Derivatives

Question

The atmospheric CO2 level A(t) at Mauna Loa, Hawaii at time t (in parts per million by volume) is recorded by the Scripps Institution of

Oceanography. The values for the months January-December 2007 were

382.44 , 383.69 ,384.24, 386.25, 386.39, 385.86, 384.4, 381.77, 380.74, 380.8, 382.33, 383.69

Assuming that the measurements were made on the first of each month, you have to estimate A′(t) on the 15th of the months January-November.

The estimates are:

Answer 1

Well, you can use the SECANT line to approximate the derivatives.

 

Basically, you calculate the slope of the secant line between each pair of

consecutive points.

 

For x=0,1,2,3,4,5,6,7,8,9,10,11 representing the month number, we

are interested in approximating f'(0.5), f'(1.5), f'(2.5) etc

 

Since x increases by 1 along the interval, the slope of the secant

line between each pair ( 0,382.44) ,  (1,383.69),  (2,384.24) ... etc... (11,383.69)

 

is just the differences between each y-value.

 

To estimate the derivative, we can just cut the difference in half and add it

to the previous y-value.

 

Specifically, f'(x+1/2) is approximated by f(x) + 1/2( f(x+1) - f(x)) =

                                                      f(x) + 1/2 f(x+1) - 1/2f(x) =

                                                       1/2 f(x+1) + 1/2 f(x) =

                                                        [f(x+1) + f(x) ]/2

 which is the AVERAGE of each y-value

 

The results are shown in the following table

 

x    f(x)       f(x+1)-f(x)         approximation = [f(x)+f(x+1)]/2

0  382.44
1  383.69         1.25                     383.065
2   384.24       0.55                      383.965
3    386.25      2.01                       385.245
4    386.39      0.14                      386.32
5    385.86     -0.53                      386.125
6     384.44     -1.42                     385.15
7     381.77     -2.67                     383.105
8     380.74     -1.03                    381.255
9     380.8       0.06                     380.77
10    382.33     1.53                     381.565
11    383.69     1.36                      383.01