Differentiable Functions

1. Seminar „Hands-On Math for Computer Scientists“ Saarbrücken, Feb. 2nd 2005 Daniel Beck, Sebastian Blohm Differentiable Functions

2. Outline • Solving exercises intuitively • Difficulties when solving the exercises • General rules for differentiability • Applying the rules

3. The exercise Determine which of the following functions are differentiable: • f(x)=x² • f(x)=1/x • f(x)=|x-1| • f(x)= √x with x≤4 and f(x)=x/4 + 1 with x>4

4. When is a function f differentiable? • A function f is differentiable at a point x0if: • It is continuous at • There exists a limit one limit of the difference quotient: • A function “f” is called differentiable (in I ) if it is diffenrentiable at every x0∈ I. • When ist a function differentiable (over his domain I) ?

5. How do I check for differentiability at x0? • If I have a plot of the function: • Check if x0 has exactly one tangent. • In the general case: • Check if f is continuous (in particular: no jump) • Check ifand both exist.

6. f(x)=x² • Differentiable over R

7. f(x) = 1/x • Differentiable over

8. f(x)=|x-1| • Differentiable over

9. f(x)= √x with x≤4 and f(x)=x/4 + 1 with x>4 Depending on the visualization, non-differentiable point might not be visible at all.

10. Applying the definition • Example : f(x)=x² • So, the limes exists for all • This was very easy! • But what about sin(x²) ? • This is clearly the wrong way!

11. Difficulties Does anyone dare to calculate ? PLUS: We cannot possibly calculate the limit of the difference quotient for all elements 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”

12. General rules for checking differentiability • Notation : • Predicates • f is differentiable at • f is differentiable over I • Functions: • Range of the function f on interval I • : for • : for

13. General rules for checking differentiability Addition: Substraction : Multiplication: Division:

14. General rules for checking differentiability Some special cases

15. General rules (continued) • Chain rule: • Case splits:

16. Example sin x²

17. Example 1/x

18. Example √x with x≤4 and f(x)=x/4 + 1 with x>4

19. Example |x-1| FAILS!

20. How to explain these rule in Active Math? • With well explained sentances! • Example : • f+g is differentiable if f an g are differentiable 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.

21. Discussion • Limits of the rule approach: • Counter example: • is differentiable over ! • Do we want to stop when the first non-differentiable point is found? (Or do we want to modify the rules respectively.) • One rule needed for each operation to be covered.