Main content

## AP®︎/College Calculus AB

### Course: AP®︎/College Calculus AB > Unit 9

Lesson 3: AP Calculus AB 2011 free response- 2011 Calculus AB free response #1a
- 2011 Calculus AB Free Response #1 (b, c, & d)
- 2011 Calculus AB free response #2 (a & b)
- 2011 Calculus AB free response #2 (c & d)
- 2011 Calculus AB free response #3 (a & b)
- 2011 Calculus AB free response #3 (c)
- 2011 Calculus AB free response #4a
- 2011 Calculus AB free response #4b
- 2011 Calculus AB free response #4c
- 2011 Calculus AB free response #4d
- 2011 Calculus AB free response #5a
- 2011 Calculus AB free response #5b
- 2011 Calculus AB free response #5c
- 2011 Calculus AB free response #6a
- 2011 Calculus AB free response #6b
- 2011 Calculus AB free response #6c

© 2023 Khan AcademyTerms of usePrivacy PolicyCookie Notice

# 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.

## Want to join the conversation?

- at9:20shouldn't it be 6pi/6 + pi/6? ... just my small ocd(4 votes)
- 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..?(6 votes)
- 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.(1 vote)

- I don't understand how the equation is set up in part d. Could someone explain?(3 votes)
- 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.(1 vote)

- 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?(3 votes) - Would this have been a question where a calculator is permitted?(1 vote)
- yeah, it even says you need a graphing calculator(3 votes)

- how did he calculated fast 7(pi)/6?9:36:D(1 vote)
- When he did pi + pi/6, he changed the common denominator, but failed to indicate that when he wrote 6pi + pi/6. It should be 6pi/6 + pi/6, which is how he ended up with 7pi/6.(3 votes)

- 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.(2 votes)
- 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)
- Sal had erased part of the answer to make room for b, c, and d. The text remaining was simply part of his previous answer that said the acceleration was decreasing.(2 votes)

- 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) - At11:26, 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.(1 vote)

## Video transcript

- [Instructor] Alright,
now lets tackle part B. Find the average velocity of the particle for the time period from zero is less than or equal to t is less
than or equal to six. So our average velocity, that's just going to be
our change in position, which we could view as our displacement, divided by our change in time. Well, what is our change
in position going to be? Well they don't give us
our position function but they do give us our velocity function. And so to figure out the average velocity, we could figure out our displacement, which is going to be equal to
the integral of our velocity. The integral of v of t, dt. And we wanna find the average
velocity for this time period. So we're gonna go t equals
zero to t equals six and then we're going to divide that by the amount of time that goes by. Well our change in time is
going to be equal to six and so, this is going to be
equal to the integral from zero to six of two sine of e to the t over four power plus one, dt and then all of that divided by six. Where did I get this from? Well they tell us, what our velocity as
a function of time is, its that right over there. So we can get our calculator
out to solve this part. So on our calculator we would hit math and then we would wanna do number 9 which is definite
integral, so I'll hit nine. And we are going to go from zero to six of, this is going to be two sine of e to the.. So let me do, second e to the And I'll just use the
variable x instead of t cause its easier to type in. X divided by four power, so that's my sine and then let me close the
parentheses for the sine, plus one and then I'm integrating
here with the respect to x just cause it was more convenient. And I am going to get that,
and then I divide by six. Divide by six. And I get And when you're taking an AP Calculus exam its important to round
to three decimal places unless they tell you otherwise,
that's what they expect. So its approximately equal
to one point nine four nine. Approximately one point nine four nine. Now lets do part C. So part C Find the total distance traveled by the particle from time t
equals zero to t equals six. So you might say, hey didn't
we just figure that out? No, this is displacement. And to remember the
difference between distance and displacement, if I have
something that starts here, it goes over there and then it comes back
to right over there. The distance traveled
is the total path length so it would be that, plus this. While the displacement would
just be this right over here. So if you wanna figure out distance, what you wanna do is take the integral of the absolute value of velocity, you could think about it as you're taking the integral of the speed function. So this is just going to be the integral from zero to six of the absolute value of v of t, dt. And then we can type this, actually let me just write it out, since you would wanna do this if you're actually taking the test. So its the integral
from zero to six of the absolute value of two sine
of e to the t over four, plus one. Close the absolute value, dt. Which is going to be
approximately equal to... So once again, we hit
math, definite integral. We're going from zero to six of... Now we'll hit math again, and then we'll go to number,
we see absolute value. The absolute value of two sine of e to the, I'll use x again just cause its easier to
use on the calculator. And then I'll close the parentheses on... Whoops, let me make sure I do this right. So let me close the
parenthases on the sine and then I have plus one and then there ya go. And then I'm integrating
with respect to x. Approximately twelve
point five seven three. Now lets do part d. I'll do it right over here. Part d. For t between zero and six the particle changes
direction exactly once. Find the position of the
particle at that time. So the way that I'm gonna tackle it is first I need to
think about at what time does the particle change direction and then, I can figure out its position by taking the integral
of the velocity function. So how do I think about where that particle changes direction? Well you're going to change direction if your velocity is at zero and right before that
your velocity was positive and then right after that
your velocity is negative or the other way around, right before that your
velocity was negative and then your velocity right
after that is positive. So we just have to figure out
when does v of t equal zero? So lets just say two sine
of e to the t over four, plus one, is equal to zero. And there's multiple
ways you can solve this, but because we're allowed to use the calculator we might as well do that. So we hit math. And then we hit B right over there, I'll just hit enter. Equation solver. So I could just say, look the left side of my
equation is two sine of, we are going to go e to the, I'll just use x as what I'm solving for cause its easier to type
in, in the calculator. And close that parentheses, plus one. The right hand side is equal to zero. And now I just have to
give an initial guess and I'll just go right
in between that interval between t equals zero and t equals six, so I'll just put a three right over there. And now I just need to solve it and I can do that by hitting
alpha and then solve. And then you see, it got us
to five point one nine six. So, t is approximately equal
to five point one nine six. But remember, they're not asking us for at what time does the
particle change direction. They wanna know, what's the position of
the particle at that time. Well we know how to find
our change in position, our change in position is
going to be our displacement from time equals zero to time five point one nine six and the displacement, we're integrating the velocity function. So that's going to be two sine of e to the t over four, plus one, dt. Now this is going to be
our change in position, but where did we start? Well they say at x equals zero, or at time zero our position was two. So this is our change,
we would start here, this would give us our total position. Where we started plus our change. And so this is going to be
approximately equal to... We are going to have two plus and then I go to math, definite integral, from zero to five point one nine six and I've typed this in many times already. Two sine of e, of e, to the x divided by four power and then close that parentheses. Plus one, and then I'm
integrating with respect to x and so this gives us, fourteen point one three five. Approximately fourteen
point one three five, is its position. And let me make sure that I put those parentheses there
in the right place. And we are done.