How would I start this?
The amount that workers contribute monthly for health insurance premiums can be modeled by
A(t)=0.07t^3-3.1t^2+54.3t-230, where A is the monthly amount contributed and t is the number of years after 1980. Find the instaneous rate of change in monthly contribution in 2000.
Don't let all the digits and decimals throw you off.
The number of years after 1980 just means that in 1981, t = 1, and 1982, t = 2, and so on.
You need to differentiate
0.07t^3 - 3.1t^2 + 54.3t - 230
Which, really, is just differentiating
t^3, t^2, and t
and multiplying them by their respective constants.
Then find the valuw of the derivative function at t=20.
Is that enough of a start?
I'm using the formula for instantaneous rate of change, but I'm getting -38.4 as my answer..what am I doing wrong?
560-1240+1086-230-560-1240+1086-230/20
-2480+2172-460/20
A(t)=0.07t^3-3.1t^2+54.3t-230
A'=.21t^2-6.2t+ 54.3 Now you are a t=20 so
A'=84-12.4+54.3
How did you get -38?
If the question says use the formal definition of a derivative aren't I supposed to use f(x)=f(x+h)-f(x)/h?
To find the instantaneous rate of change in the monthly contribution in 2000, we need to find the derivative of the function A(t) with respect to t and evaluate it at t = 2000.
We are given that A(t) = 0.07t^3 - 3.1t^2 + 54.3t - 230.
To find the derivative, we can use the power rule. The power rule states that if we have a function of the form f(t) = at^n, where a and n are constants, then the derivative is given by f'(t) = n * a * t^(n-1).
Applying the power rule to each term of the function A(t), we have:
A'(t) = d/dt (0.07t^3) - d/dt (3.1t^2) + d/dt (54.3t) - d/dt (230)
Differentiating each term, we get:
A'(t) = 0.21t^2 - 6.2t + 54.3
Now, we can evaluate A'(t) at t = 2000 to find the instantaneous rate of change in the monthly contribution in the year 2000.
A'(2000) = 0.21(2000)^2 - 6.2(2000) + 54.3
Calculating this expression,
A'(2000) = 840000 - 12400 + 54.3
A'(2000) = 827653.7
Therefore, the instantaneous rate of change in the monthly contribution in 2000 is approximately 827653.7.