Skip to main content
Ch. 3 - Derivatives
Briggs - Calculus: Early Transcendentals 3rd Edition
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Non è quello che usi tu?Cambia libro di testo
Tutti i libri di testoBriggs 3rd EditionCh. 3 - DerivativesProblema 55d
Capitolo 3, Problema 55d

An object oscillates along a vertical line, and its position in centimeters is given by y(t)=30(sint - 1), where t ≥ 0 is measured in seconds and y is positive in the upward direction.
At what times and positions is the velocity zero?

Guida verificata passo dopo passo
1
To find when the velocity is zero, we first need to determine the velocity function. The velocity is the derivative of the position function y(t) with respect to time t.
Differentiate y(t) = 30(sin(t) - 1) with respect to t. The derivative of sin(t) is cos(t), and the derivative of a constant is zero. Therefore, the velocity function v(t) is v(t) = 30cos(t).
Set the velocity function equal to zero to find the times when the velocity is zero: 30cos(t) = 0.
Solve the equation cos(t) = 0. The cosine function is zero at odd multiples of π/2, i.e., t = (2n+1)π/2, where n is an integer.
Substitute these values of t back into the original position function y(t) to find the corresponding positions. Calculate y(t) = 30(sin((2n+1)π/2) - 1) for each n to find the positions when the velocity is zero.

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:
5m

Concetti chiave

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

Velocity

Velocity is the rate of change of an object's position with respect to time. In calculus, it is often represented as the derivative of the position function. For the given function y(t) = 30(sin(t) - 1), the velocity can be found by differentiating this function with respect to time t, yielding v(t) = dy/dt.
Video consigliato:
Percorso guidato
06:29
Derivatives Applied To Velocity

Critical Points

Critical points occur where the derivative of a function is zero or undefined. In the context of the velocity function, finding when the velocity is zero will help identify the times at which the object is momentarily at rest. These points are essential for analyzing the motion of the object and determining its behavior over time.
Video consigliato:
04:50
Critical Points

Oscillation

Oscillation refers to the repetitive variation in position around a central point, often described by sinusoidal functions. In this case, the position function y(t) = 30(sin(t) - 1) indicates that the object oscillates vertically, with its motion influenced by the sine function, which varies between -1 and 1. Understanding oscillation helps in predicting the object's motion and identifying key characteristics such as amplitude and period.
Video consigliato:
03:07
Cases Where Limits Do Not Exist
Pratica correlata
Domanda del libro di testo

A capacitor is a device in an electrical circuit that stores charge. In one particular circuit, the charge on the capacitor Q varies in time as shown in the figure. <IMAGE>

b. Is Q′ positive or negative for t≥0?

175
views
Domanda del libro di testo

The energy (in joules) released by an earthquake of magnitude M is given by the equation E = 25,000 ⋅ 101.5M. (This equation can be solved for M to define the magnitude of a given earthquake; it is a refinement of the original Richter scale created by Charles Richter in 1935.)

Compute the energy released by earthquakes of magnitude 1, 2, 3, 4, and 5. Plot the points on a graph and join them with a smooth curve.

482
views
Domanda del libro di testo

Evaluate and simplify y'.

x = cos (x−y)

286
views
Domanda del libro di testo

An object oscillates along a vertical line, and its position in centimeters is given by y(t) = 30(sin t - 1), where t ≥ 0 is measured in seconds and y is positive in the upward direction.

The acceleration of the oscillator is a(t) = v′(t). Find and graph the acceleration function.

220
views
Domanda del libro di testo

A capacitor is a device in an electrical circuit that stores charge. In one particular circuit, the charge on the capacitor Q varies in time as shown in the figure. <IMAGE>

a. At what time is the rate of change of the charge Q' the greatest?

158
views
Domanda del libro di testo

Calculate the derivative of the following functions.

y = cos7/4(4x3)

357
views