4. In lectures, we looked at several functions to model the Keeling curve (atmospheric C02 con- centration). Two functions that were used to model the data (ignoring oscillations) were the exponential function yE(t) = 280 + 3530-0222 and the power function 1 1.37 The gure below shows plots of the two functions (left), and the dz'erence between the two functions (right, with the power function subtracted from the exponential function). 4 Power ' Exponential ' E 2 D. x e 'r ' E -' Q) a: '5 2 4 10 20 30 40 50 6O 0 10 20 30 40 50 60 Time since January 1, 1958 (years) Time since January 1, 1958 (years) (a) Using the gure provided, estimate the time(s) when the power function and the exponential function would give exactly the same concentration of 002 in the atmosphere. (b) We can use Newton's method to nd an approximate value for the time(s) at which the two functions intersect. To do this, we need to introduce a new function which is the difference between the exponential function and the power function 1 yp(t) = yE(t) yp(t) = (280 + 35.20-022') (5151-37 + 315). The derivative of this new function is 31330:) = 0.77.2002\" 0.4623037. Use one step of Newton's method with an initial guess of 12 years to obtain a better estimate for the time of the intersection point (i.e. to nd t for when yD(t) = O). (2 marks)