Problem 49 - Calculus - AP Edition

Question

Verify that \(y=\ln (x+e)\) satisfies \(d y / d x=e^{-y},\) with \(y=1\) when \(x=0\)

Step-by-Step Solution

Answer

Yes, the function satisfies \(\frac{dy}{dx} = e^{-y}\) with initial condition \(y=1\) at \(x=0\).

Step 1: Verify the Given Condition

First, we need to verify whether the function satisfies the given condition. We have \(y = \ln(x+e)\) and we need to find \(\frac{dy}{dx}\). Using the chain rule, we have:\[\frac{dy}{dx} = \frac{1}{x+e}\]Now, substitute \(y = \ln(x+e)\):\[e^{-y} = e^{- ext{ln}(x+e)} = \frac{1}{x+e}\]Thus, we see that indeed \(\frac{dy}{dx} = e^{-y}\).

Step 2: Verify Initial Condition

Now, we need to verify that the initial condition \(y = 1\) when \(x = 0\) holds. For the function \(y = \ln(x+e)\) at \(x = 0\), we compute:\[y = \ln(0 + e) = \ln e = 1\]This confirms the initial condition given as \(y = 1\) when \(x = 0\).

Key Concepts

Understanding Calculus and Its Role

Calculus is a branch of mathematics that deals with the study of change. It provides the tools needed to analyze and explore various types of functions, such as the one we are dealing with in this exercise: the function defined as \(y = \ln(x+e)\). This function describes a natural logarithm, which is a common type of continuous function in calculus.
To solve problems in calculus, you often use derivatives to measure how one variable changes in relation to another. For this exercise, we are particularly looking at how \(y\) changes as \(x\) changes, or \(\frac{dy}{dx}\). This derivative gives us the rate of change and is essential in verifying how the function behaves and satisfies the given conditions.
Calculus allows us to solve many practical problems, such as finding the slope of a curve at any point or predicting the future behavior of a variable. These skills are foundational for sciences, engineering, economics, and many fields where understanding dynamic systems is crucial.

Diving into the Chain Rule

The chain rule is a fundamental concept in calculus used to differentiate composite functions. A composite function occurs when you have a function inside another function, like \(y = \ln(x+e)\). In this exercise, you're asked to find the derivative of the given logarithmic function with respect to \(x\).
Here's how the chain rule works in our example:

• The outer function is the natural logarithm, \(\ln(u)\).
• The inner function is \(u = x+e\).

To differentiate using the chain rule, you first find the derivative of the outer function with respect to the inner function, then multiply it by the derivative of the inner function. In formal terms, this is expressed as:
\[\frac{dy}{dx} = \frac{d}{du}[\ln(u)] \cdot \frac{du}{dx}\]
After applying the chain rule, we arrive at:
\[\frac{dy}{dx} = \frac{1}{x+e}\]
This process shows how the chain rule simplifies the differentiation of complex functions, making it easier to deal with problems involving layers of functions nested within each other.

Understanding Initial Conditions

Initial conditions are critical in differential equations as they provide specific information needed to find particular solutions to the equations. In this example, the initial condition given is \(y = 1\) when \(x = 0\). This means that at \(x = 0\), the value of the function must be 1.
To verify this initial condition for the function \(y = \ln(x+e)\), you substitute \(x = 0\):
\[y = \ln(0+e) = \ln e = 1\]
This calculation confirms that the function meets the initial condition, as \(\ln e = 1\).
Initial conditions often serve as a starting point for solving differential equations because they help determine unique solutions among many possible solutions. They are particularly important in scenarios involving real-world applications, where they can represent initial states or configurations in a physical system.