Kurs:Mathematik für Anwender (Osnabrück 2011-2012)/Teil I/Arbeitsblatt 19/en

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Warm-up-exercises

The following task should be solved both directly and through the derivation rules.

Exercise

Determine the derivative of the function

for all .


Exercise

Determine the derivative of the function

for all .


Exercise

Determine the derivative of the function

for all .


Exercise

Determine directly

(without the use of derivation rules) the derivative of the function
at any point .


Exercise

Prove that the real absolute value

is not differentiable at the point zero.


Exercise

Determine the derivative of the function


Exercise

Prove that the derivative of a rational function is also a rational function.


Exercise

Consider and . Determine the derivative of the composite function directly and by the chain rule (Theorem 19.8).


Exercise

Prove that a polynomial has degree (or it is ), if and only if the -th derivative of is the zero poynomial.


Exercise

Let

be two differentiable functions and consider

a) Determine the derivative from the derivatives of and .

b) Let now
Compute in two ways, one directly from and the other by the formula of part .

For the “linear approximation” of differentiable maps we need the definition of affine-linear maps.


Let be a field and let and be vector spaces over . A map

where is a linear map and is a vector, is called affine-linear.


Exercise

Let be a field and let be a -vector space. Prove that given two vectors there exists exactly one affine-linear map

sucht that and .


Exercise

Determine the affine-linear map

such that and .




Hand-in-exercises

Exercise (3 points)

Determine the derivative of the function

where is the set where the denominator does not vanish.


Exercise (4 points)

Determine the tangents to the graph of the function , which are parallel to .


Exercise (4 points)

Let and . Determine the derivative of the composite directly and by the chain rule (Theorem 19.8).


Exercise (2 points)

Determine the affine-linear map

whose graph passes through the two points and .


Exercise (3 points)

Let be a subset and let

be differentiable functions. Prove the formula




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