MTH 251 – Differential Calculus Chapter 3 – Differentiation Section 3.4 The Derivative as a Rate of Change Copyright © 2010 by Ron Wallace, all rights.

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Presentation transcript:

MTH 251 – Differential Calculus Chapter 3 – Differentiation Section 3.4 The Derivative as a Rate of Change Copyright © 2010 by Ron Wallace, all rights reserved.

What does the derivative measure? Difference Quotient... given that y = f(x)  i.e. The average rate of change of the function Derivative  The instantaneous rate of change of the function  Or simply … the Rate of Change of f(x) The “change” in y with respect to a “change” in x.

Example: Linear Motion Let s = f(t) be a function the specifies the distance an object travels along a line.  aka: displacement The rate of change of the displacement is called the velocity  speed is the absolute value of the velocity The rate of change of the velocity is called the acceleration. The rate of change of the acceleration is called the jerk.

Example – Free Fall An object dropped from any height will fall a distance given by the formula … Find the velocity, acceleration, & jerk at t = 2 sec. If the object started 100 feet from the ground, when will it hit the ground and how fast will it be going?

Rate of Change in Politics The rate of increase of inflation is going down. President Nixon Fall of 1972 This was the first time a sitting president used the third derivative to advance his case for reelection. Hugo Rossi P

Rate of Change in Politics The rate of increase of inflation is going down. President Nixon Fall of 1972 P Inflation CPI 1971 – 1986 (with and without energy) view/what-causes-inflation-lessons-from-the-1970s-vol-3/553/ 90 day price controls imposed August 15, 1971, some of which ran until April of 1974 (~1000 days later)