Mathematics for Applied Sciences (Osnabrück 2023-2024)/Part I/Exercise sheet 14



Exercises

Sketch the slope triangle and the secant for the function

in the points and .


Determine the affine-linear map

whose graph passes through the two points

and .


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.


Let be an even function, and suppose that it is differentiable in the point . Show that is also differentiable in the point and that the relation

holds.


Please try to solve the following exercise in a direct way aswell as with the help of derivation rules.

Determine the derivative of the functions

for all .


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


Determine for a polynomial

the linear approximation (including the remainder function ) in the zero point.


Show, using limits of functions, that a function , which is differentiable in a point , is also continuous in this point.


Prove the product rule for differentiable functions, using limits of functions, applied to the difference quotient.


Show that the exponential function is differentiable in every point , and determine its derivative.
Hint: Apply the definition about the limit of functions to the fraction of differences. The function equation for the exponential function is helpful.


Determine the linear approximation (including the remainder function ) for the exponential function in the zero point.


Determine the derivative of the function

for all .


Determine the derivative of the function


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


Let

denote differentiable functions, and set

. Show that the derivative of can be written as a fraction, with as denominator.


Let

denote differentiable functions. Prove, by induction over , the relation


Consider and . Determine the derivative of the composite function directly and by the chain rule.


Let and . We consider the composition .

  1. Compute (the result must be in the form of a rational function).
  2. Compute the derivative of , using part 1.
  3. Compute the derivative of , using the chain rule.


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 .


Determine the derivative of the function

for all .


Let

be a bijective differentiable function with for all , and the inverse function . What is wrong in the following "Proof“ for the derivative of the inverse function?

We have

Using the chain rule, we get by differentiating on both sides the equality

Hence,


Give an example of a continuous, not differentiable function

fulfilling the property that the function is differentiable.




Hand-in-exercises

Exercise (2 marks)

Determine the affine-linear map

whose graph passes through the two points

and .


Exercise (2 marks)

Let be an odd differentiable function. Show that the derivative is even.


Exercise (3 marks)

Let be a subset and let

be differentiable functions. Prove the formula


Exercise (4 marks)

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


Exercise (3 marks)

Determine the derivative of the function

where is the set where the denominator does not vanish.


Exercise (7 (2+2+3) marks)

Let

and

Determine the derivative of the composite

directly and by the chain rule.



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