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The acceleration of an object is given by the equation $a(t)=3 e^{t}+2 t-1$ Determine its velocity and position in 3 seconds if $s(0)=4$ and $v(0)=2$

(a) 62.26 ft/sec(b) $62.76 \mathrm{ft}$

Calculus 1 / AB

Chapter 5

Integration and its Applications

Section 2

Applications of Antidifferentiation

Integrals

Campbell University

Oregon State University

Baylor University

University of Michigan - Ann Arbor

Lectures

05:53

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.

40:35

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.

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Okay, so here are going to find the velocity, acceleration and speed. I'm at time t 0 to 0 of our vector function here are F T. Is equal to tee times I had, plus t minus one squared times J had plus the quantity t minus three cubed times K had. So um if we have that our is our position function um position vector of a particle moving along a smooth curve in space than velocity is just a to be a derivative of position. So therefore velocity here V f G is going to be equal to well one times I, so that's I. And then plus two times T minus oops G minus one times ah J plus three times t minus three squared times K. And then we have at T is equal to zero. Well this just becomes uh this is V. F. T. And envy of zero is just equal to while we have I minus again, abc T is input zero. So we have I Hat -2 times J. Hat plus ah He gets three times 0 -3. So that's three times 27 uh squares positive plus 27. Okay. Hat Okay, so there's our velocity then um acceleration is while the derivative of velocity. So here is our velocity at um times at time T 0 to 0, acceleration is jen is then well the derivative of velocity, so A F T is equal to DV DT the derivative of velocity with respect to T. So that's going to give us two J Plus six times T -3 times. Okay, okay. And then again we have at T is equal to zero while we get that A of zero is going to be equal to just to J. And then plus well six times zero minus three becomes six times minus three to minus 18. Okay, yet. Okay, so there is our acceleration at time T is equal to zero and then speed is just the magnitude of velocity. So our speed is the magnitude of our VF t the velocity function. So that is going to be equal to while the square root of um one square which is just one. And then plus two times t minus one squared, gives us a plus four times t minus one squared. And then plus the quantity three times t minus three squared. All squared gives us a plus nine times t minus three to the fourth. Okay then at again t is equal to zero. Well we would have the square root of so again this is a T. Is equal to zero now um we have one plus what? Plus four times zero minus one squared. That's four times a negative one squared. That's plus four times one. And then plus nine times zero minus three to the fourth. Well Monastery to the fourth is in 81. So we get plus nine times 81 which gives us the square root here of well one plus four plus uh, 729 which is the square root of 734. Therefore, our speed here is equal to the square root Of 734.

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