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Derivatives as Rates of Change- DA1

Objectives

  • Determine a new value of a quantity from the old value and the amount of change.
  • Calculate the average rate of change and explain how it differs from the instantaneous rate of change.
  • Apply rates of change to displacement, velocity, and acceleration of an object moving along a straight line.
  • Predict the future population from the present value and the population growth rate.
  • Use derivatives to calculate marginal cost and revenue in a business situation.

Summary

The derivative of a given function y = f ( x ) measures the instantaneous rate of change of the output variable with respect to the input variable.

The units on the derivative function y = f ′ ( x ) are units of y per unit of x . Again, this measures how fast the output of the function f changes when the input of the function changes.

The central difference approximation to the value of the first derivative is given by

f ′ ( a ) ≈ f ( a + h ) − f ( a − h ) 2 h .

This quantity measures the slope of the secant line to y = f ( x ) through the points ( a − h , f ( a − h ) ) and ( a + h , f ( a + h ) ) . The central difference generates a good approximation of the derivative's value.

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