Problem 38 - Thomas Calculus in SI Units

Question

Find the function with the given derivative whose graph passes through the point \(P\). $$g^{\prime}(x)=\frac{1}{x^{2}}+2 x, \quad P(-1,1)$$

Step-by-Step Solution

Answer

The function is \(g(x) = -\frac{1}{x} + x^2 - 1\).

Step 1: Integrate the Derivative to Find the General Function

To find the function, we start by integrating \(g'(x) = \frac{1}{x^2} + 2x\). The integral of \(\frac{1}{x^2}\) is \(-\frac{1}{x}\), and the integral of \(2x\) is \(x^2\). Therefore, the indefinite integral is \(g(x) = -\frac{1}{x} + x^2 + C\), where \(C\) is the constant of integration.

Step 2: Use the Point to Determine the Constant of Integration

We know that the graph of the function passes through the point \(P(-1, 1)\). We can substitute \(x = -1\) and \(g(x) = 1\) into the equation \(g(x) = -\frac{1}{x} + x^2 + C\) to find \(C\). This gives us: \[1 = -\frac{1}{-1} + (-1)^2 + C\]. Simplifying, we get: \[1 = 1 + 1 + C\] which gives us \(C = -1\).

Step 3: Write the Specific Function

Now that we know \(C = -1\), the specific function that satisfies both the derivative and the point condition is: \[g(x) = -\frac{1}{x} + x^2 - 1\]. This is the desired function that passes through the point \(P(-1, 1)\).

Key Concepts

Integration

Integration is a fundamental concept in calculus. It allows us to find the original function from its derivative. This is useful for determining areas under curves or solving differential equations. In this exercise, we began with the given derivative, \(g'(x) = \frac{1}{x^2} + 2x\), and needed to find the original function, \(g(x)\). The process of finding \(g(x)\) involves taking the indefinite integral of the derivative.

The indefinite integral of a function represents a family of functions. It is generally represented as \(F(x) + C\), where \(F(x)\) is any antiderivative of the original function, and \(C\) is the constant of integration. Understanding how to integrate basic functions is crucial. For instance, integrating \(\frac{1}{x^2}\) results in \(-\frac{1}{x}\), and integrating \(2x\) results in \(x^2\). These laid the foundation of our general function \(g(x)\).

Bullet points to remember about integration:

• Integration is the reverse process of differentiation.
• Indefinite integrals include a constant \(C\), because differentiating a constant gives zero.
• Every integral computation generally requires understanding basic anti-derivatives.

Constant of Integration

The constant of integration, \(C\), appears in the process of indefinite integration. When we integrate a function, the result is not a single function, but a family of functions. The constant \(C\) represents the infinite possibilities for vertical shifts of the function. This happens because the derivative of a constant is zero.

In the context of our exercise, after integrating the derivative to get \(g(x) = -\frac{1}{x} + x^2 + C\), the constant \(C\) had to be determined for the function to be fully specified. We used the information that the function graph passes through point \(P(-1, 1)\). By substituting \(x = -1\) and \(g(x) = 1\) into our general form, we solved for \(C\) and found it to be \(-1\).

Key points about the constant of integration include:

• Every indefinite integral results in a constant \(C\).
• To find \(C\), additional conditions, such as a point on the graph, are necessary.
• Without determining \(C\), the function stays indefinite and non-specific.

Function Determination

Function determination is the process of finding the specific function that aligns with given conditions. In this particular problem, we began with a derivative and were required to find the function \(g(x)\) whose graph passes through a specific point \(P(-1, 1)\).

After integrating the derivative, the challenge remained to use the given point to determine \(C\), the constant of integration. This point provided concrete data to tailor the indefinite integral result into a definitive function. By plugging \(x = -1\) and \(g(x) =1\) into our generalised equation \(g(x) = -\frac{1}{x} + x^2 + C\), we solved the equation to determine \(C = -1\). Thus, we defined the unique function \(g(x)\) as \(-\frac{1}{x} + x^2 - 1\).

Steps in determining a function include:

• Start by integrating the given derivative to form a general function.
• Use specific points provided to solve for any constants, such as \(C\).
• Plug in all known values to adjust the general solution to the accurate one.