Average Rate of Change

Transcription

1 Te Derivative Tis can be tougt of as an attempt to draw a parallel (pysically and metaporically) between a line and a curve, applying te concept of slope to someting tat isn't actually straigt. Te slope of a line is essentially ow fast te y-value is increasing as te corresponding x-value increases. We would like to determine an equivalent value for some point on a curve. Tis as numerous practical applications from te statistical concept of correlation to te pysical concepts of speed or steepness Te trick is to draw a line tat just barely touces te curve at te point you are interested in and at least at tat point is in te same direction (parallel, equal slope). In fact if you were to zoom in sufficiently close you would find te curve and te line to coincide, being virtually indistinguisable Wile we migt be able to do tis pysically, wit te elp of a straigt-edge, wat we want to do is calculate tis line, or at least te slope, matematically.

2 2 Derivative.nb Average Rate of Cange An average can be tougt of in a number of different ways. Typically, it is te single value tat best represents a wole set of different values. If we assume equal weigt or spacing between te values tis is just calculated by adding up te total of te different values and dividing by te number of different values. Now, if te values are in a particular order and we want to calculate te average CHANGE between tem, we recognize tat adding up te individual canges (bot positive and negative) is equivalent to te total cange between te first and last value, wic gets divided by te number of values minus 1. To find te average RATE of cange we assume equal spacing so tat we add up te individual canges in y divided by te same cange in x. Since tese are fractions wit a common denominator we can combine tem into one fraction wit te sum in te numerator. Te end result is tat te average rate of cange is te total cange in y divided by te total cange in x between te two endpoints (x = a and x = b). AROC 1 n 1 y i n 1 y i n 1 i 1 i 1 x i n 1 x Y X f b f a b a Average of n 1 individual rates of cange or slopes between a and b Tis is true regardless of ow small te individual cange in x is as long as tey are all te same. Tey can even be infinitesimally small Tis Average Rate Of Cange (AROC) between two points on a curve ten is te same as te slope of te straigt line between tose two points. Tis line tat "cuts" across te curve at two points is referred to as te secant line of te curve at tose two points (from te Latin - to cut), so we also call te AROC te secant slope. A common example of a rate of cange would be te cange in position over some period of time. Tis is wat we tink of as velocity or speed. Suppose we travel at a constant velocity for one second, ten we cange sligtly for te next second, and again for te next and so on for 30 seconds. We could plot our position as a series of 30 connected line segments, were te slope of eac one is our speed for tat particular second.

3 Derivative.nb To find te average velocity we could add up all te individual velocities and divide by 30. However tis would be equivalent to adding up all te canges in position (in meters for instance), since te time between eac point is 1 sec, ten dividing by 30 (to get meters per second). Tis is te total cange in position divided by te total cange in time (if eac time increment was 2 secs and tere were only 15 canges in velocity we would divide by 2 wen adding up te canges in position to get te sum of velocities, but ten we would only divide by 15 - so te end result is still dividing by 30). It turns out not to matter ow frequently we cange our velocity (ow small te time increments are), as long as we end up at te same final position at te same time te average velocity remains te same. Instantaneous Rate of Cange Te average rate of cange of a function is all well and good, but it isn't quite wat we are looking for. Te direction of a curve is constantly canging and our goal is to determine wat direction it is eaded at any ONE point, wic can't necessarily be represented by any TWO points required by an AROC. Imagine traveling a curving pat (or a track) wen suddenly te pat disappears from under you and you continue in a straigt line in te direction you were traveling at te point were te pat disappeared. We want to know wat te slope of tat line is. Tis line is referred to as te tangent line (from Latin - to touc) to te curve, wic only touces te curve at a single point, rater tan cutting across it at two points. Te tangent line may start on one side of te curve and continue on te oter if te curvature canges direction at tat exact point and it may continue and cut across te curve at some oter point if te curvature canges direction back, but tese conditions do not negate te tangency as long as tere is no angle between te line and te curve at te point of intersection (tey are parallel). Wile tere isn't a specific secant line tat is te same as te tangent line, it is clear tat if one of te points is te one we are interested in, and te oter is moved closer and closer to it, te secant line will eventually become almost indestinguisable from te tangent line.

4 4 Derivative.nb 15 5 In te picture above we ave eld one point fixed at b = 1 and moved te oter point closer from a = 0 to a =.9 (wit eac secant line getting ligter as it gets closer to te tangent line). Looking at te calculation for te slope of te secant line we can see tat it doesn't make any sense wen b = a = 1, so we can't actually calculate te slope of te tangent directly tis way. AROC f b f a b a Noneteless we can see tat conceptually, and matematically as well as it turns out, te slope of te tangent line is te LIMIT of te slopes of te secant lines as te two points get closer togeter. It doesn't matter if b is approacing a or a is approacing b, it just depends wic end we consider fixed. Tecnically we need to look at te limit from bot sides in order to be complete, but as long as tere are no sudden canges it will be te same. Tis slope of te tangent line is te slope of te curve AT te single point and is wat we call te Instantaneous Rate Of Cange (IROC) of te function. We can calculate tis value matematically using te principles we ave learned for limits. IROC lim x a IROC lim 0 f x f a for te slope at a single point x a x a f a f a alternate version Using te alternate version we can calculate te IROC of te function we ave been using at te point x = 1 f x_ : x 4 4 x 3 2 x 2 4 f Expand f Expand f 1 f f 1 f 1 Expand Wen 0 we get te slope of te tangent line = 4. Given tat f(1) = 5 we can also calculate te equation of te tangent line as y = (x-1) = 4x + 1.

5 Derivative.nb 5 Differentiability Te IROC for some function f(x) at te point x = a as a more official designation, wic is te "derivative" of f at a, or f'(a). Tere are a number of situations wen it is impossible to calculate te derivative at a particular point (it doesn't exist). In tese cases we say tat te function is NOT differentiable at tat point. We can look at te limit definition of te derivative to identify te different potential problems f ' a lim 0 f a f a First of all, if f(a) does not exist it is impossible to perform tis calculation. Secondly, f(a+) as 0 is te limit of te function as x a, so if tis does not exist from one direction or te oter it is also impossible to perform tis calculation. Tird, if te limit f(a+) from eiter direction is different from f(a) ten te denominator will go to zero wile te numerator approaces some constant, wic produces an infinite limit. Tese tree conditions are te same ones tat describe continuity, so we can state tat if a function is NOT continuous, ten it will NOT be differentiable. However, tere are a couple oter circumstances under wic a function is not differentiable as well. If te limit from te rigt is not te same as te limit from te left, meaning te curve is approacing te point from eiter side wit a different slope and producing a "cusp," ten te derivative does not exist. Tis can appen wit piecewise functions, absolute values and certain root functions If te numerator ends up wit some root of tat doesn't go to zero as quickly as does ten it is possible for te function to go vertical at some point were te derivative doesn't exist. Tis also appens wit some root functions

6 6 Derivative.nb Derivative Function We ave looked at te derivative at a single point f'(a). However, tis can be evaluated at any point a were te function is differentiable. Wat tis gives us is a wole set of derivative values corresponding to all te x values. As long as tere is only one derivative value for every x ten tis describes a function of x as well, wic we call y ' f ' x y x x f x lim 0 f x f x Te only difference ere is tat te a as been replaced wit an x to indicate tat we can calculate it at some arbitrary value. Te calculation is a bit trickier because te x means you can't evaluate certain operations and te end result will generally be some expression involving x rater tan a particular value. f x 4 2 x 2 4 x 3 x 4 Expand f x x 12 2 x 4 3 x 2 x 2 12 x x 2 4 x 3 4 x 3 x 4 Expand f x f x x 12 2 x 4 3 x 12 x x 2 4 x 3 f x f x Expand x 12 x 4 2 x 12 x 2 6 x 2 4 x 3 Now wen we take te limit as 0, we are left wit several terms, f'(x) = 4 x 12 x 2 4 x 3. Tis function can be plotted alongside te original f(x) to demonstrate some useful correlations You can see tat te purple grap representing f'(x) is 0 wenever te blue one representing f(x) levels out (a local maximum or minimum of f). f'(x) is positive wenever f(x) is increasing and negative wenever f(x) is decreasing. Also it reaces a local maximum or minimum wenever f(x) is at its steepest point.