Problem 22 - Essential Calculus Early Transcendentals

Question

\(15-22=\) Sketch the graph of \(f\) by hand and use your sketch to find the absolute and local maximum and minimum values of \(f .\) (Use the graphs and transformations of Sections \(1.2 . )\) $$f(x)=\left\\{\begin{array}{ll}{4-x^{2}} & {\text { if }-2 \leqslant x<0} \\\ {2 x-1} & {\text { if } 0 \leqslant x \leqslant 2}\end{array}\right.$$

Step-by-Step Solution

Answer

Absolute maximum at (0,4); absolute minimum at (0,-1).

Step 1: Understand the Piecewise Function

The function \( f(x) \) is defined in two parts. For \( -2 \leq x < 0 \), \( f(x) = 4 - x^2 \), which is a downward-opening parabola. For \( 0 \leq x \leq 2 \), \( f(x) = 2x - 1 \), which is a linear function with a positive slope. Analyze each piece separately.

Step 2: Sketch the Graph of the First Segment

For the first segment, \( f(x) = 4 - x^2 \) for \( -2 \leq x < 0 \). This is a downward-opening parabola with a vertex at \( (0,4) \). Plot points such as \((-2,0)\) and see the curve form as \( x \to 0 \).

Step 3: Sketch the Graph of the Second Segment

For the second segment, \( f(x) = 2x - 1 \) for \( 0 \leq x \leq 2 \). Start at the point \((0, -1)\) since it passes through this point and moves linearly with a slope of 2 until \( x = 2 \) at \((2, 3)\). This is a straight line connecting these points.

Step 4: Identify Absolute and Local Extrema

Analyze both segments: The highest point (absolute maximum) is at the vertex of the parabola \( (0,4) \). The lowest value (absolute minimum) occurs at \( (0,-1) \). These transition points are also local extrema.

Key Concepts

Graph Sketching

When sketching the graph of a piecewise function like \( f(x) =\begin{cases} 4 - x^2 & \text{if} \ -2 \leq x < 0 \ 2x - 1 & \text{if} \ 0 \leq x \leq 2 \end{cases} \),it's helpful to break down each segment to understand its shape and behavior.

• First Segment: Consider the function \( 4 - x^2 \). This is a parabola opening downwards. The vertex, or the peak, of this parabola is at the point \( (0, 4) \). Plotting points such as \( (-2, 0) \) to \( (0, 4) \) will help in forming the curve.
• Second Segment: Next is the linear part, \( 2x - 1 \). This is a straight line that begins at \( (0, -1) \) with a positive slope of 2, going up to the point \( (2, 3) \).

Remember that a piecewise graph combines these separate sections together, so it's important to note any continuity or gaps between the segments. As you sketch, make sure to clearly label points and transitions to better visualize the function's behavior.

Local Extrema

Local extrema refer to the highest or lowest points in a specific section of the graph. For the given function, we need to consider each segment separately.

In the first piece, \( 4 - x^2 \), we have a local maximum at the vertex \( (0, 4) \), which is the highest point within the interval \([-2,0)\). This is because as \( x \) approaches zero from the left, \( f(x) \) increases towards this peak.

For the second part, \( 2x - 1 \), since it is a linear function that continuously increases, it does not have any local maxima or minima within \( [0, 2] \). However, the transition point at \( x = 0 \) where both parts connect might be considered when examining the overall behavior of the function.

Understanding local extrema helps in analyzing how the function behaves in small sections, giving insight into changes in direction or slope.

Absolute Maximum and Minimum Values

When identifying the absolute maximum and minimum values of a function, we look for the highest and lowest points across the entire domain.

In our example:

• Absolute Maximum: The highest value of the piecewise function occurs at \( (0, 4) \), which is the vertex of the parabolic segment. This is because no other point in the defined intervals exceeds the value of 4.
• Absolute Minimum: The lowest value can be found at the transition between segments, specifically at \( (0, -1) \). This point is part of the linear segment \( 2x - 1 \), and it marks the start of an increasing trend.

By analyzing both parts together, the absolute extrema tell us about the function's overall range and directional trends, ensuring we fully capture the function's highest and lowest outputs.