Problem 54 - University Calculus: Early Transcendentals

Question

How do \(d y / d x\) and \(d x / d y\) seem to be related? Explain the relationship geometrically in terms of the graphs. $$x^{3}+y^{2}=\sin ^{2} y$$

Step-by-Step Solution

Answer

\( \frac{dy}{dx} \) and \( \frac{dx}{dy} \) are reciprocals. Geometrically, they represent inverse slopes on a curve.

Step 1: Differentiate Implicitly to Find \( \frac{dy}{dx} \)

To find \( \frac{dy}{dx} \), use implicit differentiation. Differentiate both sides of the equation \( x^3 + y^2 = \sin^2(y) \) with respect to \( x \). The result is:\[3x^2 + 2y \frac{dy}{dx} = 2\sin(y)\cos(y) \frac{dy}{dx}\]Solve for \( \frac{dy}{dx} \):\[ \frac{dy}{dx} = \frac{3x^2}{2\sin(y)\cos(y) - 2y} \]

Step 2: Differentiate Implicitly to Find \( \frac{dx}{dy} \)

Similarly, update the differentiation by treating \( y \) as the independent variable this time. Differentiate both sides of the equation \( x^3 + y^2 = \sin^2(y) \) with respect to \( y \):\[ 3x^2 \frac{dx}{dy} + 2y = 2\sin(y)\cos(y) \]Solve for \( \frac{dx}{dy} \):\[ \frac{dx}{dy} = \frac{2\sin(y)\cos(y) - 2y}{3x^2} \]

Step 3: Analyze the Relationship Between \( \frac{dy}{dx} \) and \( \frac{dx}{dy} \)

Observe the expressions for \( \frac{dy}{dx} \) and \( \frac{dx}{dy} \). They are reciprocals of each other:\[ \frac{dy}{dx} = \frac{1}{\frac{dx}{dy}} \]The geometric interpretation is that the slope of the tangent line to the curve at any point is inversely related depending on which variable you consider the dependent variable.

Step 4: Geometric Interpretation

Geometrically, \( \frac{dy}{dx} \) represents the slope of the tangent to the curve at a point if you think of \( y \) as a function of \( x \). Conversely, \( \frac{dx}{dy} \) is the reciprocal slope, illustrating how \( x \) changes with \( y \). On graph plots, a steep curve vertically in \( dy/dx \) will appear shallow in terms of \( dx/dy \) and vice versa.

Key Concepts

Inverse Functions

Inverse functions are a fascinating part of calculus that help us understand how two variables relate to one another in a reversible way. They essentially flip the role of the input and output. For instance, if you start with a function \( f(x) \), the inverse function, denoted as \( f^{-1}(x) \), will reverse that process, turning outputs back into inputs. This creates a symmetry that is quite insightful when examining graphs and equations.

When exploring implicit differentiation, the concept of inverse functions comes into play when you recognize that \( \frac{dy}{dx} \) and \( \frac{dx}{dy} \) are inversely related. This relationship can be encapsulated in the equation:

• \( \frac{dy}{dx} = \frac{1}{\frac{dx}{dy}} \)

This means that the rate at which \( y \) changes with \( x \) is the reciprocal of the rate at which \( x \) changes with \( y \). In terms of their functions, if you have \( y \) as a function of \( x \), the inverse function will describe \( x \) as a function of \( y \). This interplay is crucial in analyzing different real-world situations like motion, economics, and geometry.

Tangent Line

The tangent line is a key concept in calculus. It touches a curve at exactly one point and represents the immediate rate of change of the function at that point. When we talk about implicit differentiation and slopes such as \( \frac{dy}{dx} \), we are fundamentally discussing the slope of the tangent line to a curve.

The equation for a tangent line can be expressed as:

• \( y = m(x - x_0) + y_0 \)

Here, \( m \) stands for the slope of the tangent line, and \( (x_0, y_0) \) is the point of tangency. Understanding the tangent line helps in visualizing how steep a function is at any given point, and therefore how rapidly the function's value is changing.

Geomtrically, when you consider the statements made for \( \frac{dy}{dx} \) and \( \frac{dx}{dy} \), they are both slopes of tangent lines but just in different perspectives. They allow us to comprehend how the function behaves, whether \( y \) is changing with respect to \( x \) or the reverse.

Slope

The slope is a vital measurement in mathematics and represents how steep a line is. It is a handy way to describe how one variable changes in relation to another. Mathematically, the slope \( m \) of a line is typically expressed as \( \frac{rise}{run} \).

In the context of implicit differentiation, \( \frac{dy}{dx} \) and \( \frac{dx}{dy} \) are both expressions of slopes depending on which variable is considered as the dependent one. For \( \frac{dy}{dx} \), it's the "traditional" slope we often refer to in calculus. It indicates how much \( y \), the dependent variable, changes with a small change in \( x \). Conversely, \( \frac{dx}{dy} \) flips this notion, measuring the slope when \( x \) depends on \( y \).

• For a steep curve, \( dy/dx \) will have a significant value when \( x \) changes slightly and \( y \) changes considerably.
• Whereas \( dx/dy \), in this case, will appear less steep because \( x \) changes less with respect to \( y \).

In graphs, these slopes sometimes give the visual impression of the curve flattening or steepening depending on their direction, offering deep geometric insights into the function's behavior.