91Ó°ÊÓ

Concavity is a property of a curve that tells us about its shape or direction. When a curve is said to be concave upward, it means that the curve opens upwards, similar to a cup or a bowl. This occurs when the second derivative of the curve, denoted as \( \frac{d^2y}{dx^2} \), is positive. It indicates that the slope of the tangent lines is increasing.

To determine the concavity of a curve described by parametric equations, like the given \( x = t + \ln t \) and \( y = t - \ln t \), we first find the second derivative \( \frac{d^2y}{dx^2} \). This involves differentiating \( \frac{dy}{dx} \) using techniques like the quotient rule. If the resulting function is positive for most or all values of \( t \), the curve is concave upward. In this case, \( \frac{d^2y}{dx^2} = \frac{2}{(t+1)^2} \), which is always greater than zero for \( t eq -1 \), confirming upward concavity for most of the curve.

The chain rule is a fundamental technique in calculus used to differentiate composite functions. When dealing with parametric equations, like \( x = t + \ln t \) and \( y = t - \ln t \), we often apply the chain rule to connect derivatives with respect to one variable to another. This is achieved by determining \( \frac{dy}{dx} \) indirectly through \( t \).

To apply the chain rule for parametric differentiation, we find \( \frac{dy}{dt} \) and \( \frac{dx}{dt} \) first. Then, we use the relation: \[ \frac{dy}{dx} = \frac{\frac{dy}{dt}}{\frac{dx}{dt}} \] This converts derivatives with respect to \( t \) back to conventional derivatives with respect to \( x \), as needed to analyze the curve's behavior, like its slope at different points.

The quotient rule is a method in calculus used to find the derivative of a quotient of two functions. When we have \( \frac{dy}{dx} = \frac{t-1}{t+1} \), and need to compute its derivative to find the second derivative \( \frac{d^2y}{dx^2} \), the quotient rule becomes necessary. The formula for the quotient rule is:

• \( \frac{d}{dt}\left( \frac{u}{v} \right) = \frac{v \frac{du}{dt} - u \frac{dv}{dt}}{v^2} \)

Here, \( u = t - 1 \) and \( v = t + 1 \). By applying the quotient rule, we find:

• \( \frac{d^2y}{dx^2} = \frac{(t+1)(1) - (t-1)(1)}{(t+1)^2} = \frac{2}{(t+1)^2} \)

This result helps us describe the concavity of the curve by evaluating whether the second derivative is positive or negative across the domain of the curve.

Parametric equations offer a way to describe curves by expressing the coordinates \( (x, y) \) as functions of an independent parameter \( t \). Instead of using an equation relating \( x \) and \( y \) directly, the curve is defined by equations like \( x = f(t) \) and \( y = g(t) \).

For these equations, we derive \( \frac{dy}{dx} \) - the rate of change of \( y \) with respect to \( x \) - by using the derivatives \( \frac{dx}{dt} \) and \( \frac{dy}{dt} \). This approach is quite powerful, particularly in complex shapes or curves that aren't simple functions of \( x \). With the example \( x = t + \ln t \) and \( y = t - \ln t \), we find:

• \( \frac{dx}{dt} = 1 + \frac{1}{t} \)
• \( \frac{dy}{dt} = 1 - \frac{1}{t} \)

By dividing these, \( \frac{dy}{dx} = \frac{t-1}{t+1} \), simplifies our study of how \( y \) changes with \( x \), crucial for analyzing the curve's features like its slope and concavity.