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An in-depth exploration of implicit differentiation and related rates, fundamental calculus concepts. It introduces situations resolved using the same mathematical model - describing curves and relating quantities. The document delves into the derivative's interpretations in physics and economics, providing examples. It also poses questions about the derivative's meaning, such as negative velocity and particle motion. This comprehensive guide is valuable for students and researchers across disciplines.
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IMPLICIT DIFFERENTIATION AND RELATED RATES Recall the two separate and apparently distinct situations that, not surprisingly, are resolved with the same mathematical model. Situation no. 1: Some very nice looking and useful curves in the plane are NOT describable as (i.e., graphs of functions) but rather as i.e. solution sets of equations. Compute equations of tangent lines.
Situation no. 2: Two quantities and are related to each other via some formula (easiest example: is the surface area of a sphere, is the volume of the same sphere. You should be able to show that , the formula that relates and ) You know how fast changes. How fast does change? (Pump air in the sphere at a certain known rate. How fast does the surface area expand?)
The ratio (called the difference quotient, measures the average amount of change in when changes from to. The limit is then the instantaneous rate of change of relative to , when. Recognize any of this? That’s right, we are talking about! , the derivative (of with respect to )
As a mathematician I don’t have to tell you what and stand for, just two numerical quantities related to each other by some formula This is the beauty of Mathematics, as your textbook puts it, on p. 73, A single mathematical concept (such as the derivative ) can have different interpretations in each of the sciences. Joseph Fourier (you’ll meet his work eventually) put it this way: Mathematics compares the most diverse phenomena and discovers the secret analogies that unite them.
each one has a favorite and! Your textbook gives you several examples, from physics, chemistry, thermodynamics, biology, economics. I will give you two, from physics (actually elementary dynamics) and economics (du jour in today’s economic environment!), but I expect you to read all the rest in your textbook, as well as in the exercises. Here we go: In elementary dynamics (first investigated by Galileo Galilei) we are interested in the motion of a particle that moves along a straight line.
The only thing that’s needed on that line is a “coordinate system” So we can talk numerically about the position of the particle on the line. You have been brainwashed into thinking that the straight line is either horizontal or vertical, but actually it could be anywhere, even Galileo was studying how a ball rolls down on a slope! And what quantities and are we interested in? the numerical position of the particle
So is simply the coordinate of the particle on the line. Without the time sequence of the five positions, you still don’t know how it moves.
That observation tells us what ought to be. That’s right, time! Remark. We already see here the importance of appropriate units: you may use, but good luck with the computations, “centuries” to describe the motion of a mouse in a maze! Same comment applies if you use “nanoseconds” to describe the flight of a rocket towards the moon. So now I will show you the time sequence of the particle in the figure, then we will guess the formula
Usual procedure is to plot the values of on the horizontal axis of a Cartesian Coordinates system and the corresponding values of on the vertical axis (the particle still moves on the same slanted slope it was on ) Then we rename and as and respectively and get the usual formula The graph and formula in our example are shown in the next slide
The formula is (roughly) And the graph (color coded to match the other figure)
Let’s return to the elementary dynamics problem, where a particle is moving on a straight line (endowed with a coordinate system.) The position coordinate of the particle at any time is given by the equation We will dispense with the standard easy questions such as Find the average velocity over the time interval or (answers in green) Find the instantaneous velocity when.
We ask instead
is the speed at time. Following tradition we call the second derivative the acceleration. It is now time to do several exercises, at least the following recommended ones, on pp. 173 – 176 of
10, 13, 14, 17, 18 20, 22. 23, 25a, 27, 28, 30, 31, 32a
. I wll add 1% to the homework score to everyone who hands in all 23 answers by 9/28.
