Math 432 – HW 7.2 Solutions

 

Assigned: 2, 5, 7, 9, 12, 13, 17, 20, 21, 23, 24, 27, 29(a, c, e, g, and i), and 30

 

Selected for grading:  2, 13, 20, 24

 

Solutions.

2.   The Laplace transform of  f (t) = t 2.

 

6.   The Laplace transform of  f (t) = cos t:

To find the indefinite integral , you could use the integral table at the front of the book, or you could do the following.

 

First, you use integration by parts:

u = cos t,  du = –sin t dt,  dv = e –st dt, 

And you use integration by parts again:

u = sin t,  du = cos t dt,  dv = e –st dt, 

Adding that last term to both sides of the equation gives

And so

Taking the limit of the definite integral gives

 

 


 

7.   The Laplace transform of  f (t) = e 3t cos 3t is

A substitution similar to the one made in #5 yields

And the limit of this expression as  N → ∞  is

 

9.   The given function is defined piecewise by

The Laplace transform is

 

12. The function whose Laplace transform we want is given by

The Laplace transform is

 


 

13.

And the domain for this function is the smallest interval common to all four domains.  That's  s > 0.

 

17.

 

20.

 

 

 

21. Here is the sketch of the function given by

Since each formula in the definition is continuous on its respective interval of definition, the only question is whether  f  is continuous at  t = 1.  As you can check, f (t) → 1 = f (1)  as  t → 1, so  f  is continuous on the interval [0, 10].

 


 

23. The function given by

is not continuous on [0, 10] since it is not continuous at  t = 1  and not continuous at  t = 3.

The one-sided limits  and  both exist and are finite.

The one-sided limits  and  both exist and are finite.

So this function is piecewise continuous on [0, 10].

Here is its graph.

 

 

24. The function

is not continuous at  t = 2  and not continuous at  t = –2 (but this second value is outside the range of our consideration).

The one-sided limits  are both finite.

So this function is piecewise continuous on [0, 10].

Here is its graph.

 


 

27. And here's one more.

This function is neither continuous not piecewise continuous on [0, 10] because the limit  is not finite.  Here is its graph.

 

29. (a)  t 3 sin t  is of exponential order α = 1.  (In fact, any α > 0 would do.)

(c)  No matter how large the constants α and M, the absolute value    will eventually (i.e., for large enough  t) be larger than Me αt, so this function is not of exponential order.

(e)  Since   for large  t, then this function is not of exponential order.

(g)  sin(t 2) + t 4e 6t  is of exponential order α = 6.

(i)    for all  t > 0 and

So this function is of order α = 2.

 

30. For all the transforms in Table 7.1,