Problem 58 - Single Variable Calculus

Question

Sketch the graph of a function having the given properties. $$ \begin{array}{l} f(0)=\pi / 2, f^{\prime}(0) \text { does not exist, } f(-1)=f(1)=0 \text { , }\\\ f^{\prime \prime}(x)>0 \text { on }(-\infty, 0) \cup(0, \infty)\\\ \lim _{x \rightarrow-\infty} f(x)=\lim _{x \rightarrow \infty} f(x)=-\pi / 2 \end{array} $$

Step-by-Step Solution

Answer

Plot the points (-1, 0), (1, 0) and (0, π/2). Since the function is concave up everywhere except x=0, and the limit as x approaches ±∞ is -π/2, create a cusp or vertical tangent at x=0. Keep the graph concave up and approach the horizontal asymptote y=-π/2 as x goes to ±∞.

Step 1: Function values at specific points

Plot the points f(-1)=f(1)=0 and f(0)=π/2. These will be the base points for our sketch.

Step 2: Absence of the first derivative at x=0

Since f'(0) does not exist, there is either a sharp corner (cusp) or a vertical tangent at x=0. The function is not smooth at this point, so there will be a sharp change in direction or a vertical tangent line.

Step 3: Positive second derivative

Since f''(x) is positive on (-∞, 0) ∪ (0, ∞), the function is concave up on these intervals. This means the graph will be curving upward to the left and right of x=0.

Step 4: Limit as x approaches ±∞

As x goes to -∞ and ∞, f(x) approaches -π/2. This means the graph will approach a horizontal asymptote at y=-π/2 as we move far to the left and far to the right. Based on these properties, you should be able to sketch the graph of the function. The graph should have points at (-1, 0), (1, 0), and (0, π/2), a cusp or a vertical tangent at x=0, concave up on both sides of x=0, and approach a horizontal asymptote at y=-π/2 as x approaches ±∞.

Key Concepts

Concavity and Second Derivatives

Understanding the role of second derivatives in the concavity of functions is essential when graphing. The second derivative of a function, denoted as \( f''(x) \), gives us information about the curvature of the function's graph. If \( f''(x) > 0 \), the function is said to be concave up, which visually means the graph looks like a cup that can hold water. Conversely, if \( f''(x) < 0 \), the graph is concave down, resembling an upside-down cup.

When sketching graphs, the concavity tells us how the function transitions between points. For instance, in the given exercise, \( f''(x) > 0 \) for \( x \) in \( (-\text{∞}, 0) \cup (0, \text{∞}) \), indicating that on either side of the point \( x = 0 \), the graph is concave up. This will make the graph resemble a 'U' shape, curving upwards as it moves away from \( x = 0 \). Keep in mind, the actual 'depth' of the curve is not determined by the second derivative, but rather its presence is signified by the positivity of \( f''(x) \).

To enhance your understanding of concavity, it's important to visualize several examples of functions with known second derivatives and sketch their graphs accordingly. Practice identifying areas of concavity and inflection points, where the concavity changes, to solidify your grasp on this concept.

Limits at Infinity

The concept of limits at infinity is crucial when you need to understand the end behavior of functions. A limit as \( x \) approaches infinity (\( x \to \text{∞} \)) gives us a glimpse into how a function behaves as it heads towards positive or negative infinite values. When we say \( \text{lim}_{x \to \text{∞}} f(x) = L \), we mean that as \( x \) becomes larger and larger, the values of \( f(x) \) get closer and closer to the number \( L \).

In the given exercise, both limits as \( x \) approaches negative and positive infinity are the same, \( - \frac{\text{π}}{2} \). This tells us that the graph of the function has a horizontal asymptote at \( y = - \frac{\text{π}}{2} \). No matter if you go far to the left (negative infinity) or far to the right (positive infinity), the graph will eventually get closer to that horizontal line but will never actually touch it.

To truly integrate the idea of limits at infinity into your graph sketching ability, practice with functions of various types, such as rational, exponential, and logarithmic functions. Compare their end behavior by calculating the limits at infinity and observing how these limits correspond to horizontal asymptotes on their graphs.

Undefined First Derivative

An undefined first derivative at a particular point has significant implications for the graph of a function. The first derivative, \( f'(x) \), represents the slope of the tangent line to the graph at any given point. If \( f'(x) \) does not exist at a point, this indicates that the graph may have a cusp (a point where the graph is continuous, but the direction changes sharply) or a vertical tangent (where the tangent line is vertical).

In our exercise, the first derivative at \( x=0 \) is undefined, which means we might expect a sharp corner or a vertical tangent at that point on the graph. This lack of smoothness implies that the graph cannot be smoothly traced through \( x=0 \) without lifting the pencil or encountering a vertical line. Note that this does not necessarily tell us the precise shape at that point; additional information would be needed for a full description. However, it's a critical feature that defines the structure of the graph.

When you encounter an undefined first derivative, consider both the left-hand and right-hand behavior of the function to predict if you'll see a cusp or a vertical tangent. Practice sketching graphs with these features by examining functions known to have undefined derivatives at specific points, such as the absolute value function at the origin, to refine your interpretation skills.