Assignment 21

course Mth 272

......!!!!!!!!...................................

20:25:27

Query problem 6.6.14 integral from -infinity to infinity of x^2 e^(-x^3)

......!!!!!!!!...................................

RESPONSE -->

confidence assessment:

.................................................

......!!!!!!!!...................................

20:25:40

Does the integral converge, and if so what is its value? Explain why the integral does or does not converge.

......!!!!!!!!...................................

RESPONSE -->

The integral converges at 0.

int. of x^2 e^(-x^3) is -x^2e^(-x^3) - 2xe^(-x^3) - 2e^(-x^3)

lim a approaches -infinity [f(0) - f(a)] 0,a + lim b approaches infinity[f(b) - f(0)]b,0 = 0

confidence assessment: 2

** The integral as stated here converges.

You need to take the limit as t -> infinity of INT(x^2 e^(-x^3), x from -t to t ).

Using the obvious substitution we see that the result is the same as the limiting value as t -> infinity of INT( 1/3 e^(-u), u from -t to t ). Using -1/3 e^(-u) as antiderivative we get -1/3 e^(-t)) - (-1/3 e^(-(-t))); the second term is 1/3 e^t, which approaches infinity as t -> infinity. The first term approaches zero, but that doesn't help. The integral approaches infinity.

Note that the integral from 0 to infinity converges: We take the limit as t -> infinity of INT(x^2 e^(-x^3), x from 0 to t ), which using the same steps as before gives us the limit as t -> infinity of -1/3 e^(-t) - (-1/3) e^0. The first term approaches zero, the second is just 1/3. So the limiting value is 1/3. **

.................................................

......!!!!!!!!...................................

20:25:46

Query problem 6.6.40 (was 6.6.38) farm profit of $75K per year, 8% continuously compounded, find present value of the farm for 20 years, and forever.

......!!!!!!!!...................................

RESPONSE -->

confidence assessment:

.................................................

......!!!!!!!!...................................

20:25:59

What is the present value of the farm for 20 years, and what is its present value forever?

......!!!!!!!!...................................

RESPONSE -->

For 20 years the value is $748,222

Forever the value is $937,500

confidence assessment: 2

.................................................

......!!!!!!!!...................................

20:26:09

What integrals did you evaluate to get your results?

......!!!!!!!!...................................

RESPONSE -->

[(75000/-.08)(e^-.08t)]20,0 = 748,222

lim b approaches infinity [(75000/-.08)(e^-.08t)]b,0 = 937,500

confidence assessment: 2

.................................................

......!!!!!!!!...................................

20:26:11

Query Add comments on any surprises or insights you experienced as a result of this assignment.

......!!!!!!!!...................................

RESPONSE -->

confidence assessment:

.................................................

Right idea on the first problem, but see my note. The second problem looks fine.

&#

Let me know if you have questions. &#