Problem 20
Question
The equation of the curve for which the cartesian subtangent varies as the reciprocal of the square of the abscissa, is (A) \(x=c e^{y^{1 / 3 k}}\) (B) \(x=c e^{y^{2} / 3 k}\) (C) \(y=c e^{x^{\prime} / 3 k}\) (D) None of these
Step-by-Step Solution
Verified Answer
None of the provided options match the derived equation.
1Step 1: Define the Subtangent
The subtangent is defined as the projection of the tangent line of a graph on the x-axis. Mathematically, the subtangent, denoted by \(T\), is given by \(T = y' \cdot x\), where \(y'\) is the derivative of the function with respect to \(x\).
2Step 2: Establish the Proportionality Relationship
According to the problem statement, the subtangent varies as the reciprocal of the square of the abscissa, \(x\). Thus, the equation for this relationship is \(y' x = \frac{k}{x^2}\), where \(k\) is a constant of proportionality.
3Step 3: Transform and Simplify the Equation
Rearrange the equation to isolate the derivative: \(y' = \frac{k}{x^3}\).
4Step 4: Integrate to Find the Curve
Integrate the derivative with respect to \(x\) to find the original function. \[ y = \int \frac{k}{x^3} \, dx = \int kx^{-3} \, dx = -\frac{k}{2x^2} + C \] where \(C\) is the integration constant.
5Step 5: Simplify the Expression
After integration, we find that \(y = -\frac{k}{2x^2} + C\). Simplify it to see if it matches any given options. However, the simplified version does not simplify to match any of A, B, or C directly.
Key Concepts
Equation of CurveSubtangentIntegrationCartesian Coordinates
Equation of Curve
An equation of a curve describes a mathematical relationship between variables, typically involving two dimensions like x and y. In this exercise, the curve's equation reflects a specific condition: the subtangent must vary inversely with the square of the abscissa. This means as we solve problems, we look for how different elements of the curve relate to this condition.
When solving for an equation of a curve given certain conditions, it's essential to set up an initial equation based on these conditions. Here, that condition was the relationship between subtangent and the square of the x-coordinate. Therefore, knowing how to express relationships like these in mathematical terms is a key part of curve equation problems.
When solving for an equation of a curve given certain conditions, it's essential to set up an initial equation based on these conditions. Here, that condition was the relationship between subtangent and the square of the x-coordinate. Therefore, knowing how to express relationships like these in mathematical terms is a key part of curve equation problems.
Subtangent
The subtangent in calculus is the segment of the x-axis cut off by the tangent at a given point on a curve. The subtangent is crucial for understanding the rate at which the curve rises or falls. When you have the equation of a curve, the subtangent helps determine how steep or gradual the tangent line is relative to the x-axis.
In the exercise, the subtangent is derived from the derivative of the function. It is mathematically represented as \(T = y' \cdot x\), where \(y'\) is the derivative of the function with respect to x. Understanding this allows us to relate geometric intuition with algebraic functions, ensuring a comprehensive grasp of curve behavior. In our specific problem, understanding how the subtangent changes with x is central to finding the equation of the curve.
In the exercise, the subtangent is derived from the derivative of the function. It is mathematically represented as \(T = y' \cdot x\), where \(y'\) is the derivative of the function with respect to x. Understanding this allows us to relate geometric intuition with algebraic functions, ensuring a comprehensive grasp of curve behavior. In our specific problem, understanding how the subtangent changes with x is central to finding the equation of the curve.
Integration
Integration is the mathematical process used to find a function when its derivative is known. In this problem, we started with a derivative, \(y' = \frac{k}{x^3}\), and our goal was to find the original function \(y\).
Integration involves reversing the differentiation process. Thus, integrating \(\frac{k}{x^3}\) with respect to x gives us \(y = -\frac{k}{2x^2} + C\), where \(C\) is the constant of integration. This process is fundamental in calculus and is used to determine the function's original form or to solve differential equations.
When applying integration, it's often helpful to recognize integration patterns and rules, such as power rule integration, to efficiently solve problems and reach the curve's equation.
Integration involves reversing the differentiation process. Thus, integrating \(\frac{k}{x^3}\) with respect to x gives us \(y = -\frac{k}{2x^2} + C\), where \(C\) is the constant of integration. This process is fundamental in calculus and is used to determine the function's original form or to solve differential equations.
When applying integration, it's often helpful to recognize integration patterns and rules, such as power rule integration, to efficiently solve problems and reach the curve's equation.
Cartesian Coordinates
Cartesian coordinates are used to precisely locate points on a two-dimensional plane using two numbers: the x-coordinate (horizontal axis) and the y-coordinate (vertical axis). This system is fundamental to calculus problems involving curves, as it provides a graphical context for mathematical equations.
In calculus, we often examine how changes in one variable (like x) affect the changes in another (like y). Understanding these relationships in a Cartesian plane provides visual clarity and aids in the comprehension of abstract mathematical relationships.
In the provided exercise, both x and y were used to understand how the subtangent behaves with respect to the x-coordinate, demonstrating the utility of the Cartesian coordinate system in solving calculus problems involving curve equations.
In calculus, we often examine how changes in one variable (like x) affect the changes in another (like y). Understanding these relationships in a Cartesian plane provides visual clarity and aids in the comprehension of abstract mathematical relationships.
In the provided exercise, both x and y were used to understand how the subtangent behaves with respect to the x-coordinate, demonstrating the utility of the Cartesian coordinate system in solving calculus problems involving curve equations.
Other exercises in this chapter
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