2.8 The Derivative as a Function

Transcription

1 .8 Te Derivative as a Function Typically, we can find te derivative of a function f at many points of its domain: Definition. Suppose tat f is a function wic is differentiable at every point of an open interval (a, b). Its derivative is te function f () = f ( + ) f () f () Te domain of f must include te interval (a, b). A function and its derivative are drawn: f is increasing f > 0 f is decreasing f < 0 f as a orizontal tangent line f = 0 Eample If f () = 3, ten its derivative is f () 4 f f ( + ) f () [( + ) () = = 3 ( + )] [ 3 ] [ = ] [ 3 ] 3 = = = 3 3 Te domains of bot f and f are te real line R. Note ow te graps correspond: wen f is increasing, te derivative is positive, wen f is decreasing, te derivative is negative. We may compute similarly for many oter functions. You sould draw te graps of tese: do te graps fit wit your calculations?. f () = = f () = bot wit domain R \ 0} = (, 0) (0, ).. f () = = f () =. Te former as dom( f ) = [0, ) wile te latter as dom( f ) = (0, ). 3. Draw te grap of y = sin. Sketc underneat te grap of its derivative, just by tinking about wen sin is increasing and were it is decreasing. Te new grap sould look very familiar... For now tis means using te it definition. Nice formulæ suc as te power law will ave to wait until after te midterm...

2 Notation If y = f () tere are many notations for te derivative function: f dy (), d, d f d, d d f (), y, D f (), D f () Te value of te derivative function at = a is denoted f dy (a), d f d, =a d, y (a), D f (a), D f (a) =a d Te symbol d may be tougt of as an operator: turning a function into its derivative. For eample, if y = f () = ten we know tat f () = dy. We could instead write d =, or d d =. Moreover f (4) = = 4 4 and d f d = =9 = 9 6. Higer Derivatives We can differentiate derivatives! For eample, te second derivative of f is te derivative of f : tat is f () = f ( + ) f () if te it eists. Leibniz s alternative notation for second derivatives reads as if one is squaring te derivative operator: d f d = d ( ) d f d d d = f d Tis can elp wen trying to understand units. We can similarly compute iger order derivatives: Tird: Fourt: Fift: f () = d3 f d 3 f (4) () = d4 f d 4 f (5) () = d5 f d 5 and so on. Te bracket notation f (n) () is preferred for derivatives iger tan tird because of te increased difficulty counting multiple prime symbols. Eample f () = 3 + as f () = 6 + and f () = 6. Ten f (n) () = 0 for all n 3. Acceleration Wen s(t) is te distance traveled by a particle at time t, te derivative v(t) = s (t) is te particle s velocity. Te second derivative a(t) = s (t) = v (t) is te acceleration of te particle. Units: remember tat eac differentiation appends a per unit time to te units. Acceleration is terefore measured as distance-per-time-per-time: for eample, m/s =ms = meters per second per second ft/r =ft r = feet per our per our In tis contet, te tird derivative s is referred to as te jerk.

3 Eample After t seconds, a ball as eigt s(t) = + 0t 4.9t meters. Its velocity is v(t) = s (t) = 0 9.8t m/s. Its acceleration is a(t) = s (t) = 9.8 m/s. Note tat tis last is te gravitational constant. Wat does a differentiable function look like? So muc for calculating wit its. We want an intuitive idea 3 of wat to epect from te graps of differentiable and non-differentiable functions. Similarly to ow we understood te concept of continuity, we consider all te ways in wic a function migt fail to be differentiable. Te most obvious way turns out to be related to continuity! Teorem. If f is differentiable at = a, ten f is continuous at = a. Proof. Suppose tat f differentiable at = a. If = a, ten f () f (a) = f () f (a) ( a) a f () f (a) = ( f () f (a)) = ( a) = f (a) 0 = 0 a a a a Terefore f is continuous at a. An equivalent statement of te Teorem is: If f is discontinuous at = a ten f is non-differentiable at = a. Tus differentiable functions can be drawn witout taking your pen off te page. Te converse to tis is false owever. It is possible for a function to be continuous but non-differentiable. For tis to f (a+) f (a) appen, te it cannot eist. Tere are two common possibilities. 4. Corners For eample f () = is continuous at = 0. Wat about its derivative? If = 0, te derivative is f > 0 () = < 0 f () To find f (0) we would need to calculate = wic does not eist. Terefore is continuous at = 0, but not differentiable. 0 f () 3 Similarly to ow a continuous function sould be drawable witout taking your pen off te page. 4 Tere are more esoteric eamples, suc as te blancmange curve wic is continuous everywere and differentiatiable nowere, but suc tings are well-beyond te scope of tis course!

4 . Vertical tangents For eample, f () = 3 = /3 is continuous everywere. If we want to searc for a derivative at = 0 we must compute te it f () f (0) = /3 = + f () By computing its we can see tat f f ( + ) f () () = = 3 /3 f () provided = 0. Tus f is continuous at = 0, but not differentiable at = 0: it as a vertical tangent line. 0 Eample: Coosing to make a function differentiable a + b if < f () = if is differentiable for all. Find constants a, b suc tat y 4 As te possible graps of f sow, to te left of = te function is a straigt line. Wic coice of line will make te function differentiable at =? 0 We can answer tis in words: firstly, a differentiable function must be continuous, so te straigt line we coose must pass troug te point (, f ()) = (, ). Secondly, a differentiable function must ave te same rate of cange wen calculated as a left- or a rigt-it, wence te required straigt line must ave te same slope as y = as =. Now we calculate: Continuity at = : We require f () = f () = f () = a + b = + Any function f wit a + b = will ave te straigt line intersecting te parabola at (, ). Differentiability at = : We require f ( + ) f () = + f ( + ) f () = b = Putting tese togeter, we see tat f is differentiable if and only if a = and b =. Indeed its derivative is f if < () = if >

5 Homework. Compute all of te derivatives not eplicitly found above: use te it definition! if 0. Let f () = = if < 0 (a) Calculate f () for f () =. (b) Wat about f ()? For wat values of does tis make sense? (c) Can you guess a formula for a function wic is twice-differentiable at every value of but not tree-times differentiable everywere? Compute its first, second and tird derivatives. 3. Te binomial teorem states tat if n is a positive integer, ten ( + ) n = n k=0 ( ) n k n k = n + n n + k n(n ) n + + n n + n ( ) n n! were = is te binomial coefficient. k k!(n k)! Use tis to prove te power law for differentiation. If n is a positive integer, ten d d n = n n