The temperature T (in ° F ) for Kansas City, Missouri, over a several day period in April can be approximated by T ( t ) = − 5.9 cos ( 0.262 t − 1.245 ) + 48.2 , where t is the number of hours since midnight on day 1. a. What is the period of the function? Round to the nearest hour. b. What is the significance of the term 48.2 in this model? c. What is the significance of the factor 5.9 in this model? d. What was the minimum temperature for the day? When did it occur? e. What was the maximum temperature for the day? When did it occur?
The temperature T (in ° F ) for Kansas City, Missouri, over a several day period in April can be approximated by T ( t ) = − 5.9 cos ( 0.262 t − 1.245 ) + 48.2 , where t is the number of hours since midnight on day 1. a. What is the period of the function? Round to the nearest hour. b. What is the significance of the term 48.2 in this model? c. What is the significance of the factor 5.9 in this model? d. What was the minimum temperature for the day? When did it occur? e. What was the maximum temperature for the day? When did it occur?
Solution Summary: The author explains the significance of the term 48.2 in this model. The temperature T for Kansas city, Missouri, over a several day period in April can be approximated by T(t)=-5.9cos
The temperature T (in
°
F
) for Kansas City, Missouri, over a several day period in April can be approximated by
T
(
t
)
=
−
5.9
cos
(
0.262
t
−
1.245
)
+
48.2
, where t is the number of hours since midnight on day 1.
a. What is the period of the function? Round to the nearest hour. b. What is the significance of the term 48.2 in this model? c. What is the significance of the factor 5.9 in this model? d. What was the minimum temperature for the day? When did it occur? e. What was the maximum temperature for the day? When did it occur?
Let
2
A =
4
3
-4
0
1
(a) Show that v =
eigenvalue.
()
is an eigenvector of A and find the corresponding
(b) Find the characteristic polynomial of A and factorise it. Hint: the answer to (a)
may be useful.
(c) Determine all eigenvalues of A and find bases for the corresponding eigenspaces.
(d) Find an invertible matrix P and a diagonal matrix D such that P-¹AP = D.
(c) Let
6
0 0
A =
-10 4 8
5 1 2
(i) Find the characteristic polynomial of A and factorise it.
(ii) Determine all eigenvalues of A and find bases for the corresponding
eigenspaces.
(iii) Is A diagonalisable? Give reasons for your answer.
most 2, and let
Let P2 denote the vector space of polynomials of degree at
D: P2➡ P2
be the transformation that sends a polynomial p(t) = at² + bt+c in P2 to its derivative
p'(t)
2at+b, that is,
D(p) = p'.
(a) Prove that D is a linear transformation.
(b) Find a basis for the kernel ker(D) of the linear transformation D and compute its
nullity.
(c) Find a basis for the image im(D) of the linear transformation D and compute its
rank.
(d) Verify that the Rank-Nullity Theorem holds for the linear transformation D.
(e) Find the matrix representation of D in the standard basis (1,t, t2) of P2.
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