Question

In Exercises \(35-38\) , find the function with the given derivative whose graph passes through the point \(P\) . $$f^{\prime}(x)=-\frac{1}{x^{2}}, \quad x>0, \quad P(2,1)$$

Short answer

The function is \(f(x) = 1/x + 1/2\).

Step-by-step solution

Find the Indefinite Integral

First, integrate the derivative to get the function. The integral of \(-1/x^2\) is \(1/x\). So, the function \(f(x)\) is \(1/x + C\), where \(C\) is the constant of integration.

Substitute the Point into the Function

Substitute the point \(P(2,1)\) into the function to solve for \(C\). So, we substitute \(x = 2\) and \(f(x) = 1\) into the function to get \(1 = 1/2 + C\).

Solve for the Constant of Integration

From the last step we have \(1 = 1/2 + C\). Solving for \(C\), we get \(C = 1 - 1/2 = 1/2\). Therefore, the original function is \(f(x) = 1/x + 1/2\).

Key concepts

Antiderivative

Understanding the antiderivative is crucial when dealing with integration. It is essentially the reverse process of taking a derivative. While a derivative represents the rate of change of a function, an antiderivative represents a function whose derivative is the given function. In simpler terms, if you're given a function's rate of change, can you determine the original function? That's what finding an antiderivative is all about. For instance, if the rate of change or derivative is given by \(-\frac{1}{x^2}\), the antiderivative would be a function whose derivative equals \(-\frac{1}{x^2}\). Finding antiderivatives, also known as indefinite integrals, is like solving a puzzle where you are trying to discover the original shape (function) from its pattern of change (derivative).

It is vital to acknowledge that the antiderivative is not unique. Since the derivative of a constant is zero, any constant added to a function doesn't affect the rate of change represented by the derivative. Therefore, the antiderivative is usually expressed as the general form of the function plus an arbitrary constant, often denoted as \(C\). In the given exercise, the antiderivative of \(-\frac{1}{x^2}\) was identified as \(\frac{1}{x}\), but we must not forget to append \(+ C\) to represent any possible constants that were lost during differentiation.

Constant of Integration

The concept of the constant of integration is a fundamental part of integration. When we find the antiderivative of a function, we are essentially retrieving a function from its derivative. However, because differentiation eliminates constant values (since their derivative is zero), when we integrate, we must consider all potential constant values that could have been present in the original function. This is why we add the constant of integration, denoted as \(C\), to the result of an indefinite integral. It represents all the constant values that would yield the same derivative.

In practice, if an initial condition is provided, such as a point through which the graph of the function must pass, we can use this information to solve for the constant. For example, in the given exercise, the function needed to pass through the point \(P(2, 1)\). By substituting these values into the integrated function, the constant \(C\) could be calculated, thus pinpointing the exact original function that not only has the specified rate of change but also fits the given condition.

Integration Techniques

When dealing with integrals, there are several techniques we can employ to find the antiderivative of a function. Common techniques include substitution, integration by parts, partial fractions, and trigonometric integration, among others. The best technique to use often depends on the form of the function we're integrating.

For simple power functions, such as \(x^n\), we apply the power rule for integration, which involves adding one to the exponent and dividing by the new exponent. In the exercise provided, the integral of \(-\frac{1}{x^2}\), or \(-x^{-2}\), was found using the power rule, resulting in \(1/x + C\). As the function is a simple power function, no advanced techniques were necessary. However, it's essential to recognize the integrand's form to choose the most appropriate integration technique. This step is crucial to simplifying the problem and solving the integral correctly.