Use the model below to estimate the average annual growth rate of a certain​ country's population for​ 1950, 1988, and​ 2010, where x is the number of years after 1900.

Question

y=-0.0000081x^3+0.00193x^2-0.214x+8.324
The estimated average annual growth rate of the​ country's population for 1950 is
  
enter your response here.
​(Round to three decimal places as​ needed.)

Answer 1

To find the estimated average annual growth rate for 1950, we need to find the slope of the function at that point.

We can do this by finding the derivative of the function and evaluating it at x = 1950.

Taking the derivative of y = -0.0000081x^3 + 0.00193x^2 - 0.214x + 8.324:

dy/dx = -0.0000081(3x^2) + 0.00193(2x) - 0.214

Simplifying:

dy/dx = -0.0000243x^2 + 0.00386x - 0.214

Evaluating at x = 1950:

dy/dx = -0.0000243(1950^2) + 0.00386(1950) - 0.214

≈ -0.0000243(3,802,500) + 0.00386(1,950) - 0.214

≈ -92.479 + 7.527 - 0.214

≈ -92.166

Therefore, the estimated average annual growth rate of the country's population for 1950 is approximately -92.166.