Ch.12 - Parametric and Polar Curves

Briggs - Calculus: Early Transcendentals 3rd Edition

Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Non è quello che usi tu?Cambia libro di testo

Briggs 3rd Edition

Ch.12 - Parametric and Polar Curves

Problema 12.1.114

Capitolo 12, Problema 12.1.114

Second derivative Assume a curve is given by the parametric equations x=f(t) and y=g(t), where f and g are twice differentiable. Use the Chain Rule to show that y″x=(fʹ(t)g″(t)−gʹ(t)f″(t))/(fʹ(t))³.

Guida verificata passo dopo passo

Recall that for parametric equations \(x = f(t)\) and \(y = g(t)\), the first derivative of \(y\) with respect to \(x\) is given by the chain rule as \(\frac{dy}{dx} = \frac{\frac{dy}{dt}}{\frac{dx}{dt}} = \frac{g'(t)}{f'(t)}\).

To find the second derivative \(\frac{d^2y}{dx^2}\), we differentiate \(\frac{dy}{dx}\) with respect to \(x\). Using the chain rule, this is \(\frac{d}{dx} \left( \frac{dy}{dx} \right) = \frac{d}{dt} \left( \frac{g'(t)}{f'(t)} \right) \cdot \frac{dt}{dx}\).

Calculate \(\frac{d}{dt} \left( \frac{g'(t)}{f'(t)} \right)\) using the quotient rule: \(\frac{d}{dt} \left( \frac{g'(t)}{f'(t)} \right) = \frac{g''(t) f'(t) - g'(t) f''(t)}{(f'(t))^2}\).

Since \(\frac{dt}{dx} = \frac{1}{\frac{dx}{dt}} = \frac{1}{f'(t)}\), substitute this into the expression for the second derivative to get \(\frac{d^2y}{dx^2} = \frac{g''(t) f'(t) - g'(t) f''(t)}{(f'(t))^2} \cdot \frac{1}{f'(t)}\).

Simplify the expression to obtain \(\frac{d^2y}{dx^2} = \frac{f'(t) g''(t) - g'(t) f''(t)}{(f'(t))^3}\), which is the desired formula.

Risposta video verificata per un problema simile:

Questa soluzione video è stata consigliata dai nostri tutor come utile per risolvere questo problema.

Durata del video:

Concetti chiave

Ecco i concetti essenziali che devi comprendere per rispondere correttamente alla domanda.

Parametric Equations and Derivatives

Parametric equations express coordinates as functions of a parameter, typically t. To analyze curves defined parametrically, derivatives of x and y with respect to t are used. Understanding how to compute first and second derivatives of these functions is essential for studying the curve's behavior.

Video consigliato:

08:02

Parameterizing Equations

Chain Rule in Differentiation

The Chain Rule allows differentiation of composite functions by relating the derivative of the outer function to the derivative of the inner function. In parametric contexts, it helps find derivatives of y with respect to x by linking dy/dt and dx/dt, crucial for expressing higher-order derivatives like y''.

Video consigliato:

05:02

Intro to the Chain Rule

Second Derivative of Parametric Curves

The second derivative y'' with respect to x measures the curvature of a parametric curve. It is found by differentiating dy/dx with respect to x, often using the chain rule to convert derivatives with respect to t into derivatives with respect to x. The formula involves first and second derivatives of both x and y with respect to t.

Video consigliato:

Percorso guidato

06:49

Differentiation of Parametric Curves

Pratica correlata

Domanda del libro di testo

53–56. Circular motion Find parametric equations that describe the circular path of the following objects. For Exercises 53–55, assume (x, y) denotes the position of the object relative to the origin at the center of the circle. Use the units of time specified in the problem. There are many ways to describe any circle.

A bicyclist rides counterclockwise with constant speed around a circular velodrome track with a radius of 50 m, completing one lap in 24 seconds.

views

Domanda del libro di testo

31–38. Equations of parabolas Find an equation of the following parabolas. Unless otherwise specified, assume the vertex is at the origin.

A parabola that opens to the right with directrix x = -4

views

Domanda del libro di testo

90–94. Focal chords A focal chord of a conic section is a line through a focus joining two points of the curve. The latus rectum is the focal chord perpendicular to the major axis of the conic. Prove the following properties.

Let L be the latus rectum of the parabola y ² =4px for p>0. Let F be the focus of the parabola, P be any point on the parabola to the left of L, and D be the (shortest) distance between P and L. Show that for all P, D+|FP|+ is a constant. Find the constant.

views

Domanda del libro di testo

Given three polar coordinate representations for the origin.

views

Domanda del libro di testo

57–64. Graphing polar curves Graph the following equations. Use a graphing utility to check your work and produce a final graph.

r² = 4 sin θ

views

Domanda del libro di testo

25–30. Converting coordinates Express the following polar coordinates in Cartesian coordinates.

(4, 5π)

views