin(t^2) \) is negative, \(h(x)\) is accumulating negative area, which in turn makes \(h(x)\) negative.

Integration can also inform us about the average value of functions over intervals and the total accumulation of quantities, such as distance traveled over time. The integral function \(h(x)\), given by the integral of \( \sin(t^2) \), showcases these concepts clearly, particularly when investigating the behavior of \(h(x)\) on different intervals like \( (0, \infty) \).

Function Behavior Analysis

Analyzing a function's behavior involves studying its symmetry and its increasing or decreasing nature over specific intervals. For the function \(h(x)\), since it is not symmetric with respect to the y-axis (not defined for \(x < 0\)), it is neither even nor odd. Understanding the symmetrical properties of functions is useful in simplifying calculations, predicting behavior, and confirming function identity.

Moreover, \(h(x)\) increases when the rate of change is positive—this occurs when \( \sin(t^2) \) is positive—indicating a positive slope. Likewise, \(h(x)\) decreases when the rate of change is negative. Identifying where the function is increasing or decreasing is vital for various applications, such as optimizing functions or solving real-world problems where such behavior is critical. Precise analysis of function behavior, both graphically and numerically, empowers students to derive meaningful conclusions about the behavior of more complex functions.

Extreme Value Theorem

The Extreme Value Theorem guarantees that a continuous function on a closed interval will have both maximum and minimum values on that interval. When applied to \(h(x)\), we know that there must exist absolute maximum and minimum values on the interval [0,3]. This theorem not only assures the existence of extrema but also provides a foundation for finding them, for example, through critical point analysis or numerical approximation.

For the function \( \sin(t^2) \) itself, we can immediately identify the absolute maximum and minimum values, which are 1 and -1, respectively. Yet, finding where \(h(x)\) reaches its maximum and minimum on [0,3] can be challenging without further analysis. If the domain changes, the location of the extrema can shift or even disappear if the function extends infinitely in any direction. In many cases, numerical methods or calculus techniques are applied to approximate these values, elucidating the importance of the Extreme Value Theorem in practical calculus problems.