Differentiabl Functions ActiveMath可微函數(shù) ActiveMath

1、intuition|difficulties | rules | examplesdifferentiable functions seminar hands-on math for computer scientists“saarbrcken, feb. 2nd 2005daniel beck, sebastian blohmintuition|difficulties | rules | examplesoutline solving exercises intuitively difficulties when solving the exercises general rules fo

2、r differentiability applying the rulesintuition|difficulties | rules | examplesthe exercisedetermine which of the following functions are differentiable: f(x)=x f(x)=1/x f(x)=|x-1| f(x)= x with x4 and f(x)=x/4 + 1 with x4 intuition|difficulties | rules | exampleswhen is a function f differentiable?

3、a function f is differentiable at a point xdifferentiable at a point x0 0 if: it is continuous at there exists a limit one limit of the difference quotient: a function “f” is called differentiable (in differentiable (in i i ) ) if it is diffenrentiable at every x0 i. when ist a function differentiab

4、le (over his domain i) ?000( ( )()lim()xxf xf xxx-intuition|difficulties | rules | exampleshow do i check for differentiability at x0 ? if i have a plot of the function: check if x x0 0 has exactly one tangent. has exactly one tangent. in the general case: check if f is continuous (in particular: no

5、 jump) check ifand both exist. 000000,00( ( )()( ( )()limlim()()-xxx xxxx xf xf xf xf xxxxxintuition|difficulties | rules | examplesf(x)=x differentiable over r intuition|difficulties | rules | examplesf(x) = 1/x differentiable over 0intuition|difficulties | rules | examplesf(x)=|x-1| differentiable

6、 over 1intuition|difficulties | rules | examplesf(x)= x with x4 and f(x)=x/4 + 1 with x4 depending on the visualization, non-differentiable point might not be visible at all.intuition|difficulties | rules | examplesapplying the definition example : f(x)=x so, the limes exists for all this was very e

7、asy! but what about sin(x) ? this is clearly the wrong way! 0002200000000( ( )()()()()limlimlim2()()()-xxxxxxf xf xxxxxxxxxxxxxx0 xintuition|difficulties | rules | examplesdifficultiesdoes anyone dare to calculate ? plus: we cannot possibly calculate the limit of the difference quotient for all elem

8、ents of the domain. how do i determine which points are crucial? how do i prove that i did not miss a non-differentiable point? solution : apply some cooking recipe”02200(sinsin)lim()-xxxxxxintuition|difficulties | rules | examplesgeneral rules for checking differentiability notation : predicates f

9、is differentiable at f is differentiable over i functions: range of the function f on interval i : for : for 000( ( )()lim()-xxf xf xxx( ):irangeg:id0:xd0 xx0 x0()-xdf0()xdf0 xx0 x000( ( )()lim()-xxf xf xxxintuition|difficulties | rules | examplesgeneral rules for checking differentiabilityaddition:

10、substraction :multiplication:division: ()iiid fd gdfg ()iiid fd gdfg- ()iiid fd gdf g ( )iiid fd gfdgintuition|difficulties | rules | examplesgeneral rules for checking differentiability ()idxx()()-iidfdf(cos )idx(sin)idx() where ()jidfjidfsome special casesintuition|difficulties | rules | examplesg

11、eneral rules (continued) chain rule: case splits: ( ) ( ) ( )()ijirange gjd gdfd gf( ) ( ) ( )( ) lim( ( )lim( ( )(if xk then else )-jlkkxkxkidgdfdfdff xg xdfgintuition|difficulties | rules | examplesexample sin x 22()() () (sin( ) (sin()dxxdxxdxxdxxdxkompsinmulididintuition|difficulties | rules | e

12、xamplesexample 1/x 000 (1) ( ) 1ddxdivdxintuition|difficulties | rules | examplesexample x with x4 and f(x)=x/4 + 1 with x4 44444444044( )(4)(1)( )4 (1)() lim()lim(1) 44()(1)4414- xxxxxxxxdx dxddxxddxxxdxdxdif xthen xelseintuition|difficulties | rules | examplesexample |x-1| 11111111111( )(1)(1) ( )

13、(1) (1)( (1) lim(1)lim( (1)( (1) (1) (if x1 then else )1-xxxxxxxxxidxddxdxddxdxxxdxdxdfgdx( )0) -iid if f xthenf else fabsdffails!intuition|difficulties | rules | exampleshow to explain these rule in active math? with well explained sentances! example : f+g is differentiable if f an g are differenti

14、able on i g(f) is differentiable if g is differentiable over the range of f and f is differentiable on i “if x=k then f else g” is differentiable if f and g are differentiable, and if they have the same value and the same derivation then aproching k the calculations are a good visualization of the reasoning.intuition|difficulties | rules | examplesdiscussion limits of the rule appr