Mathematics for Applied Sciences (Osnabrück 2011-2012)/Part I/Exercise sheet 21
- Warm-up-exercises
Exercise
Determine the derivatives of hyperbolic sine and hyperbolic cosine.
Exercise
Determine the derivative of the function
Exercise
Determine the derivative of the function
Exercise
Determine the derivatives of the sine and the cosine function by using fact.
Exercise
Determine the -th derivative of the sine function.
Exercise
Determine the derivative of the function
Exercise
Determine the derivative of the function
Exercise
Determine for the derivative of the function
Exercise
Determine the derivative of the function
Exercise
Prove that the real sine function induces a bijective, strictly increasing function
and that the real cosine function induces a bijective, strictly decreasing function
Exercise
Determine the derivatives of arc-sine and arc-cosine functions.
Exercise
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 .
Exercise
Let
be two differentiable functions. Let . Suppose we have that
Prove that
Exercise
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 .
Exercise
Consider the function
Determine the zeros and the local (global) extrema of . Sketch up roughly the graph of the function.
Exercise
Discuss the behavior of the function graph of
Determine especially the monotonicity behavior, the extrema of , and also for the derivative .
Exercise
Prove that the function
is continuous and that it has infinitely many zeros.
Exercise
Determine the limit of the sequence
Exercise
Determine for the following functions if the function limit exists and, in case, what value it takes.
- ,
- ,
- ,
- .
Exercise
Determine for the following functions, if the limit function for , , exists, and, in case, what value it takes.
- ,
- ,
- .
- Hand-in-exercises
Exercise
Determine the linear functions that are tangent to the exponential function.
Exercise
Determine the derivative of the function
The following task should be solved without reference to the second derivative.
Exercise
Determine the extrema of the function
Exercise
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