Mathematics for Applied Sciences (Osnabrück 2011-2012)/Part I/Exercise sheet 21
- 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
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