Problem 57 - Single Variable Calculus

Question

In Exercises \(57-60\), sketch the graph of a function having the given properties. $$ \begin{array}{l} f(0)=0, f^{\prime}(0)=1, f^{\prime \prime}(x)>0 \text { on }(-\infty, 0), f^{\prime \prime}(x)<0 \text { on } \\ (0, \infty), \lim _{x \rightarrow-x} f(x)=-1, \lim _{x \rightarrow x} f(x)=1 \end{array} $$

Step-by-Step Solution

Answer

To sketch the function with the given properties, follow these steps: 1. Begin at the point (0, 0) since \(f(0) = 0\). 2. Head upward as the function is increasing at x = 0 (\(f'(0) = 1\)). 3. Curve concave up on the interval \((-\infty, 0)\) (\(f''(x) > 0\)). 4. Curve concave down on the interval \((0, \infty)\) (\(f''(x) < 0\)). 5. As x approaches negative infinity, the function should approach the horizontal line at y = -1 (\(\lim_{x \to -\infty} f(x) = -1\)), and as x approaches infinity, the function should approach the horizontal line at y = 1 (\(\lim_{x \to \infty} f(x) = 1\)).

Step 1: Set up the coordinate axis and plot the given point

Create a set of coordinate axes (x-axis and y-axis) and plot the given point \(f(0) = 0\). This point serves as a reference point for the rest of the properties of the function.

Step 2: Determine the function's behavior at x = 0

The function's first derivative is given at x = 0, i.e., \(f'(0) = 1\). This indicates that the function is increasing at x = 0 because the derivative is positive.

Step 3: Determine the function's concavity

The function's second derivative provides information about the concavity of the function. We are given that the function is concave up (\(f''(x) > 0\)) on the interval \((-\infty, 0)\) and concave down (\(f''(x) < 0\)) on the interval \((0, \infty)\). This information helps us understand the curvature of the graph in these intervals.

Step 4: Incorporate limits at infinity

The function has given limits as \(x \to -\infty\) and \(x \to \infty\). As \(x \to -\infty\), the function approaches -1, and as \(x \to \infty\), it approaches 1. These properties will help us determine the end behavior of the function.

Step 5: Sketch the graph

Using the information from steps 1-4, we can now draw the graph of the function as follows: 1. Begin at the point (0, 0) since \(f(0) = 0\). 2. Head upward as the function is increasing at x = 0 (from step 2). 3. Curve concave up on the interval \((-\infty, 0)\) (from step 3). 4. Curve concave down on the interval \((0, \infty)\) (from step 3). 5. As x approaches negative infinity, the function should approach the horizontal line at y = -1, and as x approaches infinity, the function should approach the horizontal line at y = 1 (from step 4). By drawing the graph with these properties, we have successfully sketched a function with the given properties.

Key Concepts

Function Properties

In this exercise, we're dealing with a function characterized by specific properties. These properties include known values and behavior at particular points and intervals. The value at a point, denoted by \(f(0) = 0\), means when \(x\) is zero, the function value is also zero. This serves as an anchor point for sketching the graph.

The first derivative, \(f'(0) = 1\), indicates that the function is increasing at \(x = 0\). This derivative provides crucial insights into the function's slope at specific points, helping to dictate the direction of the graph at those points.

Additionally, the second derivative being positive on one side \((-\infty, 0)\) and negative on the other \((0, \infty)\), describes the change in the slope of the function, which relates intimately to its concavity. These specific properties guide our understanding of how to sketch the graph accurately.

Concavity

Concavity gives us a clue about the function's curvature. From the given properties:

• The function is concave up on the interval \((-\infty, 0)\), meaning it curves like an upward-facing cup. This is because the second derivative \(f''(x) > 0\) in this interval.
• Conversely, on \((0, \infty)\), the graph is concave down. This forms a downturned cup as the second derivative \(f''(x) < 0\).

Understanding concavity helps us avoid mistakes in sketching. When a function is concave upward, it indicates that the rate of increase of the function itself is increasing. If it’s concave down, the increase is slowing down. Recognizing these shifts helps in drawing smooth curves and transitions in the graph across different sections.

Derivative Analysis

The derivative of a function provides insight into how the function behaves. At \(x = 0\), we know \(f'(0) = 1\), so the function is increasing at this point. The derivative tells us about the steepness and direction. A positive derivative means an upward slope, while a negative derivative would mean a downward slope.

Analyzing the derivatives allows us to predict the general behavior of the function. The sign and magnitude of the derivative indicate whether the function is growing or shrinking and how sharply these changes occur.

The second derivative, as previously mentioned, helps assess the concavity, which directly affects how we visualize changes in the slope, adding another layer of information for sketching.

End Behavior

End behavior describes what happens to the function as \(x\) approaches infinity or negative infinity. Based on the limits provided:

• As \(x\to-\infty\), \(f(x)\) approaches -1. This means the function levels out near \(y = -1\).
• As \(x\to\infty\), \(f(x)\) approaches 1, indicating a leveling near \(y = 1\).

These limits represent horizontal asymptotes, guiding the function's convergence towards specific values as \(x\) moves far from zero. This understanding allows us to sketch parts of the graph that extend far to the left or the right, ensuring they align with these limiting behaviors.

End behavior is crucial for completing the graph, ensuring it not only looks accurate locally but also aligns with how it would theoretically behave over a larger span of \(x\)-values.