Problem 111 - Thomas Calculus

Question

For \(x>0,\) sketch a curve \(y=f(x)\) that has \(f(1)=0\) and \(f^{\prime}(x)=1 / x .\) Can anything be said about the concavity of such a curve? Give reasons for your answer.

Step-by-Step Solution

Answer

The curve is \( y = \ln(x) \) and is concave down for \( x>0 \).

Step 1: Identify the function

We start with the derivative of the function, which is given as \( f'(x) = \frac{1}{x} \). This suggests that the function \( f(x) \) might be a logarithmic function, since the derivative of \( \ln(x) \) is \( \frac{1}{x} \).

Step 2: Integrate to find the function

To find \( f(x) \), we integrate \( f'(x) = \frac{1}{x} \), which gives us \[ f(x) = \int \frac{1}{x} \, dx = \ln|x| + C. \] For \( x > 0 \), this simplifies to \[ f(x) = \ln(x) + C. \]

Step 3: Apply the initial condition

We are given that \( f(1) = 0 \). Substitute \( x = 1 \) into the function to find \( C \):\[ f(1) = \ln(1) + C = 0, \]since \( \ln(1) = 0 \), it follows that \( C = 0 \). Thus, the function is \( f(x) = \ln(x) \).

Step 4: Determine the concavity

To determine the concavity of the curve, we find the second derivative of \( f(x) \). Start with \( f'(x) = \frac{1}{x} \). The second derivative is:\[ f''(x) = \frac{d}{dx}\left(\frac{1}{x}\right) = -\frac{1}{x^2}. \]Since \( f''(x) = -\frac{1}{x^2} < 0 \) for \( x > 0 \), the curve is concave down.

Key Concepts

Derivative

The derivative of a function is a fundamental concept in calculus that represents the rate at which a function changes at any given point. In simpler terms, it's like finding the speed of a moving object at a specific moment in time. For a given function, if you know the derivative, you can determine the behavior of the function, such as whether it is increasing or decreasing.

In the exercise, we were given the derivative of the function: \( f'(x) = \frac{1}{x} \). This tells us that as \( x \) grows larger, the value of the derivative decreases, indicating a slowing rate of increase. Conversely, as \( x \) approaches zero from the positive side, \( f'(x) \) becomes very large, suggesting a rapid increase. Understanding derivatives helps us visualize how the curve's slope behaves across different intervals, which is critical for sketching graphs.

Logarithmic Function

A logarithmic function is the inverse operation of an exponential function and is widely used in many areas of mathematics and real-world applications, such as measuring the intensity of earthquakes or sound. Typically represented as \( \ln(x) \) for natural logarithms, they simplify to simpler multiplication when dealing with repeated multiplicative processes.

In this exercise, we integrate the derivative \( \frac{1}{x} \), which is typical for logarithmic functions, to find \( f(x) = \ln(x) + C \). This equality implies that our function is indeed logarithmic. Since \( \ln(1) = 0 \), when given \( f(1) = 0 \), we conclude that \( C \) must be zero, resulting in our function being simply \( f(x) = \ln(x) \). The logarithmic function often has specific characteristics like slowly increasing as \( x \) grows and having a vertical asymptote at \( x = 0 \).

Integration

Integration can be thought of as the reverse operation of finding a derivative. While derivatives deal with rates of change, integration is about finding the accumulated value or area under a curve. It helps in determining the original function from its rate of change.

In our task, we used integration to find \( f(x) \) from its derivative \( f'(x) = \frac{1}{x} \). By integrating, we obtained \( f(x) = \ln|x| + C \). For the positive domain \( x > 0 \), this simplifies to \( f(x) = \ln(x) + C \). Understanding integration allows us to piece together the whole function from its differential components, giving us a comprehensive view of the curve.

Concavity

Concavity is a property of a curve that describes how its direction changes. Specifically, it tells us whether a curve is opening upwards (concave up) or downwards (concave down). This is determined using the second derivative.

For the given function \( f(x) = \ln(x) \), the second derivative was calculated as \( f''(x) = -\frac{1}{x^2} \). Since \( f''(x) < 0 \) for \( x > 0 \), the curve is concave down. In practical terms, this means that the slope of the curve is decreasing; hence, the curve opens downward like a frown. Recognizing concavity is crucial for understanding the broader shape and form of a graph, allowing predictions about its future behavior in terms of rates of increase or decrease.