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# Find functions satisfying given conditions

Find functions satisfying the given conditions in each of the following cases.

1. for and .
2. for all , and .
3. for all and .
4. and .

1. We make the substitution . Since this gives us . Therefore,

Since we are given we can solve for ,

Therefore,

2. We make the substitution . Since we then have . Therefore,

Since we are given that we can solve for ,

Therefore,

This formula is valid for since and for all .

3. We make the substitution . Since we then have . So,

Since we are given that we can solve for ,

Therefore,

This is valid for since and for all .

4. We make the substitution . Then, and so we have

So, we consider the two cases separately. If then we have and

If then we have and

Therefore, we have the function

Now, to solve for we use the condition that . (Here we’re going to assume we want to make the function continuous at , i.e., that the two pieces of this piecewise definition take the same value at 0 so that the limits from the left and right would be equal.) Therefore, we have

and

Thus, the function is given by

# Prove that different functions may have the same average

Let be a continuous, strictly monotonic function on with inverse , and let be given positive real numbers. Then define,

This is called the mean of with respect to . (When for , this coincides with the th power mean from this exercise).

Show that if with , then .

Proof. Let with . Then, has an inverse since it is strictly monotonic (since it is the composition of and the linear function , both of which are strictly monotonic for ). Its inverse is given by

So,

# Show the mean of a strictly monotonic function lies in an interval

Let be a continuous, strictly monotonic function on with inverse , and let be given positive real numbers. Then define,

This is called the mean of with respect to . (When for , this coincides with the th power mean from this exercise).

Show that

Proof. Since is strictly monotonic on the positive real axis and are positive reals, we know is strictly increasing or strictly decreasing, and correspondingly we have,

First, assume is striclty increasing, then

Since is strictly increasing so is its inverse (by Apostol’s Theorem 3.10); thus, we have

If is strictly decreasing then

# Formula for counting lattice points in the ordinate set of a function

For a nonnegative function defined on an interval define the set

(i.e., the region enclosed by the graph of the function and the -axis between the vertical lines at and ). Then prove

where is the greatest integer less than or equal to .

We can help ourselves by drawing a picture to get a good idea of what is going on, then turn that intuition into something more rigorous. The picture is as follows:

In the picture, we can see the number of lattice points in the ordinate set of (not including the -axis since the question stipulates contains the points ). At each integer between and , we count the number of lattice points beneath , the greatest integer less than or equal to . Then we need to turn this intuition from the picture into a proof:

Proof. Let with . We know such an exists since with . Then, the number of lattice points in with first element is the number of integers such that . But, by definition, this is (the greatest integer less than or equal to ). Summing over all integers we have,

# Deduce and prove a formula for [nx]

Use the previous exercise (1.11, #4 parts (d) and (e)) to deduce a formula for and prove this formula is correct.

Claim:

Proof. Let , then we have

Thus, there exists some with such that

Hence, . Thus,

Then, for each with we have

Then, for , we have

So,

# Prove some properties of the greatest integer function

For any we denote the greatest integer less than or equal to by . Prove the following properties of the function :

1. for any integer .
2. or .
3. .
4. .

1. Proof. Let for some integer . Then,

But, we defined . Thus, for any

2. Proof. If , then for some . Hence, and

On the other hand, if , then let . This gives us,

3. Proof. Let and , then we have

So, adding, we obtain,

Thus,

4. Proof. By part (c) we have,

If , then let ,

Thus, .
On the other hand, if , then let , and

Thus,

5. Proof. By part (c) we have

And,

So, putting these together we have,

If , then, let , so

Thus, .
Next, if , then let , giving us,

Thus,

Finally, if , then let , and we have

So,

# Draw the graphs of some functions

Draw the graphs of the functions defined below on the interval , and if it is a step function find a partition such that the function is constant on the open subintervals of .

1. .
2. .
3. .
4. .
5. .
6. .

1. This is not a step function. The graph is below.

2. This is not a step function. The graph is below.

3. This is a step function and it is constant on the open subintervals of the partition, . The graph is below.

4. This is a step function and it is constant on the open subintervals of the partition, . The graph is below.

5. This is a step function and it is constant on the open subintervals of the partition, . The graph is below.

6. This is a step function and it is constant on the open subintervals of the partition, . The graph is below.

# Draw the graphs of the functions and points of intersection for x^2 -2 and 2x^2 + 4x + 1

Let and . Find the points of intersection and draw the graph.

The points of intersection are at,

Thus, they intersect at and . These correspond to the points and . The graph is below.

# Draw the graphs and find the points of intersection of x and x^3

Let and . Draw the graphs of and and show the points at which they intersect.

First, we can determine algebraically the points of intersection. They are the points such that , i.e.,

The function values at these points are,

Thus, the points of intersection are . The graph is below.

# Verify some formulas for the function g(x) = (4-x^2)^(1/2)

Let for .

1. .
2. .
3. .
4. .
5. .
6. .

1. Proof.
. Valid for
2. Proof.
. Valid for .
3. Proof.
. Valid for
4. Proof.
. Valid for
5. Proof.
. Valid for
6. Proof.
. Valid for