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# 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

# Calculate the average power in an electrical circuit

Let and denote the current and voltage, respectively, in a circuit at time . Define

Then, we define the average power by the integral equation

where is the period of the voltage and current. Find and calculate the average power.

First, since the voltage is given by we know it is periodic with period , so . Then, we compute the average power,

# Compute average voltage and root-mean-square of voltage

Denote the voltage in a circuit at time by . Let

Calculate the following:

1. Average voltage over the interval .
2. The root-mean-square of the voltage

1. Denote the average of by and compute,

2. The root-mean-square is given by the square root of the function over the interval . This gives us

Where we used the formula from the solution of Example 3, p. 101 of Apostol, to compute the integral .

# Prove some more properties of the averages of functions on an interval

Define to be the average of a function on the interval ,

1. If with , prove there exists with such that

2. Prove part (a) holds for weighted averages of functions where for a nonnegative weight function we define the weighted average of on by

1. Proof. Let

Then, since , we have . Furthermore,

So,

2. Proof. Let

Then,

Thus, (since since and since is nonnegative both of these are nonnegative).
Then,

# Prove some properties for weighted averages of functions

With reference to the previous exercise which of the following properties are valid for weighted averages of a function on an interval . Denote the weighted average of with a weight function on by .

2. Homogeneous property: for all .
3. Monotone property: if on .

All of these properties are valid for weighted averages.

1. Proof. We compute,

2. Proof. We compute,

3. Proof. Assume on , then since is nonnegative (definition of a weight function) we have for all . Next, by the monotone property of the integral we have

Then, since is nonnegative, is also nonnegative and we have

# Prove some properties of averages of functions

Prove that the average has the following properties:

2. Homogeneous property: for a constant .
3. Monotone property: if for all .

1. Proof. We compute using the formula for the average of a function on an interval,

2. Proof. Again, we compute,

3. Proof. Assume for all . Then, by the monotone property of the integral we have,

# Find weight functions so that the weighted average has given values

Let for . We know the average of on is . Find a nonnegative function such that the weighted average of on is

1. ,
2. ,
3. ,

First, we recall that given functions and (with nonnegative) defined on an interval , the weighted average is given by

1. Let , then

2. Let , then

3. Let , then

# Find a value at which functions are equal to their mean on an interval

1. Let on the interval . Find with such that equals the average of on the interval.
2. Do part (a) with for .

1. We want to equal the average of on , so we set them equal and solve:

2. Again, we set equal to the average of on :

# Compute the average value of the function on the interval

Compute the average value of

Recalling that , we compute the average ,