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Ch. 3 - Derivatives
Hass - Thomas' Calculus 15th Edition
Hass15th EditionThomas' CalculusISBN: 9780137616077Non è quello che usi tu?Cambia libro di testo
Tutti i libri di testoHass 15th EditionCh. 3 - DerivativesProblema 3.4.5b
Capitolo 3, Problema 3.4.5b

Motion Along a Coordinate Line


Exercises 1–6 give the positions s = f(t) of a body moving on a coordinate line, with s in meters and t in seconds.


b. Find the body’s speed and acceleration at the endpoints of the interval.


s = 25/t² − 5/t, 1 ≤ t ≤ 5

Guida verificata passo dopo passo
1
First, find the velocity function v(t) by differentiating the position function s(t) = \(\frac{25}{t^2} - \frac{5}{t}\) with respect to time t. Use the power rule and the chain rule for differentiation.
The velocity function v(t) is the derivative of s(t), so calculate v(t) = \(-\frac{50}{t^3} + \frac{5}{t^2}\).
Next, find the acceleration function a(t) by differentiating the velocity function v(t) with respect to time t. Again, apply the power rule and the chain rule.
The acceleration function a(t) is the derivative of v(t), so calculate a(t) = \(\frac{150}{t^4} - \frac{10}{t^3}\).
Evaluate the velocity v(t) and acceleration a(t) at the endpoints of the interval t = 1 and t = 5 to find the body's speed and acceleration at these points.

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Derivative

The derivative of a function represents the rate of change of the function with respect to a variable. In the context of motion, the derivative of the position function s = f(t) with respect to time t gives the velocity of the body. Calculating the derivative is essential to find the speed and acceleration at specific points.
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Percorso guidato
05:44
Derivatives

Velocity and Speed

Velocity is the derivative of the position function with respect to time, indicating the direction and rate of motion. Speed, however, is the magnitude of velocity, representing how fast an object is moving regardless of direction. To find speed at the endpoints, compute the absolute value of the velocity at those points.
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Percorso guidato
06:29
Derivatives Applied To Velocity

Second Derivative and Acceleration

Acceleration is the rate of change of velocity with respect to time, found by taking the second derivative of the position function. It indicates how quickly the velocity is changing. For this problem, calculate the second derivative of s = f(t) to determine the acceleration at the interval's endpoints.
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06:02
The Second Derivative Test: Finding Local Extrema
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