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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 if and decreasing if .
3. Taking the second derivative,

Thus, is increasing for all .

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 all since for all .
3. Taking the second derivative,

Thus, is increasing if and is 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 all since for all .
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,

for an integer. (We found the zeros of sine in this exercise, Apostol Section 2.8 Exercise #1.)

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

Thus, is increasing for or and is 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 if and decreasing if .
3. Taking the second derivative,

Thus, is increasing if or and is decreasing if or .

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 if or and is decreasing if or . (We have to take some care here to leave out the points and since the function is not defined at these points.)
3. Taking the second derivative,

Thus, is increasing for and and is 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. is increasing if or , and is decreasing if .
3. Taking the second derivative,

Thus, is increasing for all (and is undefined 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 for all (and is undefined at ).

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 if and is decreasing if .
3. Taking the second derivative,

Thus, is increasing for all since is positive for all .

4. We sketch the curve,