Calculating acceleration AP Calculus AB BC 2015 3c
About this video
This video solves 2015 AP Calculus question 3(c): Bob's velocity is B(t) = t^3 - 6t^2 + 300 metres per minute. Acceleration is the derivative of velocity, so B'(t) = 3t^2 - 12t and B'(5) = 75 - 60 = 15 metres per minute per minute.
You'll be able to
- explain that acceleration is the derivative of velocity with respect to time.
- use the power rule to differentiate a polynomial velocity function and evaluate it at a time.
- give the correct units for acceleration, such as metres per minute squared.
Key words
- velocity
- How fast position is changing, with direction; here measured in metres per minute.
- acceleration
- The rate of change of velocity with respect to time, the derivative of velocity.
- power rule
- The derivative of t^n is n times t^(n-1).
Transcript
- [Instructor] Bob is riding his bicycle along the same path. For zero is less than or equal to t, is less than or equal to 10, Bob's velocity is modeled by B of t is equal to t to third minus 60 squared plus 300, where t is measured in minutes and B of t is measured in meters per minute.
Find Bob's acceleration at time t equals five. Well, acceleration, this is a velocity function right over here. So the acceleration is gonna be the derivative of the velocity function with respect to time. What is the rate of change of velocity with respect to time?
That's acceleration. So we really just want to evaluate, with this Bob's acceleration at t equals five, that's going to be B prime of five. So let's first figure out what B prime of t is. B prime of t is equal to, we'll take the derivative here. It's pretty straightforward. Just use the power rule.
So it's gonna be 3t squared minus 12t, 2 times -6 is 12, or -12. And then the derivative of 300. 300 is a change with respect to time, so it's just zero. And so B prime of five is going to be equal to 3 times 5 squared minus 12 times 5, which is equal to 75 minus 60, which is equal to 15.
And the units here, this is acceleration, so this is going to be, his velocity was in meters per minute. And so this is going to be meters per minute per minute because, remember, time is in terms of minutes. So we could write as meters, let me write it out, meters per minute per minute, which is the same thing as meters per minute, meters per minute squared.
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