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# Use derivatives to sketch the graph of a function

Let

1. Find all points such that ;
2. Determine the intervals on which is monotonic by examining the sign of ;
3. Determine the intervals on which is monotonic by examining the sign of ;
4. Sketch the graph of .

1. We take the derivative,

Thus,

2. is increasing for and , and decreasing for .
3. Taking the second derivative,

Thus, is increasing if , and decreasing if .

4. We sketch the curve,

# Use derivatives to sketch the graph of a function

Let

1. Find all points such that ;
2. Determine the intervals on which is monotonic by examining the sign of ;
3. Determine the intervals on which is monotonic by examining the sign of ;
4. Sketch the graph of .

1. We take the derivative,

Thus,

2. is increasing for , and decreasing for .
3. Taking the second derivative,

Thus, is increasing if , and decreasing if .

4. We sketch the curve,

# Use derivatives to sketch the graph of a function

Let

1. Find all points such that ;
2. Determine the intervals on which is monotonic by examining the sign of ;
3. Determine the intervals on which is monotonic by examining the sign of ;
4. Sketch the graph of .

1. We take the derivative,

Thus,

2. Since when and when we have is increasing if and is decreasing if .
3. Taking the second derivative,

Thus, is increasing for ; and decreasing for .

4. We sketch the curve,

# Use derivatives to sketch the graph of a function

Let

1. Find all points such that ;
2. Determine the intervals on which is monotonic by examining the sign of ;
3. Determine the intervals on which is monotonic by examining the sign of ;
4. Sketch the graph of .

1. We take the derivative,

Thus,

2. Since when and when we have is decreasing if and is increasing if .
3. Taking the second derivative,

Since this is positive for all , this means is increasing for all .

4. We sketch the curve,

# 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,

# 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 increasing on .

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 increasing on .

Next,