In the 2nd lecture's problems this question
f(t)= -5t^2 +60t+120
we compute the rate of change by
the first derivative
then in this question
he solved it by the equation of the 2nd derivative
2. Compute f'(10)
Does the account balance continue to increase through
out the year? How do you know?
f(10) = −10 · 10+60
= −40dollars per month
why did we use the 2nd derivative here???
OCW Scholar - Single Variable Calculus
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Remember that a derivative of a function is just the rate at which the function is changing at any given point. You can apply that concept recursively to see how fast the rate of change itself is changing.
In nature, changing quantities often change in sporadic or dynamic ways. One of the best ways to fix this concept in your mind is to use the measure of distance. If you are moving, say in a car or bicycle, your distance is changing over time. That's f(x). However, you can also measure your "speed", which is how fast your distance is changing over time. That's the derivative, f'(x). In addition, you can measure your "acceleration", which is how fast your "speed" is changing over time. That's the second derivative, f''(x) -- your acceleration is a measure of the rate-of-change of the rate-of-change of your distance from your starting point over time.
In the problem, you are doing something similar. You have a function that tells you how your balance changes from $0 over time. If you want to know how much your balance is changing over time, you take the first derivative, f'(x). However, your balance is not changing at a steady rate - the balance increases quickly in the beginning of the year and ends up decreasing near the end. If we want to see how your rate-of-change-of-balance is itself changing, we take the second derivative. Whenever the second derivative is positive, then the *rate* at which your balance changes is increasing. When the second derivative is less than zero, then the *rate* at which your balance changes is decreasing. Note that your balance rate-of-change can be positive but still decreasing; it may go from 50 -> 65 -> 75 -> 80 -> 75 -> 65 -> 50; the second derivative changes at t = 3, but the first derivative will continue to be positive for several more months.