Problem 30 - Calculus Early Transcendentals: Pearson New International Edition

Question

Sketch the graph of a function \(g\) that has the following properties: (a) \(g\) is everywhere smooth (continuous with a continuous first derivative); (b) \(g(0)=0\) (c) \(g^{\prime}(x)<0\) for all \(x\); (d) \(g^{\prime \prime}(x)<0\) for \(x<0\) and \(g^{\prime \prime}(x)>0\) for \(x>0\).

Step-by-Step Solution

Answer

The function is a smooth, continuously decreasing curve passing through the origin, concave down for \(x<0\) and concave up for \(x>0\).

Step 1: Analyze property (a)

The function \(g\) is described as smooth everywhere, which means that it is continuous and has a continuous first derivative, \(g^{\prime}(x)\). Therefore, the function does not have any breaks, corners, or cusp points.

Step 2: Understand property (b)

We know that \(g(0)=0\). This tells us that the graph of the function passes through the origin, that is the point \((0,0)\).

Step 3: Interpret property (c)

The property \(g^{\prime}(x)<0\) indicates that \(g(x)\) is a decreasing function for all \(x\). Since the first derivative \(g^{\prime}(x)\) is negative at every point, the graph should always slope downwards as \(x\) increases.

Step 4: Apply property (d) to the left of origin

For \(x<0\), the second derivative \(g^{\prime\prime}(x)<0\). This implies that the graph is concave down on the left side of the origin. The slope of the decreasing function becomes steeper as \(x\) approaches 0 from the left.

Step 5: Apply property (d) to the right of origin

For \(x>0\), the second derivative \(g^{\prime\prime}(x)>0\). This means the graph is concave up on the right side. The slope of the decreasing function becomes less steep as \(x\) moves away from the origin to the right.

Step 6: Sketch the graph

Drawing the graph, it starts at the origin \((0, 0)\). To the left of the origin, the graph arcs downward becoming steeper, indicating negative slope and negative curvature (concave down). To the right of the origin, the graph remains decreasing but curves upward, indicating negative slope and positive curvature (concave up).

Key Concepts

Continuous Functions

When we talk about continuous functions in calculus, we are referring to functions that do not have any interruptions in their graph. Essentially, this means the graph can be drawn without lifting your pencil from the paper.

Continuous functions are important because they allow predictions and calculations without any sudden changes or jumps. If the function is continuous, it means that:

• For every point within the domain, the function has a value.
• As you approach any value from either side, the function approaches a specific output value, known as the limit.
• This makes continuous functions smooth and predictable.

In addition, it's important to note that a function may not only be continuous but also have a continuous first derivative. This indicates the function is smooth and does not have any corners or sharp edges, which can be felt if you tried to trace the graph with your finger.

Derivatives

Derivatives in calculus measure how a function changes as its input changes. They are essentially the slope of a function at a given point.

When discussing the first derivative, which is often noted as \(g'(x)\), you are looking at how much and in what direction the function is changing. If \(g'(x) < 0\), it means the function is decreasing - the graph slopes downwards as you move to the right. This is a crucial insight when trying to understand the behavior of the graph.

• If \(g'(x) > 0\), the function is increasing.
• If \(g'(x) = 0\), it suggests a potential extremum or turning point.

The second derivative, represented as \(g''(x)\), tells us about the concavity of the function. Understanding both derivatives together helps in analyzing the dynamic nature of how a function's curve is shaped and its direction.

Concavity

Concavity refers to the "curvature" of the function's graph. It's like determining if the function's graph is "smiling" or "frowning." This is defined by the second derivative, \(g''(x)\).

If \(g''(x) > 0\), the function's graph is concave up ("smiling"). This suggests that as you move from left to right, the graph becomes less steep if it is decreasing, or steeper if it is increasing. For our exercise, on the right side of the origin where \(x > 0\), the function does exactly this.

• If \(g''(x) < 0\), it's concave down ("frowning"), meaning the graph steepens if decreasing or lessens if increasing. The left side of our graph exhibits this behavior under \(x < 0\).

Acknowledging concavity helps us anticipate how the shape of the function might change across different intervals of \(x\), offering a deeper insight into its graph.

Graphing Functions

Graphing functions allow us to visualize mathematical relationships concretely. Knowing how to graph based on derivatives and concavity insights is key to understanding and predicting behavior.

For sketching a graph based on the given function properties:

• Start by plotting known points like \(g(0) = 0\), ensuring the graph passes through the origin.
• With \(g'(x) < 0\), maintain a downward slope across all \(x\).
• Incorporate concavity: to the left of the origin, the graph falls steeper (concave down), and to the right, the graph, though declining, curves upwards (concave up).

By methodically applying each property, you can draw a function accurately reflecting its mathematical behavior, offering a clear picture of the function's dynamic nature.