Question

2) Annual coal production in the US 1975 1980 1985 (in quadrillion BTU per year) is given in the table. This is the rate of coal production and it is always increasing. It is NOT a total amount (note that the rate changes). a) Draw a graph that represents this information te 10.82 13.06 14.61 14.99 18.6019.33 22.46 e Annual coal production per year is a rate of change (i.e. derivative). Describe the function you would take the derivative of to get r(t). That is, describe a function F(t) such that F (t)-r(t). The description should include what tand F(t) represent. where the t-axis represents the number of years since 1960 and the y-axis represents r(t) the rate of coal production at t years (since 1960) Label the axis! b) What does the ordered pair (0, 10.82) represent? c) Give both an over-estimate and an under-estimate of the amount of coal produced between 1960 and 1965. Explain how you arrived at your estimates and what they mean geometrically (as an area). Particularly, show how your area relates to the graph of r(t) d) Use a Riemann sum to estimate the coal production between 1960 and 1990 and represent the sum geometrically on your graph. Could you use a different Riemann sum to estimate the coal production (this is not simply a yes or no answer, explain)?

Answer 1

Year and rate of coal production is given as follows. b) Order par (0, 10.82) is a pt lying on y axis which denotes that initial production rate of coal was 10.82 quadrillion BTU/year. c) Amount of coal produced b/w 1960 and 1965 can be found out by area of trapezium as shaded in graph Area = 1/2 times h times (sum of parallel sides) = 1/2 times 5 times (10.82 + 13.06) = 59.7 Coal Produces = 59.7 BTU quadrillion in spam of these 5 years. Under estimate = 10.82 times 5 = 54.1 quadrillion BTU in these 5 years Over estimate = 13.06 times 5 = 65.3 quadrillion BTU in 5 years.