Déjà Vu At 8:00 a.m. on Saturday, a nun begins running up the side of a mountain to his weekend campsite (see figure). On Sunday morning at 8:00 a.m., he runs back down the mountain. It takes him 20 minutes to run up but only10 minutes to run down. At some point on the way down, he realizes that he passed the same place at exactly the same time on Saturday. Prove that he is correct. [ Hint : Let s ( t ) and r ( t ) be the position functions for the runs up and down, and apply the Intermediate Value Theorem to the function f ( t ) = s ( t ) − r ( t ) .]
Déjà Vu At 8:00 a.m. on Saturday, a nun begins running up the side of a mountain to his weekend campsite (see figure). On Sunday morning at 8:00 a.m., he runs back down the mountain. It takes him 20 minutes to run up but only10 minutes to run down. At some point on the way down, he realizes that he passed the same place at exactly the same time on Saturday. Prove that he is correct. [ Hint : Let s ( t ) and r ( t ) be the position functions for the runs up and down, and apply the Intermediate Value Theorem to the function f ( t ) = s ( t ) − r ( t ) .]
Déjà Vu At 8:00
a.m.
on Saturday, a nun begins running up the side of a mountain to his weekend campsite (see figure). On Sunday morning at 8:00
a.m.,
he runs back down the mountain. It takes him 20 minutes to run up but only10 minutes to run down. At some point on the way down, he realizes that he passed the same place at exactly the same time on Saturday. Prove that he is correct. [Hint: Let s(t) and r(t) be the position functions for the runs up and down, and apply the Intermediate Value Theorem to the function
f
(
t
)
=
s
(
t
)
−
r
(
t
)
.]
A factorization A = PDP 1 is not unique. For A=
7 2
-4 1
1
1
5 0
2
1
one factorization is P =
D=
and P-1
30
=
Use this information with D₁
=
to find a matrix P₁ such that
-
-1 -2
0 3
1
-
- 1
05
A-P,D,P
P1
(Type an integer or simplified fraction for each matrix element.)
Matrix A is factored in the form PDP 1. Use the Diagonalization Theorem to find the eigenvalues of A and a basis for each eigenspace.
30 -1
-
1 0 -1
400
0
0 1
A=
3 4 3
0 1 3
040
3 1 3
0 0
4
1
0
0
003
-1 0 -1
Select the correct choice below and fill in the answer boxes to complete your choice.
(Use a comma to separate vectors as needed.)
A basis for the corresponding eigenspace is {
A. There is one distinct eigenvalue, λ =
B. In ascending order, the two distinct eigenvalues are λ₁
...
=
and 2
=
Bases for the corresponding eigenspaces are {
and ( ), respectively.
C. In ascending order, the three distinct eigenvalues are λ₁ =
=
12/2
=
and 3 = Bases for the corresponding eigenspaces are
{}, }, and {
respectively.
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