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Suppose that the position of a particle as a function of time is given by the formula $s(t)=4 t^{2}-t^{4}$ for $t>0 .$ (a) Find the velocity and acceleration as functions of time. (b) Find the time at which the velocity is zero and the time at which the acceleration is zero. (c) Find the time at which the velocity is maximum. (d) Find the time at which the acceleration is maximum.

(a) $v=8 t-4 t^{3}, a=8-12 t^{2}$(b) $0, \sqrt{2} ; \frac{\sqrt{6}}{3}$(c) $\frac{\sqrt{6}}{3}$(d) 0

Calculus 1 / AB

Chapter 3

Applications of the Derivative

Section 3

Concavity and the Second Derivative

Derivatives

Campbell University

Harvey Mudd College

University of Michigan - Ann Arbor

University of Nottingham

Lectures

04:40

In mathematics, a derivative is a measure of how a function changes as its input changes. Loosely speaking, a derivative can be thought of as how much one quantity is changing in response to changes in some other quantity; for example, the derivative of the position of a moving object with respect to time is the object's velocity. The concept of a derivative developed as a way to measure the steepness of a curve; the concept was ultimately generalized and now "derivative" is often used to refer to the relationship between two variables, independent and dependent, and to various related notions, such as the differential.

30:01

In mathematics, the derivative of a function of a real variable measures the sensitivity to change of the function value (the rate of change of the value of the function). If the derivative of a function at a chosen input value equals a constant value, the function is said to be a constant function. In this case the derivative itself is the constant of the function, and is called the constant of integration.

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The position function of a…

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The position of a particle…

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A particle moves along a l…

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The acceleration function …

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Suppose that the equation …

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17:16

The function $s(t)$ descri…

01:49

question 16 states suppose the position of a particle is a function of time is given by S F T equals four T squared minus t to the fourth for T is greater than zero for part A. They want you to just find the velocity and acceleration functions. So velocity is just a derivative of position. So that would just be B f T equals 80 minus 40 cubed and acceleration is the derivative of velocity. So taking the derivative again, you would just get eight minus 12 t squared for part B. They would like you to find the time which velocity zero and the time which acceleration is zero. So setting those equations equal to zero VF t equals zero is equal to 80 minus 40 cubed. You can then factor out for tea so you have to minus t squared and T would then just be equal to zero or square it to Ah, another route is negative square too. But we are only looking at time greater than zero a. F t then is equal to zero when eight minus 12 t squared is equal to zero. So 12 t squared is equal to eight T squared is equal to to over three. So our T is going to be square to over three for part C. They would like you to find the time which velocity is maximum, so V f T will be a Max Orman when v prime of T is equal to zero. And then if the double prime of T is less than zero, it is concave down, which is equal to a max. So doing that, the prime of T is just a f t. Ah, so when that is equal to zero is at square to over three, then taking a prime of T, which is equivalent to the double prime of T that is just equal to negative 24 t so plugging in our time here, negative 24 times square root to over three comes out to about negative 19.59 which is negative, so concave down and it is a maximum. So your answer, then for C would be a maximum at T equals square root two ah square to over three. Then, for Part D, they would like you to find the time at which acceleration is maximum. So again, Max Hermann when a prime of T is equal to zero. So when negative 24 t is equal to zero, which is just t equals zero. To test this, a double prime of T, which is being negative 24 negative means concave down, which is a max. So there is a max AT T equals zero and those are all your answers for questions, 16.

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