The instantaneous rate of change at any point where x=a, can be numerically obtained by determining the slope of a typical secant to the curve y=f(x) centered around the point x=a. It is obtainable by determining the values of function f(x) in two closely spaced instants (a+h) and (a-h) and dividing by the interval, i.e. 2h. The formula is:
Rate of instantaneous change=(f(a+h)~f(a+h))/(2h)
The smaller the value of h, more accurate is the estimate of the slope and hence of the instantaneous rate of change.
If you take f(a-h) and f(a) instead, you are likely to get the instantaneous rate of change around the point (a-h/2) and not a. That is the reason I am going to use the earlier formula for this calculation.
A person’s blood pressure P(t), in mm of Hg, is modelled by the function:
The instantaneous rate of change of the person’s blood pressure at time t=0.5 s has to be determined.
The instantaneous rate of change of the person’s blood pressure at time t=0.5 s:
`= (P(0.500-0.001)- P(0.500+0.001))/(2*0.001)=580.33`
Make the interval finer by one decimal point and instantaneous rate of change of the person’s blood pressure becomes 580.415 which is very close to the actual derivative (dy/dx) value of 580.416.