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The acceleration function (in $ m/s^2 $) and the initial velocity are given for a particle moving along a line. Find (a) the velocity at time $ t $ and (b) the distance traveled during the given time interval.
$ a(t) = 2t + 3 $, $ v(0) = -4 $, $ 0 \le t \le 3 $
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Calculus 1 / AB
Indefinite Integrals and the Net Change Theorem
Oregon State University
Harvey Mudd College
University of Nottingham
Idaho State University
In mathematics, an indefinite integral is an integral whose integrand is not known in terms of elementary functions. An indefinite integral is usually encountered when integrating functions that are not elementary functions themselves.
In mathematics, integration is one of the two main operations of calculus, with its inverse operation, differentiation, being the other. Given a function of a real variable (often called "the integrand"), an antiderivative is a function whose derivative is the given function. The area under a real-valued function of a real variable is the integral of the function, provided it is defined on a closed interval around a given point. It is a basic result of calculus that an antiderivative always exists, and is equal to the original function evaluated at the upper limit of integration.
The acceleration function …
The velocity function (in …
A particle moves with acce…
Alright I think give us the acceleration is to T plus three to t plus three then. Uh And we also know that the derivative of velocity is equal to acceleration. So if I find the integral of that uh it will give me the velocity so that now I. M. V. Of T. And the anti directive we add one to the expo divided by your new X 12 by by two is one Plus three T Plus C. And that's where we're going to use B. Uh Zero is equal to negative four because as I plug in negative four and for velocity and plug in zero for these two T 00 squared plus three times zero is just zero. Your left was C. Is equal to negative four. So the equation of velocity would be T squared plus three t minus four. Now as far as total distance traveled goes um Probably what I would do is first of all we care about from 0 to 3 zero 23 Um as I would see if there's any change in direction and you find the changing direction by setting this equal to zero. If you find factors in mega for that had to be three before -1. So there will be T plus four t minus one. So this object changes directions at T equals negative for which doesn't really make any sense and T.E. is one. Um Let's see. So what would I do? I would you're gonna have to find the integral of this, which is why this problem is so complicated to find that's right T D. T. The total distance traveled. That integral from 0 to 1 of that function. GT Yeah. Starting here and then it changes directions. Uh and then the integral from 1 to 3 of t squared plus three t minus four. Yeah. Gt uh Now one of these answers to be negative because we're moving to the left I don't know which one that is. I'm just doing the anti derivative From 0 to wine and then the same thing here really all the same work but as you plug in one you'll get one third plus three halves minus four. Yeah that's going to be a negative number. So absolute value that thing because total additions traveler needs to be positive and then add to it. This I guess I didn't show plugging in zero but you get a bunch of zeros. Um One third plus three halves one third times three squared cubed times three halves times three squared minus four times three. You can see why there's so much work to be done here minus plugging in all those ones now. On third plus three hands minus four. Yeah. Yeah. Uh I'll go ahead and put absolute value there just in case. Um And let's see yeah we can go to calculate and just type that in. Mhm. See absolute value that and then we squared Plus three cubed divided by two minus slopes. I'm using a calculator and I'm struggling right now minus 12 minus the quantity. One third plus 3/2 -4 hopefully attacked us all incorrectly because I don't want to go back and change anything. Um And I can change that to a decimal or to a fraction as well. Math track will say 89 6. Here we go.
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