Differentiability

Learning tasks

A) Derivative at a point --- definition and tangent
1) Computing a derivative from the definition	Ex 1 Ex 2 Ex 3
2) Studying lateral differentiability	Ex 4 Ex 5 Ex 6
3) Using the \(DL_1\) characterization of differentiability	Ex 7 Ex 8 Ex 9
B) Algebraic operations on derivatives
4) Computing derivatives with the operation toolbox	Ex 10 Ex 11 Ex 12
5) Differentiating the inverse map	Ex 13 Ex 14 Ex 15 Ex 16
C) Local extrema and Fermat's theorem
6) Locating extrema via critical points	Ex 17 Ex 18 Ex 19
D) Rolle's theorem and the mean value theorem
7) Applying Rolle's theorem to find a zero of \(f'\)	Ex 20 Ex 21 Ex 22
8) Applying the mean value theorem (equality form)	Ex 23 Ex 24 Ex 25
E) Mean value inequality\(\virgule\) monotonicity\(\virgule\) and the limit of the derivative
9) Showing that a function is Lipschitz from a bound on \(f'\)	Ex 26 Ex 27 Ex 28
10) Studying monotonicity from the sign of \(f'\)	Ex 29 Ex 30 Ex 31
11) Applying the contraction principle to recurrent sequences	Ex 32 Ex 33 Ex 34
12) Establishing \(C^1\) at a point via the limit-of-derivative theorem	Ex 35 Ex 36 Ex 37
F) Higher-order derivatives and classes \(C^k\)
13) Computing higher-order derivatives via Leibniz	Ex 38 Ex 39 Ex 40
14) Determining the class \(C^k\) of a piecewise function	Ex 41 Ex 42 Ex 43
G) Brief extension to complex-valued functions
15) Differentiating a complex-valued function componentwise	Ex 44 Ex 45 Ex 46
16) Applying the complex mean value inequality	Ex 47 Ex 48 Ex 49

Lesson
Text book

Exercises Correction
A) Derivative at a point --- definition and tangent
    1) Computing a derivative from the definition Ex 1 Ex 2 Ex 3
    2) Studying lateral differentiability Ex 4 Ex 5 Ex 6
    3) Using the \(DL_1\) characterization of differentiability Ex 7 Ex 8 Ex 9
B) Algebraic operations on derivatives
    4) Computing derivatives with the operation toolbox Ex 10 Ex 11 Ex 12
    5) Differentiating the inverse map Ex 13 Ex 14 Ex 15 Ex 16
C) Local extrema and Fermat's theorem
    6) Locating extrema via critical points Ex 17 Ex 18 Ex 19
D) Rolle's theorem and the mean value theorem
    7) Applying Rolle's theorem to find a zero of \(f'\) Ex 20 Ex 21 Ex 22
    8) Applying the mean value theorem (equality form) Ex 23 Ex 24 Ex 25
E) Mean value inequality\(\virgule\) monotonicity\(\virgule\) and the limit of the derivative
    9) Showing that a function is Lipschitz from a bound on \(f'\) Ex 26 Ex 27 Ex 28
    10) Studying monotonicity from the sign of \(f'\) Ex 29 Ex 30 Ex 31
    11) Applying the contraction principle to recurrent sequences Ex 32 Ex 33 Ex 34
    12) Establishing \(C^1\) at a point via the limit-of-derivative theorem Ex 35 Ex 36 Ex 37
F) Higher-order derivatives and classes \(C^k\)
    13) Computing higher-order derivatives via Leibniz Ex 38 Ex 39 Ex 40
    14) Determining the class \(C^k\) of a piecewise function Ex 41 Ex 42 Ex 43
G) Brief extension to complex-valued functions
    15) Differentiating a complex-valued function componentwise Ex 44 Ex 45 Ex 46
    16) Applying the complex mean value inequality Ex 47 Ex 48 Ex 49

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