Question

True or False The slope of \(x y^{2}+x=1\) at \((1 / 2,1)\) is \(2 .\) Justify your answer.

Short answer

False

Step-by-step solution

Differentiate the given function

Given the function \(f(x) = xy^{2}+x-1\), we can rewrite this in terms of y: \(f(x) = x(y^{2}+1) - 1\). Differentiating both sides with respect to x, it gives the derivative: \(f'(x) = y^{2} + 1 + 2xy\frac{dy}{dx}\). From this equation, isolate the term \(\frac{dy}{dx}\), which represents the slope at a given point. \(\frac{dy}{dx} = \frac{- y^{2} - 1}{2xy}\).

Substituting the given coordinates

Substitute \(x = 1/2\) and \(y = 1\) into the equation for \(\frac{dy}{dx}\) to find the slope at the point (1/2, 1). This gives: \(\frac{dy}{dx}|_{x=1/2, y=1} = \frac{- (1)^{2} - 1}{2(1/2)(1)} = -2.\)

Compare the result with the given slope

The slope computed is -2, which is not the same as the given slope, 2. Thus, the statement is false.

Key concepts

Slope of a Curve

Understanding the slope of a curve is key in calculus, especially when discussing motion or the rate of change. Unlike straight lines with constant slopes, the slope of a curve varies at every point. It is the steepness or incline of the curve at a particular point and is represented mathematically by the derivative at that point.

For a curve given by an equation involving both x and y, such as \(xy^2 + x = 1\), the slope is not straightforward to calculate. We can't directly use the rise-over-run formula as we do with lines. Instead, we need a tool from calculus, called implicit differentiation, to find the derivative that reflects the slope. This derivative, represented as \(\frac{dy}{dx}\) when differentiated with respect to x, provides us with the rate of change of y with respect to x at any point on the curve.

Derivative

The derivative represents the rate at which one quantity changes with respect to another. In the context of functions, it is the instantaneous rate of change of the function's output with respect to its input. This is why it's such a powerful concept in calculating movement, growth rates, and optimization problems in various fields of study.

When you differentiate a function that's explicitly given in terms of one variable, like \(y = f(x)\), you directly apply the rules of differentiation. However, if a function involves both x and y, as with implicit functions, you use implicit differentiation. This method takes advantage of the relationship between x and y to find \(\frac{dy}{dx}\) without explicitly solving for y. The derivative is key in determining the slope of a curve at a given point, providing critical insights into the behavior of variables in a system.

Substituting Coordinates

Substituting coordinates into a derivative is a practical application of calculus. Once the general form of the derivative, or \(\frac{dy}{dx}\), is found, you can determine the slope of the function at any specific point by substituting the x and y values of that point into the derivative formula.

This process involves taking the coordinates of the given point, which are typically in the form \((x, y)\), and inserting them into the derivative to calculate the specific numerical value of the slope at that point. For instance, in our exercise, after finding that \(\frac{dy}{dx} = \frac{- y^{2} - 1}{2xy}\), we substitute \(x = 1/2\) and \(y = 1\) to find the exact slope of the curve at that point. This step can confirm or refute a proposed slope value, ensuring the accuracy of graphical models or predictions based on the slope.

Justifying Solutions in Calculus

In calculus, justifying solutions goes beyond merely arriving at an answer; it involves explaining the reasoning or the method used to reach that answer. Justification confirms the validity of the process taken and is an essential skill in mathematics because it teaches critical thinking and analytical reasoning.

When justifying whether the slope of a curve at a particular point is a specific value, as was presented in the original exercise, it's not enough to state the answer as true or false. You must show the work and reasoning that leads to the conclusion. In the given example, implicit differentiation was properly used to find the derivative, the coordinates were correctly substituted to calculate the specific slope at the point, and then it was compared against the proposed slope value. This methodical approach confirms that the given statement about the slope is false, as demonstrated through a structured and justifiable calculus method.