Problem 18 - Thomas Calculus in SI Units

Question

Sketch the graph of each function and determine whether the function has any absolute extreme values on its domain. Explain how your answer is consistent with Theorem 1. $$h(x)=\left\\{\begin{array}{ll} \frac{1}{x}, & -1 \leq x < 0 \\ \sqrt{x}, & 0 \leq x \leq 4 \end{array}\right.$$

Step-by-Step Solution

Answer

The function has an absolute minimum of 0 at \(x=0\) and an absolute maximum of 2 at \(x=4\).

Step 1: Analyze the Piecewise Function

The given function \( h(x) \) is a piecewise function. For \( -1 \leq x < 0 \), it is defined as \( \frac{1}{x} \), which is a hyperbolic shape that increases negatively toward infinity as \( x \) approaches 0. For \( 0 \leq x \leq 4 \), it is defined as \( \sqrt{x} \), which is a square root function beginning at \( (0, 0) \) and increasing to \( (4, 2) \).

Step 2: Sketch the Graph

On a coordinate plane, sketch the graph of \( h(x) \). Start by plotting the curve \( \frac{1}{x} \) for \( -1 \leq x < 0 \), which will asymptotically approach the y-axis without touching it. Then plot \( \sqrt{x} \) from \( (0, 0) \) to \( (4, 2) \), forming a curved line that starts from the origin and curves upward to the point \( (4, 2) \). The graph does not define \( h(0) \) for \( \frac{1}{x} \); thus, there is a gap at \( x = 0 \).

Step 3: Identify Absolute Extreme Values

For the subdomain \( -1 \leq x < 0 \), note that \( \frac{1}{x} \) approaches negative infinity as \( x \to 0^- \). There is no maximum or minimum here. For \( 0 \leq x \leq 4 \), \( \sqrt{x} \) has an absolute minimum at \( x = 0 \) where \( h(0) = 0 \) and an absolute maximum at \( x = 4 \) where \( h(4) = 2 \). Thus, in the domain as a whole, the function has an absolute minimum of 0 at \( x=0 \) and an absolute maximum of 2 at \( x=4 \).

Step 4: Consistency with Theorem 1

Theorem 1 states that if a function is continuous on a closed interval, then it must have both a maximum and a minimum on that interval. \( h(x) \) is continuous on its closed subdomain \( [0, 4] \), and it does attain absolute extreme values there. However, \( h(x) \) is not continuous over the entire domain \([-1, 4]\) due to the discontinuity at \( x = 0^{-} \), which explains the absence of maximum in that portion.

Key Concepts

Hyperbolic Functions

Hyperbolic functions arise in various mathematical contexts, defined similarly to traditional trigonometric functions but for the hyperbola. A hyperbolic function can appear in a piecewise function like in our analysis of the function \( h(x) \). In this case, we have \( f(x) = \frac{1}{x} \) as our hyperbolic component, which behaves characteristically with certain features:

• As \( x \) approaches 0 from the negative side, \( \frac{1}{x} \) sharply decreases, moving toward negative infinity.
• This creates a vertical asymptote at \( x = 0 \), meaning the graph comes infinitely close to the axis but never actually reaches it.
• This feature is typical for hyperbolic functions, which often describe hyperbolic growth or decay, common in physics and engineering.

Understanding these aspects of hyperbolic functions helps anticipate how they will behave, especially in piecewise contexts where multiple functions come together. By knowing that the graph approaches infinitely without touching a point, you can properly sketch and analyze functions that use this expression.

Square Root Functions

Square root functions are another elementary function form, often paired in exercises like these to demonstrate different growth behaviors. In the piecewise function provided, the segment \( \sqrt{x} \) covers the interval \( 0 \leq x \leq 4 \):

• Starting from the origin (0,0), the graph of \( \sqrt{x} \) shows a gentle curve upwards, reaching the point (4,2).
• Each value of \( x \) in its domain corresponds to a non-negative \( y \)-value, making this function strictly increasing.
• The gentle curve is an essential characteristic of square root functions, representing slower growth rates compared to linear ones.

The square root function's graph is critical in understanding how continuous functions behave and why they maintain a smooth aspect—free of abrupt breaks or leaps. In the context of absolute extreme values, which further clarify the graph’s behavior, the square root segment provides a crucial point to evaluate: it confidently reaches defined minimum and maximum values needed in continuous contexts.

Absolute Extreme Values

Absolute extreme values are important in analyzing a function's behavior within its domain—pinpointing where the function hits its highest and lowest points. For this piecewise function:

• The absolute minimum occurs at \( x = 0 \) with \( h(0) = 0 \), marking where the function hits its lowest point on the graph.
• The absolute maximum happens at \( x = 4 \) with \( h(4) = 2 \), noting the highest point.
• In detailing this, we're consistent with Theorem 1, which states there must be a max and min in continuous intervals.

However, it's important to mention that not all sections contribute to these extremes: the hyperbolic section \( ( -1 \leq x < 0 ) \) approaches negative infinity and does not reach an absolute extreme. Recognizing this helps in understanding and predicting behavior of piecewise functions particularly at boundaries where conditions or changes occur, emphasizing the role of continuity in determining extremal points.