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Now, if you want to get a little bit more of a visual understanding of this, and what I'm about to do, you would not actually have to do on the actual exam. So, the units are gonna be meters per minute per minute. AP CALCULUS AB/CALCULUS BC 2015 SCORING GUIDELINES Question 3 t (minutes) v(t)(meters per minute)0122024400200240220150Johanna jogs along a straight path.
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They give us v of 20. When our time is 20, our velocity is going to be 240. That's going to be our best job based on the data that they have given us of estimating the value of v prime of 16. We can estimate v prime of 16 by thinking about what is our change in velocity over our change in time around 16. So, let me give, so I want to draw the horizontal axis some place around here. Let's graph these points here. Johanna jogs along a straight path lyrics. Well, let's just try to graph. So, that's that point.
If we put 40 here, and then if we put 20 in-between. So, 24 is gonna be roughly over here. And we don't know much about, we don't know what v of 16 is. They give us when time is 12, our velocity is 200. Let me do a little bit to the right. But what we wanted to do is we wanted to find in this problem, we want to say, okay, when t is equal to 16, when t is equal to 16, what is the rate of change? Johanna jogs along a straight path forward. And we would be done. This is how fast the velocity is changing with respect to time. And so, these obviously aren't at the same scale. So, if you draw a line there, and you say, alright, well, v of 16, or v prime of 16, I should say. It goes as high as 240. So, we literally just did change in v, which is that one, delta v over change in t over delta t to get the slope of this line, which was our best approximation for the derivative when t is equal to 16.
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So, let's figure out our rate of change between 12, t equals 12, and t equals 20. Use the data in the table to estimate the value of not v of 16 but v prime of 16. So, this is our rate. And so, this is going to be 40 over eight, which is equal to five. But this is going to be zero. Johanna jogs along a straight path wow. We see that right over there. And so, let's just make, let's make this, let's make that 200 and, let's make that 300. We see right there is 200.
So, v prime of 16 is going to be approximately the slope is going to be approximately the slope of this line. And then, finally, when time is 40, her velocity is 150, positive 150. Estimating acceleration. And so, then this would be 200 and 100. So, that is right over there. And then our change in time is going to be 20 minus 12. So, when the time is 12, which is right over there, our velocity is going to be 200.
Johanna Jogs Along A Straight Path Forward
For good measure, it's good to put the units there. And then, when our time is 24, our velocity is -220. And we see on the t axis, our highest value is 40. But what we could do is, and this is essentially what we did in this problem.
So, they give us, I'll do these in orange. Let me give myself some space to do it. So, when our time is 20, our velocity is 240, which is gonna be right over there. So, if we were, if we tried to graph it, so I'll just do a very rough graph here. And so, what points do they give us? So, we could write this as meters per minute squared, per minute, meters per minute squared. We could say, alright, well, we can approximate with the function might do by roughly drawing a line here.
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And when we look at it over here, they don't give us v of 16, but they give us v of 12. Fill & Sign Online, Print, Email, Fax, or Download. So, at 40, it's positive 150. For 0 t 40, Johanna's velocity is given by. We go between zero and 40. So, -220 might be right over there. And so, these are just sample points from her velocity function. AP®︎/College Calculus AB.
And then, that would be 30. So, we can estimate it, and that's the key word here, estimate.