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3.32 Dynamical interpretation

Let xx be a real variable that stands for time, and f⁢(x)f(x) represent the distance along a road that a car has reached at time xx. Then we have the following quantities:

f⁢(x+h)-f⁢(x)=(distance travelled between times x and x+h);f(x+h)-f(x)=({\hbox{distance travelled between times}}\quad x\quad{\hbox{and}}\quad x+h);
f⁢(x+h)-f⁢(x)h=(average velocity between times x and x+h);{{f(x+h)-f(x)}\over{h}}=({\hbox{average velocity between times}}\quad x\quad{\hbox{and}}\quad x+h);
f′⁢(x+h)-f′⁢(x)=(change in velocity between times x and x+h);f^{\prime}(x+h)-f^{\prime}(x)=({\hbox{change in velocity between times}}\quad x\quad{\hbox{and}}\quad x+h);
f′⁢(x+h)-f′⁢(x)h=(average acceleration between times x and x+h).{{f^{\prime}(x+h)-f^{\prime}(x)}\over{h}}=({\hbox{average acceleration between times}}\quad x\quad{\hbox{and}}\quad x+h).