Pregunta
upstudy study bank question image url

  1. The basics Consider the function
    a. What is its rate of change when is 0 ?
    b. What is its rate of change when is 1 ?
    c. What is its second derivative?
    d. What is its fifth derivative?

Ask by Sherman Summers. in Canada
Mar 14,2025

Solución de inteligencia artificial de Upstudy

Respuesta verificada por el tutor

Responder

Simplified Answers:
(a) Rate of change when :
0
(b) Rate of change when :
20
© Second derivative:
(d) Fifth derivative:

Solución

Step 1. Write down the function
We have
Step 2. Compute the first derivative
Differentiate each term:
  • Derivative of is .
  • Derivative of is .
  • Derivative of is .
  • Derivative of is .
  • Derivative of is .
  • Derivative of is .
Thus,
(a) Rate of change when
Evaluate at :
(b) Rate of change when
Evaluate at :
Step 3. Compute the second derivative
Differentiate :
  • Derivative of is .
  • Derivative of is .
  • Derivative of is .
  • Derivative of is .
  • Derivative of is .
Thus,
© The second derivative is
Step 4. Compute the fifth derivative
We calculate the derivatives term by term from the original :
  • For : All derivatives are (for orders ).
  • For :
    • Higher derivatives become .
  • For :
    • Subsequent derivatives are .
  • For :
    • .
  • For :
    • .
  • For :
    • .
Now, summing the contributions only from terms that survive to the fifth derivative:
  • From and and : Their fifth derivatives are .
  • From : The fifth derivative is .
  • From : The fifth derivative is .
Thus,
(d) The fifth derivative is

Respondido por UpStudy AI y revisado por un tutor profesional

error msg
Explicar
Simplifique esta solución

Bonus Knowledge

To tackle the function , we’ll dive into each part step by step.
a. The rate of change of a function at a certain point is found using the derivative. The first derivative of our function is calculated as . Evaluating this at gives .
b. Evaluating gives . So, the rate of change when is .
c. The second derivative is the derivative of : .
d. The fifth derivative can be computed from the original function; since the highest degree is , the derivatives reduce to zero after the sixth, leading us to conclude at (as it comes from the term).
So the results are , , , and .

¡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