yego.me
💡 Stop wasting time. Read Youtube instead of watch. Download Chrome Extension

2015 AP Calculus BC 2a | AP Calculus BC solved exams | AP Calculus BC | Khan Academy


3m read
·Nov 11, 2024

At time ( T ) is greater than or equal to zero, a particle moving along a curve in the XY plane has position ( X(T) ) and ( Y(T) ). So, its x-coordinate is given by the parametric function ( X(T) ) and y-coordinate by the parametric function ( Y(T) ).

With the velocity vector ( V(T) ) equal to, and the x-component of the velocity vector is ( \cos(T^2) ), and the y-component of the velocity vector is ( e^{0.5T} ). At ( T=1 ), the particle is at the point ( (3, 5) ).

All right, find the x-coordinate of the position of the particle at time ( T=2 ). All right, so how do we think about this? Well, you could view the x-coordinate at time ( T=2 ). So, let's say, we could say ( X(2) ), which they don't give to us directly. But we could say that's going to be ( X(1) ) plus some change in x as we go from ( T=1 ) to ( T=2 ).

But what is this going to be? Well, we know what the velocity is, and so the velocity, especially the x-component, we can really focus on the x-component for this first part because we only want to know the x-coordinate of the position of the particle. Well, we know we're going—we know the x-component of velocity is a function of ( T ): ( \cos(T^2) ).

If you take your velocity in a certain dimension and then multiply it times a very small change in time, ( dT ), this would give you your very small change in x. If you multiply velocity times change in time, it'll give you a displacement. But what we can do is we can sum up all of the changes in time from ( T=1 ) to ( T=2 ).

Remember this is the change in x from ( T=1 ) to ( T=2 ). So what we have right over here, we can say that ( X(2) ), which is what we're trying to solve, is going to be ( X(1) ). They give that at time ( T=1 ), the particle is at the point ( (3, 5) ). Its x-coordinate is three, so this right over here is three.

Then, our change in x from ( T=1 ) to ( T=2 ) is going to be this integral: the integral from ( T=1 ) to ( T=2 ) of ( \cos(T^2) dT ).

Just to make sure we understand what's going on here, remember how much we are moving over a very small ( dT ). Well, you take your velocity in that dimension times ( dT ), it'll give you your displacement in that dimension, and then we sum them all up from ( T=1 ) to ( T=2 ).

In this part of the AP test, we are allowed to use calculators, and so, let's use one. All right, so there's my calculator, and I can evaluate. So let's see, I want to evaluate three plus the definite integral.

I click on math, and then I can scroll down to function integral right there, the definite integral of—and I make sure I'm in radian mode, which that's what you should assume—so ( \cos(T^2) ).

Now, I'll use ( x ) as my variable of integration, so I'll say ( \cos(x) ) of ( x^2 ), and my variable of integration is ( x ). I'm really integrating ( \cos(x^2) , dx ) but it'll give the same value from 1 until 2.

Now, I let the calculator munch on it a little bit, and I get approximately 2.557. So this is approximately 2.55. Did I—let me make sure that I added the three? Yeah, three plus that definite integral from ( 1 ) to ( 2 ) is 2.55, and I just rounded that. So there you go.

More Articles

View All
Couples Share the Happiness and Heartache of Interracial Marriage | National Geographic
That was the first time that I initially told him that I loved him was at Cairo. Do well, he likes to yodel. I can almost cry describing her. She’s the love of my life. I fell in love with her as she was getting out of a taxi the first time I ever saw her…
Stripe Head of Design Katie Dill Reviews Startup Websites
I’m Ain Epstein and welcome to another episode of Design Review. Today, I’m going to be joined by Katie Dill, who is the Head of Design at Stripe, and we’re going to be taking a look at a bunch of user-submitted websites to give them feedback on how they …
Remembering the Battle of Mogadishu | No Man Left Behind
My role in that battle was a team leader with one of the platoons that went in on the air assault. I went and originally on the helicopters. When you make it out of something where others didn’t, you’re going to spend the rest of your life thanking the pe…
Finding the end time for a movie in 24 hour time | Math | Khan Academy
We are told that Andre goes to a movie that starts at 19 hours 45 minutes, or 1945, and is 90 minutes long. What time is the movie finished? So pause this video and see if you can answer that before we work through it together. All right, so what I like …
Sal Khan on the importance of free, high-quality AI tools for teachers & district leaders
Hello everyone! Good afternoon. We are slowly welcoming folks into the room. Thanks for taking the time to join us. My name is Philipe Esamia. I’m the video manager here at KH Academy. Uh, shortly we will be joined by S on, the CEO and founder of KH Acade…
Uncut Interview with Sam Altman on Masters of Scale [Audio]
Hey, how’s it going? This is Craig Cannon, and you’re listening to Y Combinator’s podcast. So today, we have an uncut interview from the Masters of Scale podcast, and in it, Reed Hoffman, the co-founder of LinkedIn, interviews Sam Altman. All right, here …