Question

Find the equation of a curve that has a second derivative \(y^{\prime \prime}=12 / x^{3}\) if it has a slope of -6 at the point (1,0).

Short answer

The equation of the curve is \(y = \frac{6}{x} - 6\).

Step-by-step solution

Integrate the second derivative to find the first derivative

To find the first derivative of the curve, you need to integrate the second derivative with respect to x. The integral of the second derivative, which is given by \(y^{\prime\prime}=12/x^{3}\), is \(y^{\prime} = -6/x^{2} + C_1\), where \(C_1\) is the constant of integration.

Determine the constant of integration for the first derivative

Given that the slope of the curve at the point (1, 0) is -6, we can find the constant \(C_1\) by substituting \(x = 1\) and \(y^{\prime} = -6\) into the first derivative equation. Plugging these values in, we get \(-6 = -6/(1)^2 + C_1\), which simplifies to \(C_1 = 0\).

Integrate the first derivative to find the original curve equation

Now, we integrate the first derivative \(y^{\prime} = -6/x^{2}\) to find the original curve's equation. The integral of -6/x^2 with respect to x is \(y = 6/x + C_2\), where \(C_2\) is another constant of integration.

Determine the constant of integration for the original curve

To find the constant \(C_2\), we use the given point (1, 0). By substituting \(x = 1\) and \(y = 0\) into the curve equation, we obtain \(0 = 6/1 + C_2\). Solving for \(C_2\), we find that \(C_2 = -6\).

Write the final equation of the curve

Now we can write the final equation of the curve with the constants we have found. The equation is \(y = 6/x - 6\).

Key concepts

Differential Calculus

Differential calculus is a branch of mathematical analysis that deals with the study of how functions change when their inputs change. It's centered on the concept of the derivative, which measures the rate at which a quantity changes. In the exercise, the second derivative, denoted as \(y''\), describes the curvature of the function or, more specifically, the rate at which the slope of the curve changes. This concept is fundamental for understanding the behavior of functions, as it can indicate points of inflection, maxima, and minima on the curve.

In this case, the second derivative \(y'' = 12/x^{3}\) helps us find the original function in two stages of integration. Each integration step takes us one level down in the derivative hierarchy; from the second derivative we integrate to find the first derivative, which represents the slope of the curve, and another integration takes us from the slope to the equation of the curve itself.

Integration Constants

When we integrate a derivative to find the original function, we add an unknown constant called the integration constant. This constant arises because indefinite integration is the reverse operation of differentiation, and when a function is differentiated, any constant term becomes zero. Therefore, during integration, we acknowledge the possibility of a constant that was once part of the original function but was lost through differentiation.

In the exercise, two constants, \(C_1\) and \(C_2\), appear as a result of integrating the second and first derivatives respectively. These constants can be specific values instead of arbitrary numbers because we have additional information: the curve's slope at a particular point and the coordinate of a point on the curve. Finding the values for these constants is crucial as they ensure the curve's equation matches the given conditions of the problem, providing an accurate description of the curve's behavior and its original equation.

Slope of a Curve

The slope of a curve at any given point is a measure of the steepness or the incline of the curve at that point. It is represented by the first derivative of the function at that point, denoted by \(y'\). The concept of the slope is pivotal in understanding the direction in which the curve moves: an upward trend for a positive slope or a downward trend for a negative slope.

In this problem, we are given the slope of the curve at a specific point, which is -6 at the point (1, 0). This information facilitates the determination of the integration constant \(C_1\) for the first derivative. By integrating the second derivative and using the given slope, we are able to precisely define the curve's behavior around that point. This type of problem is common in real-world applications, such as in physics for motion analysis or in economics for cost functions.