Problem 77 - 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: Understanding the Derivative

The problem states that the derivative of the curve is given by \(f'(x) = \frac{1}{x}\). This means that as \(x\) increases, the rate of change of \(y\) depends on the value of \(x\), specifically the inverse of \(x\).

Step 2: Integration to Find the Curve

To find \(f(x)\), integrate the derivative \(f'(x) = \frac{1}{x}\). The integral of \(\frac{1}{x}\) is \(\ln|x|\), so \(f(x) = \ln x + C\), where \(C\) is a constant to be determined.

Step 3: Using Initial Condition to Find Constant C

We know that \(f(1) = 0\). Substitute \(x = 1\) into \(f(x) = \ln x + C\) to find \(C\): \(0 = \ln 1 + C\), with \(\ln 1 = 0\), so \(C = 0\). Therefore, \(f(x) = \ln x\).

Step 4: Sketching the Curve

The curve \(y = \ln x\) is defined for \(x > 0\). It passes through the point \((1, 0)\) and increases without bound as \(x\) increases, but increases more slowly as \(x\) becomes larger.

Step 5: Analyzing the Concavity

The concavity of the curve is determined by the second derivative. Find \(f''(x)\) by differentiating \(f'(x) = \frac{1}{x}\), which gives \(f''(x) = -\frac{1}{x^2}\). Since \(f''(x) < 0\) for \(x > 0\), the curve is concave down for all \(x > 0\).

Key Concepts

Derivative

The derivative is a fundamental concept in calculus that tells us how a function changes as we adjust its inputs. For the given exercise, the derivative of the function is specified as \( f'(x) = \frac{1}{x} \). This derivative indicates that the rate of change of the function depends inversely on \( x \).
The fact that \( f'(x) \) equals \( \frac{1}{x} \) means that as \( x \) increases, the slope of the tangent to the curve becomes less steep, illustrating a slowing rate of increase for the function. This is important for understanding the nature and behavior of the curve itself.
Derivatives can help in sketching curves, optimizing systems, and solving numerous practical problems, making them incredibly beneficial for understanding and applying mathematical concepts.

Integration

Integration is the process of finding a function given its derivative, essentially working backward from differentiation. When tackling the curve outlined in the exercise, integration of the derivative \( f'(x) = \frac{1}{x} \) leads us to the function \( f(x) = \ln|x| + C \), where \( C \) is an integration constant.
This integral arises from observing that the derivative of the natural logarithm of \( x \) is \( \frac{1}{x} \). By using this relationship, we translate the derivative into the logarithmic function, ensuring that we capture the full nature of the original function.
To apply specific conditions, such as \( f(1) = 0 \) in this exercise, solve for \( C \), resulting in the specific function \( f(x) = \ln x \).
Integration not only helps in constructing functions but also aids in areas like calculating areas under curves and solving differential equations.

Curve Sketching

Curve sketching involves drawing the graph of a function based on information provided by its derivatives and other key function properties. In our exercise, knowing \( f(x) = \ln x \), we can sketch the curve for \( x > 0 \).
The curve passes through \( (1, 0) \) as specified by the initial condition. Observing the derivative \( f'(x) = \frac{1}{x} \), which is positive for all \( x > 0 \), we know the function is increasing over its domain. However, as \( x \) becomes larger, the rate of increase slows down.
When sketching, it’s important to capture these characteristics: starting at a point, how the curve increases, and how quickly this happens, giving a comprehensive picture of the function’s behavior over its interval.
Curve sketching helps visualize mathematical relationships and can inform predictions and interpretations in real-world scenarios.

Concavity

Concavity describes the curvature direction of a graph and is pivotal in interpreting the shape of a function's graph. In this exercise, the second derivative \( f''(x) = -\frac{1}{x^2} \) tells us about the concavity of the function.
Since \( f''(x) < 0 \) for all \( x > 0 \), the curve is consistently concave down in its domain. This implies that the tangent line to the curve at any point lies above the curve itself.
Understanding concavity assists us in visualizing the nature of the graph: whether it bends upwards or downwards. Such knowledge is crucial in practical applications like economics and engineering, where one frequently needs to assess stability or optimize performance based on curvature.