- 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
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Determine the derivative of the function
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Prove that the real sine function induces a bijective, strictly increasing function
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and that the real cosine function induces a bijective, strictly decreasing function
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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
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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
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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
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The following task should be solved without reference to the second derivative.
Determine the extrema of the function
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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
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