- Warm-up-exercises
Prove that the function
-
is differentiable but not twice differentiable.
Consider the function
-
defined by
-
Examine in terms of continuity, differentiability and extremes.
Determine local and global extrema of the function
-
Determine local and global extrema of the function
-
Consider the function
-
Find the point
such that the tangent of the function at is parallel to the secant between
and .
Prove that a real polynomial function
-
of degree
has at most extrema, and moreover the real numbers can be divided into at most sections, where is strictly increasing or strictly decreasing.
Determine the limit
-
by polynomial long division.
Determine the limit of the rational function
-
at the point
.
Next to a rectilinear river we want to fence a rectangular area of , one side of the area is the river itself. For the other three sides, we need a fence. Which is the minimal length of the fence we need?
Discuss the following properties of the rational function
-
domain, zeros, growth behavior,
(local)
extrema. Sketch the graph of the function.
Consider
-
a) Prove that the function has in the real interval exactly one zero.
b) Compute the first decimal digit in the decimal system of this zero point.
c) Find a rational number
such that
.
Determine the limit of
-
at the point
,
and specifically
a) by polynomial division.
b) by the rule of l'Hospital.
Let
be a polynomial,
and
.
Prove that is a multiple of if and only if is a zero of all the derivatives .
- Hand-in-exercises
From a sheet of paper with side lengths of cm and cm we want to realize a box (without cover) with the greatest possible volume. We do it in this way. We remove from each corner a square of the same size, then we lift up the sides and we glue them. Which box height do we need to realize the maximum volume?
Discuss the following properties of the rational function
-
domain, zeros, growth behavior,
(local)
extrema. Sketch the graph of the function.
Prove that a non-constant rational function of the shape
-
(with
,
,)
has no local extrema.
Determine the limit of the rational function
-
at the point
.
Let
and
-
be a rational function. Prove that is a polynomial if and only if there is a higher derivative such that
.