Successive Approximations to Estimate Derivatives

In the chapter How Fast is CO2 Increasing?, chapter 4 we estimated the slope of the tangent line at $$x=67$$ x = 67 (or 2017) of the CO2 function in figure 4.1 using (CO2(a+h)-CO2(a-h))/(a+h-(a-h)) $$a=67$$ a = 67 and $$h=0.01$$ h = 0.01 to get 2.324202 ppm per year. The calculation is presented R Code 4.2 . The question is, how accurate is our estimate of the slope of the tangent line of CO2 at $$x=67$$ x = 67 (or 2017)? Is $$h=0.01$$ h = 0.01 small enough so that the secant line that straddles the tangent line provides an accurate slope estimation? If we look at figure 8.1 which is the CO2 function from figure 4.1 zoomed in with a tangent line at 2017, $$x=67$$ x = 67 , (dashed red line). Two square points are added at $$(67-0.01, CO2( 67-0.01 ) )$$ ( 67 - 0.01 , C O 2 ( 67 - 0.01 ) ) and $$(67+0.01, CO2( 67+0.01 ) )$$ ( 67 + 0.01 , C O 2 ( 67 + 0.01 ) ) . Note that it appears that the tangent line connects the two points and hence is the same as the second line connecting the two points. Based on this graph it would seem that our secant line slope approximation of the tangent line slope at $$x=67$$ x = 67 is good (whatever good means?).