**Problem 14.5**

a)

We will make a table of each function, it's AR plateau, its
break frequency (w_{b}), and
slope.

where N.A. means that there is no break frequency. In general,

f for 10 = tan^{-1}(0/10) =
0^{o}

f for 10s+1 = tan^{-1}(10w/1) = tan^{-1}(10w)

f for s+1 = tan^{-1}(w/1) = tan^{-1}(w)

f total = 0 - tan^{-1}(10w) - tan^{-1}(w)

We must calculate AR_{total} and f_{total},
which is shown in the table below:

The asymptotic plot is shown below (note that we were only asked to plot AR and not f):

b)

We will make a table of each function, it's AR plateau, its
break frequency (w_{b}), and
slope.

where N.A. means that there is no break frequency. In general,

f for 10 = tan^{-1}(0/10) =
0^{o}

f for 10s+1 = tan^{-1}(10w/1) = tan^{-1}(10w)

f for (s+1)^{2} = 2tan^{-1}(w/1) = 2tan^{-1}(w)

f total = 0 - tan^{-1}(10w) - 2tan^{-1}(w)

We must calculate AR_{total} and f_{total},
which is shown in the table below:

The asymptotic plot is shown below (note that we were only asked to plot AR and not f):

c)

We will make a table of each function, it's AR plateau, its
break frequency (w_{b}), and
slope.

where N.A. means that there is no break frequency. In general,

f for 10 = tan^{-1}(0/10) =
0^{o}

f for s+1 = tan^{-1}(w/1) = tan^{-1}(w)

f for 10s+1 = tan^{-1}(10w/1) = tan^{-1}(10w)

f for (0.1s+1) = tan^{-1}(0.1w/1) = tan^{-1}(0.1w)

f total = 0 + tan^{-1}(w) - tan^{-1}(10w)
- tan^{-1}(0.1w)

We must calculate AR_{total} and f_{total},
which is shown in the table below:

The asymptotic plot is shown below (note that we were only asked to plot AR and not f):

d)

We will make a table of each function, it's AR plateau, its
break frequency (w_{b}), and
slope.

where N.A. means that there is no break frequency. In general,

f for 10 = tan^{-1}(0/10) =
0^{o}

f for -s+1 = tan^{-1}(-w/1) = tan^{-1}(-w)

f for 10s+1 = tan^{-1}(10w/1) = tan^{-1}(10w)

f for (0.1s+1) = tan^{-1}(0.1w/1) = tan^{-1}(0.1w)

f total = 0 - tan^{-1}(w) - tan^{-1}(10w)
- tan^{-1}(0.1w)

We must calculate AR_{total} and f_{total},
which is shown in the table below:

The asymptotic plot is shown below (note that we were only asked to plot AR and not f):

e)

f for 10 = tan^{-1}(0/10) =
0^{o}

f for s = tan^{-1}(w/0) = tan^{-1}(infinity) = 90^{o}

f for 10s+1 = tan^{-1}(10w/1) = tan^{-1}(10w)

f total = 90^{o}
- tan^{-1}(10w)

We must calculate AR_{total} and f_{total},
which is shown in the table below:

The asymptotic plot is shown below (note that we were only asked to plot AR and not f):

f)

f for 10 = tan^{-1}(0/10) =
0^{o}

f for s+1 = tan^{-1}(w/1) = tan^{-1}(w)

f for s = tan^{-1}(w/0) = tan^{-1}(infinity) = 90^{o}

f for 10s+1 = tan^{-1}(10w/1) = tan^{-1}(10w)

f for (0.1s+1) = tan^{-1}(0.1w/1) = tan^{-1}(0.1w)

f total = tan^{-1}(w) - 90^{o} - tan^{-1}(10w) - tan^{-1}(0.1w)

The asymptotic plot is shown below (note that we were only asked to plot AR and not f):