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# 2011 Calculus AB Free Response #1 (b, c, & d)

Integration to find average value of a function. Using a graphing calculator to calculate definite integrals. Created by Sal Khan.

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• at shouldn't it be 6pi/6 + pi/6? ... just my small ocd • In part d, Sal takes the definite integral of V(t) to get the position function but includes a constant c which he solves for later. When a constant of integration isn't required for definite integrals why does Sal include one? Or is this something I don't know..? • Let me try to explain it another way. When you're integrating a velocity function from [A,B], think of it as finding the distance traveled when the particle was in between A and B, and not counting the distance that the particle already STARTED at. The constant was added because the integrating part finds out where the patricle traveled to and how far, while the constant adds in the place where the particle started to move from.
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• I don't understand how the equation is set up in part d. Could someone explain? • The integral of v(t) tells us the displacement or position of the particle. The anti-derivative is missing the constant because we don't know the constant for the initial function. However, when given the initial constant x(0)=2, we know the beginning position along the horizontal position. This should hopefully explain why the equation is set up that way.
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• b: correct me if im wrong .
for t beetween 0 to 6 the particle has some negative movement(x<0)
I think if we differ beetwen speed and velocity in part "A" we must keep the same criteria in part B.
they request the avrage velocity and not avrage Speed.So :just find the distance beetween t=0 and t=6 and devide it to 6.and forget about the displacement of particle going and coming back,just the difference beetween start taime and end Time.
AM I CORRECT? • Would this have been a question where a calculator is permitted?
(1 vote) • how did he calculated fast 7(pi)/6? :D
(1 vote) • Just a question, 1a's answer is speed is increasing from previous video right? Then why does he have the answer on a as "decreasing" in this video? It's kind of confusing. • In part a, Sal says that speed is increasing even though the velocity is decreasing, but in this video he writes at the top that the speed in part a was decreasing. How do I determine whether or not it is decreasing? Do I compare the numbers I got in the acceleration and velocity equations? If acceleration is smaller than velocity is speed increasing?
(1 vote) • At the last few seconds of the video, shouldn't the answer be 14.135 and not 14.134?
Would they mark this part wrong if we make a small rounding error like this?
(1 vote) • At , I don't understand why the position function is the integral of the velocity function. Can anyone help me?
(1 vote) • We need to ask ourselves question - what is velocity? One answer I could give you is that velocity is how fast object moves, or how fast it changes its position. Well in mathematics there is a way to describe that - derivative. Derivative of a function with respect to given variable tells us how fast value of given function changes as the variable grows.
So let's define x(t) to be function of position, with respect to time. Therefore, its derivative, will be velocity - v(t) - because it tells us how fast the position changes, which is consistent with our understanding of velocity.
Now that we know that v(t) is derivative of x(t), we can also say that x(t) is integral of v(t) - since integral is opposite of derivative.

Another way to understand it is to split time into very tiny little intervals - call them dt. Since those intervals are infinitely small, we can assume that in each of those intervals velocity is constant, and it equals v(t). Therefore distance traveled in that interval of time equals to v(t) * dt. Now we need to add distances traveled during all of those periods of time to each other - and we do it by integration.
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