Problem 32 - Essential Calculus

Question

Locate the absolute extrema of the function (if any exist) over each interval. \(f(x)=\sqrt{4-x^{2}}\) (a) [-2,2] (b) [-2,0) (c) (-2,2) (d) [1,2)

Step-by-Step Solution

Answer

The absolute extrema for the given intervals are: (a) [-2, 2] maximum at f(0) = 2, minimum at f(-2) = f(2) = 0; (b) [-2, 0) maximum at f(0) = 2, minimum at f(-2) = 0; (c) (-2, 2) the maximum at f(0) = 2, no minimum; and (d) [1, 2) maximum at f(1) = \(\sqrt{3}\), minimum at f(2) = 0.

Step 1: Finding the first derivative

To find where the absolute extrema occurs, it's necessary to find where the derivative equals 0 or is undefined. The derivative of \(f(x)=\sqrt{4-x^{2}}\), using the chain rule, is \(f'(x) = -x/\sqrt{4-x^{2}}\).

Step 2: Finding the critical points

Set the derivative to 0 and solve for x, it's impossible because the fraction is 0 when the numerator is 0, but if \(x = 0\), the denominator is non-zero. Therefore, there's no real solutions, which means there's no critical points except the end points.

Step 3: Substituting end points and critical points into original function

Substitute the endpoints of the interval and the critical points (if exist) into the original function. For the interval [-2,2], the endpoints are x = -2, 2. Therefore, the maximum and minimum values are the maximum and minimum of f(-2), f(2), which is also the maximum and minimum of -\(\sqrt{4}\), \(\sqrt{4}\). Repeat this step for each of the intervals [(a), (b), (c), and (d)].

Step 4: Identifying the absolute extrema

The absolute extrema are the maximum and minimum values calculated in Step 3. For each interval, identify the smallest and largest values.

Key Concepts

Absolute Extrema

When we talk about absolute extrema, we are referring to the highest or lowest points the function achieves on a given interval. These points are found by plugging in specific x-values into the function. Absolute extrema are crucial because they help us understand the behavior of a function over a particular range.

• Absolute Maximum: Largest y-value on the interval
• Absolute Minimum: Smallest y-value on the interval

We find absolute extrema by evaluating both the endpoints of an interval and the critical points of the function. This is important because a function could achieve its highest or lowest value at the bounds or within the interval. Hence, including all possible cases ensures accuracy in finding the extrema.

Critical Points

Critical points are the x-values where a function's derivative equals zero or is undefined. These are crucial in determining a function's behavior because they are potential points where the function's direction changes, which may correspond to a local maximum or minimum.

• Set the derivative equal to zero and solve for x
• If the derivative does not exist at a point, that point is also a critical point

It's essential to check these points as they could reveal places where the function significantly changes its behavior, like reaching a peak or a valley. For the function \(f(x)=\sqrt{4-x^{2}}\), the critical point analysis showed that there are no critical points except the endpoints, due to the nature of the derivative.

Derivative

The derivative of a function gives us the slope or rate of change at any given point. In simple terms, it tells us how the function is behaving: increasing, decreasing, or staying flat. Derivatives are indispensable in calculus, especially when it comes to finding extrema points.

• Use derivative to check for increasing/decreasing functions
• Find points where derivative is zero for potential extrema

For the given function, the derivative is found using the chain rule, resulting in \(f'(x) = -\frac{x}{\sqrt{4-x^{2}}}\). This derivative helps identify potential critical points, which in turn help locate where the extrema might occur within a given interval.

Intervals

In calculus, an interval refers to a specific range of x-values over which we examine a function. Calculating extrema over different intervals can yield different results, emphasizing the importance of context when analyzing functions. An interval can be:

• Closed: Includes its endpoints, noted by brackets \([a, b]\)
• Open: Does not include endpoints, noted by parentheses \((a, b)\)
• Half-open: Includes only one endpoint, e.g., \([a, b)\) or \((a, b]\)

Each interval is unique in that the presence or absence of endpoints changes how we evaluate the function. For instance, an absolute maximum or minimum can occur at an endpoint, but this possibility vanishes when endpoints are excluded, such as in an open interval.