Estimating rate of change 2015 AP Calculus AB BC 1b
About this video
For 2015 AP Calculus AB/BC question 1b, the video decides whether the amount of water in a pipe is increasing or decreasing at t = 3 hours. Water enters at rate R(t) and drains at rate D(t), so you compare R(3) and D(3). With a calculator in radian mode, R(3) is about 5.09 and D(3) is about 5.4, so water drains faster than it enters and the amount is decreasing.
You'll be able to
- explain that the amount of water increases when the rate in is greater than the rate out.
- evaluate rate functions at a given time with a calculator.
- give a reason for whether an amount is increasing or decreasing.
Key words
- Rate in
- How fast water flows into the pipe, R(t).
- Rate out
- How fast water drains from the pipe, D(t).
- Radian mode
- A calculator setting needed so that trig functions like sine treat angles as radians.
Transcript
- [Instructor] Now let's tackle the next part. Is the amount of water in the pipe increasing or decreasing at time T is equal to three hours? Give a reason for your answer. Well, what would make it increasing? Well, if the rate at which things are going in is larger than the rate of things going out, then the amount of water would be increasing.
But if it's the other way around, if we're draining faster at T equals 3, then things are flowing into the pipe, well then, the amount of water would be decreasing. Let me be clear. So amount, so if R of T greater than. Actually, let me write it this way. If R of 3, T equals 3, because T is given in hour, T is measured in hours.
If R of 3 is greater than D of 3, whoops, than D, D of 3. If R of 3 is greater than D of 3 that means water is flowing in at a higher rate than leaving. So that means that water in pipe, right, then, then water in pipe increasing, increasing. And then if it's the other way around, if D of 3 is greater than R of 3, then water in pipe decreasing.
Then you're draining faster than you're putting into it. Then water in pipe decreasing, decreasing. So we just have to evaluate these functions at 3. So let's see, R... Actually, I can do it right over here. So let me make a little line here. So R of 3 is equal to.
Well, let me get my calculator out. This is going to be... Oops, not that calculator. Let me get this calculator out. And I'm assuming that things are in radians here. So I already put my calculator in radian mode. So it's going to be 20 times sine of, whoops, 20 times sine of 3 squared is 9 divided by 35.
And it gives us, this is equal to approximately 5.09. So this is approximately 5.09. And D of 3, D of 3 is going to be approximately, so the calculator back out. So we have -0.04 times 3 to the third power. So times 27 plus, plus 0.4 times 9, times 9, T squared, plus 0.0.
Oh no, it's just 0.9696 times T, times 3. And this gives us 5.4. So this is equal to 5.4. So D of 3 is greater than R of 3. So water increasing, sorry, water decreasing. We're draining faster than we're getting water into it. So water, water is decreasing.
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