/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 40 Determine the \(t\) intervals on... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 8: Problem 40

Determine the \(t\) intervals on which the curve is concave downward or concave upward. $$ x=2 \cos t, \quad y=\sin t, \quad 0

Short Answer

Expert verified
Detailed computations of the second derivative and determining the sign are required to find the intervals of concavity, which can be done following the steps outlined.

Step by step solution

01

Find the Derivatives

First, we need to find \(\frac{dy}{dt}\) and \(\frac{dx}{dt}\). Given:\(x=2 \cos t\) \(y=\sin t\)Differentiating both sides with respect to \(t\), we get:\(\frac{dx}{dt} =-2 \sin t\)\(\frac{dy}{dt} = \cos t\)
02

Find First Derivative of y with Respect to x

Next, find the derivative of \(y\) with respect to \(x\) (\(\frac{dy}{dx}\)) by dividing \(\frac{dy}{dt}\) by \(\frac{dx}{dt}\):\(\frac{dy}{dx} = \frac{dy/dt}{dx/dt} = - \frac{\cos t}{2 \sin t}\)
03

Find Second Derivative (\(y\) with Respect to \(x\))

The second derivative (\(\frac{d^2y}{dx^2}\)) is given by \(\frac{d}{dt} (\frac{dy}{dx}) / \frac{dx}{dt}\) First, find \(\frac{d}{dt}(\frac{dy}{dx}): -\frac{d}{dt}(\frac{\cos t}{2 \sin t})\)Then, divide that by \( \frac{dx}{dt}\):\(\frac{d^2y}{dx^2} = \frac{\frac{d}{dt}(\frac{\cos t}{2 \sin t})}{ \frac{dx}{dt} }\)
04

Determine Intervals of Concavity

Finally, determine the values of \(t\) where \(\frac{d^2y}{dx^2}\) is positive or negative. Recall that the curve is concave upward at values of \(t\) where \(\frac{d^2y}{dx^2}\) is positive, and concave downward where it is negative.

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

Find two different sets of parametric equations for the rectangular equation. $$ y=\frac{2}{x-1} $$

Consider a projectile launched at a height \(h\) feet above the ground and at an angle \(\theta\) with the horizontal. If the initial velocity is \(v_{0}\) feet per second, the path of the projectile is modeled by the parametric equations \(x=\left(v_{0} \cos \theta\right) t\) and \(y=h+\left(v_{0} \sin \theta\right) t-16 t^{2}\) The center field fence in a ballpark is 10 feet high and 400 feet from home plate. The ball is hit 3 feet above the ground. It leaves the bat at an angle of \(\theta\) degrees with the horizontal at a speed of 100 miles per hour (see figure). (a) Write a set of parametric equations for the path of the ball. (b) Use a graphing utility to graph the path of the ball when \(\theta=15^{\circ} .\) Is the hit a home run? (c) Use a graphing utility to graph the path of the ball when \(\theta=23^{\circ} .\) Is the hit a home run? (d) Find the minimum angle at which the ball must leave the bat in order for the hit to be a home run.

In Exercises \(7-16,\) find the eccentricity and the distance from the pole to the directrix of the conic. Then sketch and identify the graph. Use a graphing utility to confirm your results. \(r=\frac{6}{2+\cos \theta}\)

Use a graphing utility to graph the curve represented by the parametric equations (indicate the orientation of the curve). Eliminate the parameter and write the corresponding rectangular equation. $$ \begin{array}{l} x=4+2 \cos \theta \\ y=-1+\sin \theta \end{array} $$

Determine any differences between the curves of the parametric equations. Are the graphs the same? Are the orientations the same? Are the curves smooth? $$ \text { (a) } \begin{array}{l} x=\cos \theta \\ y=2 \sin ^{2} \theta \\ 0<\theta<\pi \end{array} $$ $$ \text { (b) } \begin{aligned} x &=\cos (-\theta) \\ y &=2 \sin ^{2}(-\theta) \\ 0 &<\theta<\pi \end{aligned} $$
See all solutions

Recommended explanations on Math Textbooks

Probability and Statistics

Read Explanation

Calculus

Read Explanation

Logic and Functions

Read Explanation

Statistics

Read Explanation

Mechanics Maths

Read Explanation

Discrete Mathematics

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.