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Find the acceleration of an object for which the displacement $s$ (in $\mathrm{m}$ ) is given as a function of the time $t$ (in s) for the given value of $t.$$$s=3(1+2 t)^{4}, t=0.500$$

Calculus 1 / AB

Chapter 23

The Derivative

Section 9

Higher Derivatives

Derivatives

Harvey Mudd College

University of Nottingham

Boston College

Lectures

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The position of an object …

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Finding Speed A body'…

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Acceleration to position G…

02:22

An object has an accelerat…

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Use the given position fun…

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The acceleration function …

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An object's position …

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An object is moving in the…

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In this problem we need to determine the acceleration of an object for was the displacement. S. Is given as a function of the time. T. As three times one plus T. Two P. Hold to the power For now the acceleration of an object is the second derivative of the displacement. S. So that has determined the second derivative of S. First of all, let us find the first derivative that will be equal to three times the derivative of one plus to go to the power four. We use the powerful and the chain rule. So we obtained four times one plus two T. Pull to the power four minus one which is three. And due to the chain rule we multiply with the derivative of one plus two T. And that is equal to two. Hence we get this to be equal to 24 times one plus two t. Old cube. Next let us determine the second derivative. This will be equal to 24 times the derivative of one plus two T. Hold queue Once again we use the power rule and the chain rule, we have three times one plus two T. To the power three minus one which is two. And we multiply with the derivative of one plus two T, which is equal to two. So what we obtain is 24 times three times two which is 144 times one plus two T. Whole square. Now we need to determine the acceleration at P is equal to 0.500 seconds. So for that we need to find the value of the second derivative At the value of key is equal to 0.500. So we substitute equals to 0.500. to get 1 44 times one plus two times 0.500 old square. So this will be equal to 1 44 times one plus one whole square. So this will be 1 44 times two square, or 1 44 times four, and the value of that is equal to 576 since the displacement is given in mitos and the time is given in seconds. The unit for this will be meter per second square. Hence the required acceleration will be 576 m per second square.

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