Problem 24 - Thomas Calculus

Question

In Exercises \(15-30\) , find the absolute maximum and minimum values of each function on the given interval. Then graph the function. Identify the points on the graph where the absolute extrema occur, and include their coordinates. $$ g(x)=-\sqrt{5-x^{2}}, \quad-\sqrt{5} \leq x \leq 0 $$

Step-by-Step Solution

Answer

The absolute maximum is at \( (-\sqrt{5}, 0) \) and the absolute minimum is at \( (0, -\sqrt{5}) \).

Step 1: Understanding the Function

The function given is \( g(x) = -\sqrt{5 - x^2} \). This is a transformed square root function that describes a quarter of a circle positioned below the x-axis. The interval \(-\sqrt{5} \leq x \leq 0\) bounds the function.

Step 2: Determine Critical Points

To find the critical points, we first find the derivative of \( g(x) \). \[ g'(x) = \frac{d}{dx}[-\sqrt{5-x^2}] = \frac{-x}{\sqrt{5-x^2}} \]Set \( g'(x) = 0 \) and solve for \( x \):\[ \frac{-x}{\sqrt{5-x^2}} = 0 \longrightarrow x = 0 \]

Step 3: Evaluate Function at Endpoints and Critical Points

Evaluate \( g(x) \) at the critical point and the endpoints. \(g(-\sqrt{5}) = -\sqrt{5 - (-\sqrt{5})^2} = 0\)\(g(0) = -\sqrt{5 - 0^2} = -\sqrt{5}\)

Step 4: Identify Absolute Extrema

The absolute maximum is the largest value of the function and the absolute minimum is the smallest.- Maximum: \( g(-\sqrt{5}) = 0 \)- Minimum: \( g(0) = -\sqrt{5} \)

Step 5: Graph the Function and Identify Extrema

Graph \( y = -\sqrt{5-x^2} \) on the interval \( -\sqrt{5} \leq x \leq 0 \). This forms a semi-circular arc, with the endpoints at \((-\sqrt{5}, 0)\) and \((0, -\sqrt{5})\). The absolute maximum occurs at \((-\sqrt{5}, 0)\), and the absolute minimum occurs at \((0, -\sqrt{5})\).

Key Concepts

Critical Points

Understanding critical points is crucial for finding where a function's graph reaches its peaks or valleys, which are known as "extrema." These are the points where the function can potentially have a local minimum or maximum or neither. To find critical points, we start by calculating the derivative of the function. The critical points arise where this derivative is either zero or undefined. In our exercise, we have the function

• \( g(x) = -\sqrt{5 - x^2} \)

The derivative is

• \( g'(x) = \frac{-x}{\sqrt{5-x^2}} \)

Setting this derivative to zero helps us find the point where the slope of the tangent to the function is zero, such as

• \( x = 0 \)

Critical points are also checked in this interval to ensure no point where the derivative is undefined within physical constraints of the problem.

Derivative

The derivative is a fundamental concept in calculus that represents how a function changes as its input changes. Effectively, it measures the "rate of change" or the "slope" of the function at any given point. For our function,

• \( g(x) = -\sqrt{5 - x^2} \)

The derivative,

• \( g'(x) = \frac{-x}{\sqrt{5-x^2}} \)

helps us understand how the function behaves over the interval. By differentiating the function, we find

• where the function's slope is zero
• where it might reach maximum or minimum values within the interval

In this context, knowing where the derivative equals zero indicates potential points of extrema, which is pivotal for finding absolute maxima or minima.

Graphing Functions

Graphing functions allows you to visually interpret and comprehend the function's behavior, including important features like maxima, minima, and the general shape or trend of the graph. In our particular exercise, the function

• \( g(x) = -\sqrt{5 - x^2} \)

forms a semi-circle below the x-axis, bounded by the interval

• \(-\sqrt{5} \leq x \leq 0\)

When graphing:

• Mark the endpoints at \((-\sqrt{5}, 0)\) and \((0, -\sqrt{5})\)
• Identify where the absolute maximum \((x = -\sqrt{5})\)
• Identify where the absolute minimum \((x = 0)\) occur

Graphing not only supports how you determine where these extrema occur but also enriches your understanding of the function's geometry.

Interval Notation

Interval notation helps define the precise range within which we investigate a function and look for critical points or extrema. It provides a streamlined way to express domain or specific parts of a function that are important for analysis. In our function,

• \(g(x) = -\sqrt{5 - x^2}\)

we deal with the interval

• \([-\sqrt{5}, 0]\)

This tells us precisely where the function is to be evaluated for absolute maximum and minimum values. The square brackets in this notation indicate that both endpoints, \(-\sqrt{5}\) and \(0\), are included in our examination. Through interval notation, determining where to inspect a function's behavior becomes clear, ensuring no part of the domain is overlooked during analysis.