Problem 32 - Calculus Early Transcendentals: Pearson New International Edition

Question

Sketch the graph of a function \(f\) that has the following properties: (a) \(f\) is everywhere continuous; (b) \(f(-4)=-3, f(0)=0, f(3)=2\); (c) \(f^{\prime}(-4)=0, f^{\prime}(3)=0, f^{\prime}(x)>0\) for \(x<-4, f^{\prime}(x)>0\) for \(-43\); (d) \(f^{\prime \prime}(-4)=0, f^{\prime \prime}(0)=0, f^{\prime \prime}(x)<0\) for \(x<-4, f^{\prime \prime}(x)>0\) for \(-40\).

Step-by-Step Solution

Answer

Sketch a continuous curve passing through given points, increasing and decreasing as per derivatives, with inflection at given points.

Step 1: Plot Given Points

- Start by plotting the points \((-4, -3), (0, 0), \text{and} (3, 2)\) on a coordinate plane. These points represent known values of the function \(f\).

Step 2: Analyze Extrema from Derivatives

- The conditions \(f'(-4) = 0\) and \(f'(3) = 0\) suggest that these points are where the slope is zero, leading to either local minima or maxima. \(f'(x)>0\) for \(x < -4\) and \(-4 < x < 3\) indicates that the function is increasing in these intervals, while \(f'(x)<0\) for \(x > 3\) indicates decreasing behavior.

Step 3: Determine Concavity from Second Derivatives

- For \(f''(x)<0\) when \(x<-4\), the graph is concave down. \(f''(x)>0\) when \(-40\), with \(f''(x)<0\), the graph is again concave down, suggesting another inflection point at \(x=0\).

Step 4: Sketch the Graph

- Begin from the left: the graph should be decreasing concavely down before \(x=-4\). As \(x o -4\) from the left, the function approaches the point \((-4, -3)\) where it changes to increasing with concave up behavior until \(x=0\), and then reaches the origin (0, 0).- From \(x=0\) to \(x=3\), increase concavely up to \(x=3\), where you reach the point \((3, 2)\).- Beyond \(x=3\), the behavior shifts to decreasing and concave down.

Key Concepts

Understanding Calculus

Calculus is a fundamental branch of mathematics that focuses on change. This discipline is divided into two main areas - differential calculus and integral calculus. The former deals with the idea of a derivative, which represents the rate of change of a function. Calculus is essential in understanding how functions behave, particularly in terms of their slopes and how they change.

In our exercise, calculus helps us explore how the function's graph behaves around specific points and intervals. For instance, calculus tells us where the function increases, decreases, or changes its concavity.

The Role of Derivatives

Derivatives are crucial in determining the behavior of a function. The derivative of a function, denoted as \( f' \), provides us with the slope of the tangent line at any given point. This information is vital because it tells us how the function is changing or moving in a specific interval.

When looking at our exercise, the derivative conditions such as \( f'(-4)=0 \) and \( f'(3)=0 \) indicate that these points might be local extrema points, such as minima or maxima. These are points where the function changes from increasing to decreasing (or vice versa). Moreover, knowing that \( f'(x) > 0 \) suggests the function is increasing, while \( f'(x) < 0 \) suggests it is decreasing. Understanding these changes helps in sketching an accurate graph of the function.

Exploring Concavity

Concavity is an essential aspect of analyzing a graph, which involves second derivatives. This concept tells us the direction in which a graph curves. If the second derivative, \( f''(x) \), is positive in a region, the function is concave up, resembling an upward-opening cup. Conversely, if \( f''(x) \) is negative, the function is concave down, similar to a downward-opening cup.

In our task, at intervals like \( x < -4 \) and \( x > 0 \), where \( f''(x) < 0 \), the graph is concave down. Meanwhile, between \( -4 < x < 0 \), \( f''(x) > 0 \) indicates the graph is concave up. Such concavity changes are critical markers of graph behavior and often indicate inflection points where the curve changes direction.

The Importance of Continuity

Continuity of a function means that there are no breaks, holes, or jumps in its graph. A continuous function is one that you can draw without lifting your pencil from the paper. This characteristic is crucial when sketching graphs as it ensures that the function behaves predictably between known points.

In our exercise, the function is stated to be continuous everywhere, guaranteeing smooth transitions between the given points like (-4, -3), (0, 0), and (3, 2). Knowing the function is continuous helps in ensuring there are no unexpected disruptions in the graph, making it easier to visualize and plot accurately.