- Warm-up-exercises
Determine the derivatives of hyperbolic sine and hyperbolic cosine.
Determine the derivative of the function
-
Determine the
derivative
of the
function
-
Determine the derivatives of the sine and the cosine function by using
fact.
Determine the
-th derivative of the sine function.
Determine the derivative of the function
-
Determine the derivative of the function
-
Determine for
the derivative of the function
-
Determine the derivative of the function
-
Prove that the real sine function induces a bijective, strictly increasing function
-
and that the real cosine function induces a bijective, strictly decreasing function
-
Determine the derivatives of arc-sine and arc-cosine functions.
We consider the function
-
a)
Prove that
gives a continuous bijection between
and
.
b) Determine the inverse image
of
under
, then compute
and
. Draw a rough sketch for the inverse function
.
Let
-
be two differentiable functions. Let
.
Suppose we have that
-
Prove that
-
We consider the function
-
a) Investigate the monotony behavior of this function.
b) Prove that this function is injective.
c) Determine the image of
.
d) Determine the inverse function on the image for this function.
e) Sketch the graph of the function
.
Consider the function
-
Determine the zeros and the local (global) extrema of
. Sketch up roughly the graph of the function.
Discuss the behavior of the function graph of
-
Determine especially the monotonicity behavior, the extrema of
,
and also for the derivative
.
Prove that the function
-
![{\displaystyle {}f(x)={\begin{cases}x\sin {\frac {1}{x}}{\text{ for }}x\in {]0,1]},\\0{\text{ for }}x=0,\end{cases}}\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c3d48a7a4f4680c21607a6e524c032d4a530a51b)
is continuous, and that it has infinitely many zeros.
Determine the limit of the sequence
-
Determine for the following functions if the function limit exists and, in case, what value it takes.
,
,
,
.
Determine for the following functions, if the limit function for
,
,
exists, and, in case, what value it takes.
,
,
.
- Hand-in-exercises
Determine the linear functions that are tangent to the exponential function.
Determine the derivative of the function
-
The following task should be solved without reference to the second derivative.
Determine the extrema of the function
-
Let
-
be a polynomial function of degree
.
Let
be the number of local maxima of
and
the number of local minima of
. Prove that if
is odd then
and that if
is even then
-
