Problem 12 - Calculus Early Transcendentals

Question

Sketches from properties Sketch a graph of a function that is continuous on \((-\infty, \infty)\) and has the following properties. Use a sign graph to summarize information about the function. $$\begin{aligned} &f^{\prime}(-1) \text { is undefined; } f^{\prime}(x)>0 \text { on }(-\infty,-1) ; f^{\prime}(x)<0\\\ &\text { on }(-1, \infty) \end{aligned}$$

Step-by-Step Solution

Answer

a) The function has a local minimum at x=-1 b) The function has a sharp turn or cusp at x=-1 c) The function is always increasing d) The function has no discontinuity points Answer: b) The function has a sharp turn or cusp at x=-1.

Step 1: Understand the properties of the function

The properties of the function are given as: 1. \(f^{\prime}(-1)\) is undefined 2. \(f^{\prime}(x) > 0\) on \((-\infty, -1)\) 3. \(f^{\prime}(x) < 0\) on \((-1, \infty)\) From these properties, we can infer that: - There is a sharp turn or a cusp at \(x=-1\) - The function is increasing on the interval \((-\infty, -1)\) - The function is decreasing on the interval \((-1, \infty)\)

Step 2: Create a sign chart for the first derivative

To better visualize the properties, let's create a sign chart for the first derivative: Interval | Sign of \(f^{\prime}(x)\) | Behavior ------------------|------------------------|------------ \((-\infty, -1)\) | Positive | Increasing \((-1, \infty)\) | Negative | Decreasing

Step 3: Sketch the graph

Now, let's sketch the graph based on the behavior concluded in the previous step: 1. The graph should be increasing on the interval \((-\infty, -1)\). 2. There should be a sharp turn or a cusp at the point where \(x=-1\). 3. The graph should be decreasing on the interval \((-1, \infty)\). As an example, a possible graph for the function is: ^ | | * | \ | \ | \ |--------*----------> |

Key Concepts

Continuous Functions

In calculus, a function is said to be continuous if, intuitively speaking, you can draw its graph without picking up your pencil. This implies that there are no breaks, holes, or jumps in the graph. Specifically, a function is continuous on an interval if it is continuous at every point within that interval. For the function described in the exercise, we're told it's continuous on \( (-\infty, \infty) \), meaning it has no discontinuities for all real numbers. This ensures the graph will be a connected, unbroken line. Understanding continuity is vital because many of the theorems and methods in calculus, such as the Intermediate Value Theorem, require functions to be continuous.

First Derivative Test

The First Derivative Test is a powerful tool for determining whether a particular point on a function is a local maximum or minimum. It involves taking the derivative of the function, determining where the derivative is zero or undefined, and examining the sign of the derivative around those points. In this exercise, \(f^{\textprime}(-1)\) is undefined, suggesting a potential sharp turn or cusp at \(x = -1\). We use the test by analyzing the sign of the derivative before and after this point to infer that the function switches from increasing to decreasing at \(x = -1\), indicating a peak or local maximum at that point.

Sign Chart Method

The Sign Chart Method is used to illustrate the behavior of the derivatives of a function. It's a visual approach where you create a chart listing intervals on the 'x' axis and the corresponding sign (positive or negative) of the first derivative \(f^{\textprime}(x)\) for that interval. The sign of the derivative indicates whether the function is increasing (positive derivative) or decreasing (negative derivative). Drawing a sign chart as in the step-by-step solution helps to summarize and clarify this information, making it easier to understand the overall behavior of the function before graphing it.

Increasing and Decreasing Intervals

Identifying increasing and decreasing intervals of a function is crucial when sketching its graph. Intervals where the first derivative \(f^{\textprime}(x)\) is positive indicate where the function is increasing; those where it is negative, the function is decreasing. In the exercise, we deduce that the function is increasing on the interval \((-\infty, -1)\) and decreasing on \((-1, \infty)\). These intervals are determined using the first derivative of the function and the sign chart, and they greatly aid in visualizing the function's graph.