Problem 14 - Calculus Early Transcendentals

Question

True–False Determine whether the statement is true or false. Explain your answer. Every integral curve for the slope field \(d y / d x=e^{y}\) is concave up.

Step-by-Step Solution

Answer

The statement is true; all integral curves are concave up.

Step 1: Understand the Equation

The given differential equation is \( \frac{dy}{dx} = e^y \). This is a first-order ordinary differential equation indicating the slope field for the function \( y(x) \). Integral curves are solutions to this differential equation.

Step 2: Determine Concavity of Integral Curves

To determine the concavity of the integral curves, we need to find the second derivative, \( \frac{d^2y}{dx^2} \). The second derivative tells us about the concavity of the function. We differentiate \( \frac{dy}{dx} = e^y \) with respect to \( x \).

Step 3: Differentiate to Find Second Derivative

Differentiating \( \frac{dy}{dx} = e^y \) with respect to \( x \), we apply the chain rule:\[ \frac{d^2y}{dx^2} = \frac{d}{dx}(e^y) = e^y \cdot \frac{dy}{dx} = (e^y)^2. \]

Step 4: Analyze the Second Derivative

The second derivative \( \frac{d^2y}{dx^2} = (e^y)^2 \) is always positive since \( (e^y)^2 > 0 \) for all real \( y \). A positive second derivative indicates that the function is concave up.

Step 5: Conclusion Based on Analysis

Since the second derivative is always positive, every integral curve of the differential equation \( \frac{dy}{dx} = e^y \) is concave up. This means that the statement given in the problem is true.

Key Concepts

Concavity

Concavity is a property of a curve that indicates the direction it bends. It tells us whether a curve is opening upwards or downwards. For any function, the concavity is determined by the sign of its second derivative.

• If the second derivative is positive, the function is concave up, forming a U-shape.
• If the second derivative is negative, the function is concave down, forming an inverted U-shape.

In our exercise, we looked at whether the integral curves of the differential equation \( \frac{dy}{dx} = e^y \) are concave up. By computing the second derivative, we found it to be \( \frac{d^2y}{dx^2} = (e^y)^2 \), which is always positive. Therefore, these integral curves are indeed concave up. Concavity helps in understanding how the slope of the tangent line changes as we move along the curve. It gives insight into the behavior and shape of the graph of a function.

Differential Equation

A differential equation is an equation that involves one or more derivatives of a function. These equations are essential in describing various physical phenomena, ranging from motion to growth processes.There are different types of differential equations:

• Ordinary Differential Equations (ODEs): Involves derivatives of a function with respect to one independent variable, like time or space.
• Partial Differential Equations (PDEs): Involves derivatives with respect to multiple independent variables.

In our discussed exercise, we have a first-order ODE: \( \frac{dy}{dx} = e^y \), which tells us about the slope of the function at any point \( y \). Solving this differential equation gives us the integral curves, which are solutions defining how \( y \) changes with \( x \). These solutions help us model the dynamics represented by the equation.

Second Derivative

The second derivative of a function provides information on the curvature of the graph of the function. In essence, it gives a deeper insight into how a function behaves, particularly its concavity.Here’s why the second derivative is crucial:

• The first derivative \( \frac{dy}{dx} \) tells us the slope or the rate of change of the function.
• The second derivative \( \frac{d^2y}{dx^2} \) informs us about how the slope itself is changing. It detects the acceleration or the curvature.

In our example, differentiating \( \frac{dy}{dx} = e^y \) resulted in \( \frac{d^2y}{dx^2} = (e^y)^2 \). Since \( (e^y)^2 \) is always positive, the second derivative confirms that the function is concave up everywhere. Therefore, every integral curve described by this equation bends upwards. Understanding the second derivative is fundamental in calculus as it allows predictions about the geometric shape of the curve and identification of points of inflection where the curvature changes direction.