Pregunta
upstudy study bank question image url

\( 1 \leftarrow \) Use the Chain Rule to find \( \frac{d Q}{d t} \), where \( Q=\sqrt{x^{2}+y^{2}+z^{2}}, x=\sin t, y=\cos t \), and \( z=\cos t \).

Ask by Norris Valdez. in the United States
Mar 17,2025

Solución de inteligencia artificial de Upstudy

Respuesta verificada por el tutor

Responder

\[ \frac{dQ}{dt} = \frac{-\cos t\,\sin t}{\sqrt{\sin^2 t + 2\cos^2 t}}. \]

Solución

¡Inicia sesión para desbloquear respuestas gratis!

Una plataforma de aprendizaje en la que confían millones de estudiantes y profesores reales.

star-icon Descubrir

Respondido por UpStudy AI y revisado por un tutor profesional

Extra Insights

To find \( \frac{dQ}{dt} \) using the Chain Rule, we first express \( Q \) in terms of \( t \): \[ Q = \sqrt{x^2 + y^2 + z^2} = \sqrt{(\sin t)^2 + (\cos t)^2 + (\cos t)^2} \] This simplifies to: \[ Q = \sqrt{\sin^2 t + \cos^2 t + \cos^2 t} = \sqrt{1 + \cos^2 t} \] Next, we apply the Chain Rule: \[ \frac{dQ}{dt} = \frac{1}{2\sqrt{1 + \cos^2 t}} \cdot \frac{d}{dt}(1 + \cos^2 t) \] Now we differentiate \( 1 + \cos^2 t \): \[ \frac{d}{dt}(1 + \cos^2 t) = 2\cos t(-\sin t) = -2\cos t \sin t \] Substituting back, we get: \[ \frac{dQ}{dt} = \frac{-2\cos t \sin t}{2\sqrt{1 + \cos^2 t}} = \frac{-\cos t \sin t}{\sqrt{1 + \cos^2 t}} \] Thus, the final result is: \[ \frac{dQ}{dt} = \frac{-\cos t \sin t}{\sqrt{1 + \cos^2 t}} \]

preguntas relacionadas

If you invest \( \$ 1000 \) for \( t \) years at \( 6.71 \% \) compounded at the frequencies below, find the following. (a) Suppose you compound at \( 6.71 \% \) monthly. i) Report an expression equivalent to the value of \( \$ 1000 \) invested for \( t \) years at \( 6.71 \% \) compounded monthly by completing the box with the growth factor if compounded annually. 1000 \( \square \) Number \( t \) (Round to \( \underline{4} \) decimal places.) ii) Report the effective annual rate: \( \square \) Number \% (Round to \( \underline{2} \) decimal places.) (b) Suppose you compound at \( 6.71 \% \) continuously. i) You would expect \( 6.71 \% \) compounded continuously to give a \( \square \) Click for List yield than what is given in part (a). ii) Complete the boxes below to report the expression for the value of \( \$ 1000 \) invested for \( t \) years at \( 6.71 \% \) compounded continuously and the equivalent growth factor if compounded annually. \[ \begin{array}{l} 1000 e^{(\text {Number } t)} \\ \approx 1000(\text { Number })^{t} \end{array} \] (Round to \( \underline{4} \) decimal places.) iii) Report the effective annual rate: \( \square \) Number \% (Round to \( \underline{2} \) decimal places.) (c) Complete the boxes to summarize: i) From part (a) we have that 6.71 \% compounded monthly is equivalent to \( \square \) Number \( \% \) compounded annually. ii) From part (b) we have that 6.71 \% compounded continuously is equivalent to \( \square \) Number \( \% \) compounded annually.

Latest Calculus Questions

If you invest \( \$ 1000 \) for \( t \) years at \( 6.71 \% \) compounded at the frequencies below, find the following. (a) Suppose you compound at \( 6.71 \% \) monthly. i) Report an expression equivalent to the value of \( \$ 1000 \) invested for \( t \) years at \( 6.71 \% \) compounded monthly by completing the box with the growth factor if compounded annually. 1000 \( \square \) Number \( t \) (Round to \( \underline{4} \) decimal places.) ii) Report the effective annual rate: \( \square \) Number \% (Round to \( \underline{2} \) decimal places.) (b) Suppose you compound at \( 6.71 \% \) continuously. i) You would expect \( 6.71 \% \) compounded continuously to give a \( \square \) Click for List yield than what is given in part (a). ii) Complete the boxes below to report the expression for the value of \( \$ 1000 \) invested for \( t \) years at \( 6.71 \% \) compounded continuously and the equivalent growth factor if compounded annually. \[ \begin{array}{l} 1000 e^{(\text {Number } t)} \\ \approx 1000(\text { Number })^{t} \end{array} \] (Round to \( \underline{4} \) decimal places.) iii) Report the effective annual rate: \( \square \) Number \% (Round to \( \underline{2} \) decimal places.) (c) Complete the boxes to summarize: i) From part (a) we have that 6.71 \% compounded monthly is equivalent to \( \square \) Number \( \% \) compounded annually. ii) From part (b) we have that 6.71 \% compounded continuously is equivalent to \( \square \) Number \( \% \) compounded annually.
¡Prueba Premium ahora!
¡Prueba Premium y hazle a Thoth AI preguntas de matemáticas ilimitadas ahora!
Quizas mas tarde Hazte Premium
Estudiar puede ser una verdadera lucha
¿Por qué no estudiarlo en UpStudy?
Seleccione su plan a continuación
Prima

Puedes disfrutar

Empieza ahora
  • Explicaciones paso a paso
  • Tutores expertos en vivo 24/7
  • Número ilimitado de preguntas
  • Sin interrupciones
  • Acceso completo a Respuesta y Solución
  • Acceso completo al chat de PDF, al chat de UpStudy y al chat de navegación
Básico

Totalmente gratis pero limitado

  • Solución limitada
Bienvenido a ¡Estudia ahora!
Inicie sesión para continuar con el recorrido de Thoth AI Chat
Continuar con correo electrónico
O continuar con
Al hacer clic en "Iniciar sesión", acepta nuestros términos y condiciones. Términos de Uso & Política de privacidad