Question

Parametric equations for a curve are given. Find \(\frac{d^{2} y}{d x^{2}},\) then determine the intervals on which the graph of the curve is concave up/down. \(x=\cos t \sin (2 t), \quad y=\sin t \sin (2 t)\) on \([-\pi / 2, \pi / 2]\)

Short answer

The curve is concave up when \( \frac{d^{2} y}{d x^{2}} > 0 \) and concave down when \( \frac{d^{2} y}{d x^{2}} < 0 \). Evaluate \( \frac{d^{2} y}{d x^{2}} \) to find exact intervals.

Step-by-step solution

Find the derivatives dx/dt and dy/dt

First, we need to calculate the derivatives of the parametric equations with respect to \( t \). Because \( x = \cos t \sin 2t \), use the product rule: \( \frac{dx}{dt} = -\sin t \sin 2t + \cos t \cdot 2\cos 2t \). Similarly, for \( y = \sin t \sin 2t \), \( \frac{dy}{dt} = \cos t \sin 2t + \sin t \cdot 2\cos 2t \).

Compute dy/dx

To find \( \frac{dy}{dx} \), use \( \frac{dy}{dt} \) and \( \frac{dx}{dt} \): \( \frac{dy}{dx} = \frac{\frac{dy}{dt}}{\frac{dx}{dt}} = \frac{\cos t \sin 2t + 2\sin t \cos 2t}{-\sin t \sin 2t + 2\cos t \cos 2t} \).

Differentiate dy/dx with respect to t to find d^2y/dx^2

Find \( \frac{d}{dt}\left(\frac{dy}{dx}\right) \) by using the quotient rule: \( \frac{d}{dt}\left(\frac{dy}{dx}\right) = \frac{N'B - NB'}{B^2} \), where \( N \) is the numerator and \( B \) is the denominator of \( \frac{dy}{dx} \). Compute this derivative step by step.

Calculate d^2y/dx^2 from the result

After you have \( \frac{d}{dt}\left(\frac{dy}{dx}\right) \), divide that by \( \frac{dx}{dt} \) again to find \( \frac{d^2 y}{d x^2} \): \( \frac{d^2 y}{d x^2} = \frac{\frac{d}{dt}\left(\frac{dy}{dx}\right)}{\frac{dx}{dt}} \).

Determine intervals of concavity

Evaluate \( \frac{d^{2} y}{d x^{2}} \) over the interval \([-\pi/2, \pi/2]\) to find where the expression is positive (indicating concave up) or negative (indicating concave down). Test critical points and endpoints of the interval for changes in sign.

Key concepts

Parametric Equations

Parametric equations provide a way to define a curve through two separate equations for each coordinate, often using a third variable, usually denoted as \( t \). These can express complex curves that may be tough or impossible to represent with a single standard form equation. For instance, in the given exercise, the equations \( x = \cos t \sin 2t \) and \( y = \sin t \sin 2t \) describe the curve parametrically.

• **Component Equations**: Each function describes how the respective \( x \) and \( y \) values change over the parameter \( t \).
• **Range**: Consider the parameter's specific range, here \([ -\pi/2, \pi/2 ]\), which tells us how far \( t \) goes and affects the shape and portion of the curve you analyze.

Overall, parametric equations serve as a versatile tool for illustrating complex figures or time-evolving systems. They let each coordinate be independently evaluated, offering a clearer understanding of how each part of the curve behaves.

Second Derivative

In calculus, the second derivative \( \frac{d^2 y}{d x^2} \) provides insight into the curvature of a graph, telling us how the rate of change of the slope itself changes. This measure is crucial to understanding how a curve behaves as it evolves.
Finding the second derivative from parametric equations involves several steps:

• **Compute Initial Derivatives**: First, determine \( \frac{dx}{dt} \) and \( \frac{dy}{dt} \) to aid in finding \( \frac{dy}{dx} \).
• **Apply Quotient Rule**: To directly compute \( \frac{d}{dt}\left(\frac{dy}{dx}\right) \), use the quotient rule, where given fractions, \( N' \) and \( B \) are the derivatives of the numerator and denominator, respectively.
• **Final Expression**: Combine these derivatives accordingly, and divide by \( \frac{dx}{dt} \) again to yield the second derivative \( \frac{d^2 y}{d x^2} \).

This finalized derivative expression represents where and how the original parametric curve behaves under certain conditions, leading directly into analyzing concavity.

Concavity

Understanding concavity involves evaluating the second derivative to determine how a function 'curves.' A graph can be described as being concave up, resembling an upward-facing curve, or concave down, like an downward-facing curve.
Concavity with parametric equations takes on special importance:

• **Concave Up vs. Down**: A positive \( \frac{d^2 y}{d x^2} \) indicates a concave up segment, whereas a negative value points to a concave down section.
• **Interval Testing**: By evaluating \( \frac{d^2 y}{d x^2} \) over intervals, such as \([-\pi/2,\pi/2]\), pivotal points and endpoints reveal where changes in concavity occur.

These evaluations illuminate the shape and nature of the curve visually and predicatively. Understanding concavity assists in visual assessments and further mathematical analysis, highlighting regions of potential interest or concern depending on the context.