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# Find an inverse for a function defined by an integral

Define a function for by

1. Prove that is strictly increasing on the nonnegative real axis.
2. If denotes the inverse of prove that is proportional to and find the constant of proportionality.

1. Proof. To show is strictly increasing we take the derivative,

Since for all we have that is strictly increasing on the nonnegative real axis

2. Proof. If is the inverse of then we know (Theorem 6.7 on page 252 of Apostol)

But, we have defined to the function such that . Therefore,

Therefore, we have that

Taking another derivative of with respect to we then have

# Find an inverse for the function log |x|

Consider the function for . Prove that this function has an inverse, determine the domain of this inverse, and find a formula to compute the inverse .

Proof. From the discussion on page 146 of Apostol we know that a function which is continuous and strictly monotonic on an interval has an inverse on . The function is continuous and strictly monotonic on the negative real axis; therefore, it has an inverse. We know it is continuous since the log function is continuous on the positive real axis, and for all , in particular, for all . Furthermore, we know it is strictly monotonic since

Therefore, has an inverse for all . The domain of this inverse is the range of which is all of

To find a formula for the inverse we set

Therefore, valid for all .

A sketch for the graph of is given by

# Establish the formula for the derivative of arccsc x

Establish the following formula for the derivative of is correct,

For let

Then we know

Therefore, by Theorem 6.7 (p. 252 of Apostol) we have,

Where we use the trig identity in the final line. Then, since we have,

# Establish the formula for the derivative of arcsec x

Establish the following formula for the derivative of is correct,

For let

Then we know

Therefore, by Theorem 6.7 (p. 252 of Apostol) we have,

Where we use the trig identity in the final line. Then, since we have,

# Establish the formula for the derivative of arccot x

Establish the following formula for the derivative of is correct,

For let

Then we know

Therefore, by Theorem 6.7 (p. 252 of Apostol) we have,

Using a trig identity for tangent and cosecant we have

since . Therefore we conclude,

# Establish the formula for the derivative of arctan x

Establish the following formula for the derivative of is correct,

For let

Then we know

Therefore, by Theorem 6.7 (p. 252 of Apostol) we have,

Using a trig identity for tangent and secant we have

since . Therefore we conclude,

# Establish the formula for the derivative of arccos x

Establish the following formula for the derivative of is correct,

For let

Then we know

Therefore, by Theorem 6.7 (p. 252 of Apostol) we have,

Using the pythagorean identity for sine and cosine we have

since . Furthermore, since on the range of (i.e., for ) we must take the positive square root. Therefore we conclude,

# Show that a function is monotonic and find a formula for its inverse

Let

Show that is strictly monotonic on . Find the domain of the inverse of , denoted by . Find a formula for computing for each in the domain of .

First, to show is strictly increasing on we note that it is strictly increasing on each component (since and are all increasing functions on the domains given). Then we must consider the transition from one of these regions to another. (In other words, we know the function is increasing on each interval, but we need to check that it is increasing from one interval to the next.)

and,

Thus, is indeed increasing on all of .
Next,

# Show a function is monotonic and find a formala for its inverse

Let . Show that is strictly monotonic on . Find the domain of the inverse of , denoted by . Find a formula for computing for each in the domain of .

First, to show is monotonic let with . Then we consider

But, since by assumption, we have . The second term in the product is also positive since it is a sum of positive terms. Therefore,

Hence, is strictly increasing on . (Of course, there are faster ways to discover the is increasing, but we do not know yet what is a derivative.)

Next,

# Show a function is monotonic and find a formula for its inverse

Let . Show that is strictly monotonic on . Find the domain of the inverse of , denoted by . Find a formula for computing for each in the domain of .

First, to show is monotonic let with . Then

Hence, is strictly decreasing on .

Next,