Mathematics for Applied Sciences (Osnabrück 2011-2012)/Part I/Exercise sheet 19
- Warm-up-exercises
Determine the derivative of the functions
for all .
Determine the derivative of the function
for all .
Determine the derivative of the function
for all .
Determine directly (without the use of derivation rules) the derivative of the function
at any point .
Prove that the real absolute value
is not differentiable at the point zero.
Determine the derivative of the function
Prove that the derivative of a rational function is also a rational function.
Consider and . Determine the derivative of the composite function directly and by the chain rule.
Prove that a polynomial has degree (or it is ), if and only if the -th derivative of is the zero poynomial.
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 .
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 .
Determine the affine-linear map
such that and .
- Hand-in-exercises
Exercise (3 marks)
Determine the derivative of the function
where is the set where the denominator does not vanish.
Exercise (4 marks)
Determine the tangents to the graph of the function , which are parallel to .
Exercise (7 (2+2+3) marks)
Let
and
Determine the derivative of the composite
directly and by the chain rule.
Exercise (2 marks)
Determine the affine-linear map
whose graph passes through the two points
and .
Exercise (3 marks)
Let be a subset and let
be differentiable functions. Prove the formula