Let P(t) be the percentage of Americans under the age of 18 at time t. The table gives values of this function in census years from 1950 to 2000. (I didn't put the table here because it will not display well so the table has basically t as time from 1950 to 2000 and P(t) values. I found P'(t) and it asks me these questions and I have no idea about what they are asking for)
--- How would it be possible to get more accurate values for P'(t). Note the US census is only taken every 10 years so we cannot get more data point.
----what kinds of things could result in a more accurate tangent line
To get more accurate values for P'(t), one approach could be to use interpolation or extrapolation techniques. Since the US census is only taken every 10 years, you can estimate the values of P'(t) for the in-between years by making assumptions based on the available data points.
Here are a few methods you can consider:
1. Linear Interpolation: Assume that the rate of change between consecutive census years is constant. You can calculate the average rate of change between two data points and use that to estimate P'(t) for the years in between.
2. Polynomial Interpolation: Fit a polynomial function to the available data points and use that function to estimate P'(t) for the years in between. Higher-degree polynomials may provide a more accurate representation of the data, but be cautious of overfitting.
3. Moving Average: Calculate the average rate of change over a certain window of years centered around the year of interest. This can help smooth out fluctuations in the data and provide a more accurate estimate of P'(t) for a particular year.
4. Regression Analysis: Perform a regression analysis on the available data points to find a best-fit line or curve that represents the relationship between time and P(t). This can help generate more accurate tangent lines for any given year.
In general, to obtain a more accurate tangent line, it is important to have more precise and closely spaced data points. Additionally, considering factors such as demographic trends, societal changes, and other relevant data sources can help refine the estimates and improve accuracy.