Differentiation rules practice question 12
A particle moves along the -axis with position at time . Find its acceleration at .
Acceleration is the rate of change of velocity, and velocity is the rate of change of position. How many derivatives of is that?
The velocity is . Differentiate once more for the acceleration, then substitute .
An integer.
Theory UBC Math 100 §3.4CLP-1 §2.14Key results
Worked solution
Strategy
Acceleration is the second derivative of position. Differentiate the polynomial twice and evaluate.
Working
Velocity.
Acceleration.
The acceleration is negative. The velocity v(1) = 0 is changing from positive to negative there, so the particle is turning round.
Answer
The answer is negative 6.
Check
The velocity drops from 0.63 at t = 0.9 to −0.57 at t = 1.1, a change of −1.2 in 0.2, which is −6 per unit time.
Where this goes wrong
Reporting the velocity v(1) = 0 instead of the acceleration, or differentiating only once. Acceleration is two derivatives away from position.
The idea behind it
The differentiation rules lesson, a figure to move, in a few steps
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