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Final Exam Practice Problems โ Answers
Math 240 โ Calculus III
Summer 2015, Session II
Linear Algebra
- (a) {(0, 0) โ R^2 }
(b) {(โt, 4 t, t) โ R^3 : t โ R}
(c) {(1, 2 , 4) โ R^3 }
(d) This system is inconsistent.
(e) {(โ 1 โ 2 s โ 4 t, s, โ 2 โ 3 t, t) โ R^4 : s, t โ R} (Answers may vary.)
- x(t) = (3 โ 2 t, 1 โ t, t)
- (a) โ 7
(b) 29
(c) โ 2
- (a) The inverse of this matrix is 1
2
[
]
(b) This matrix is not invertible.
(c) The inverse of this matrix is (^) ๏ฃฎ
- (a) Answers may vary. Correct answers include
{v 1 , v 3 } and {(1, 2 , 1 , 3), (0, 1 , 4 , 1)}.
(b) Answers may vary. Correct answers include
{v 1 , v 2 , v 4 } and {(1, 1 , 1 , 1 , 1), (0, 0 , 1 , 3 , 0), (0, 0 , 0 , 1 , 0)}.
- One method of verification is to compute
det
([
v 1 v 2
])
and note that it is nonzero. Then
(2, โ1) = 5v 1 โ 3 v 2.
- A basis for Ker(T ) is {(โ 2 , 1 , 1)}. A basis for Rng(T ) is {(1, โ 2 , 3), (0, 1 , โ2)}. (Answers may
vary.)
- (a)
[
]
(b)
[
]
[
]
- (a) False.
(b) True.
(c) True.
(d) True.
(e) False.
More Linear Algebra
11. [D] =
๏ฃป ,^ [L] =
- x โ 2 y + z = 0
- (a) 3
[
]
[
]
[
]
[
]
(b) (x^3 + 7x + 3) โ (x^3 + 7x^2 + 3) โ (x^3 โ x^2 โ x) + (x^3 + 6x^2 โ 8 x) = 0
- (a) This is a basis.
(b) Not a basis.
(c) Not a basis โ C^0 (R) is infinite dimensional.
- We can see from the definition that T takes a 2 ร 2 matrix as input and produces a 2 ร 3
matrix. Now we check that it preserves addition and scalar multiplication:
T
([
a 1 b 1
c 1 d 1
]
[
a 2 b 2
c 2 d 2
])
= T
([
a 1 + a 2 b 1 + b 2
c 1 + c 2 d 1 + d 2
])
[
a 1 + a 2 a 1 + a 2 a 1 + a 2 โ b 1 โ b 2
c 1 + c 2 c 1 + c 2 c 1 + c 2 โ d 1 โ d 2
]
- x(t) = (^12)
[
e^3 t^ + eโt
5 e^3 t^ + eโt
]
- x(t) = c 1 et
[
cos 2t
cos 2t + sin 2t
]
[
sin 2t
sin 2t โ cos 2t
]
- x(t) =
1 5 e
โt
[
5 cos t 2 cos t + sin t
]
3 5 e
โt
[
5 sin t 2 sin t โ cos t
]
= eโt
[
cos t โ 3 sin t cos t โ sin t
]
- x(t) = c 1 et
[
]
[
2 t + 1
t
]
= et
[
2 c 1 + c 2 (2t + 1)
c 1 + c 2 t
]
- x(t) = c 1
[
]
[
3 t + 1
โt
]
[
3 c 1 + c 2 (3t + 1)
โc 1 โ c 2 t
]
- x(t) = c 1 eโ^2 t
๏ฃป (^) + c 2 eโt
2 cos
2 t
sin
2 t โ 2 cos
2 t โ sin
2 t
๏ฃป (^) + c 3 eโt
2 sin
2 t
โ cos
2 t โ 2 sin
2 t + cos
2 t
- x(t) = c 1 eโ^2 t
๏ฃป (^) + c 2 eโ^2 t
๏ฃป (^) + c 3 eโ^2 t
4 t + 1
โ 2 t โ 4 t
- (a) y(x) = c 1 x^2 + c 2 xโ^4
(b) y(x) = c 1 + c 2 x
โ (^7) + c 3 x
โ
โ 7
- (a) A(D) = D^2 (D โ 1)
(b) A(D) = (D โ 7)^4 (D^2 + 16)
(c) A(D) = (D โ 4)^2 (D^2 โ 8 D + 41)D^2 (D^2 + 4D + 5)^3
(d) A(D) = D^2 + 6D + 10
- (a) y(x) = c 1 eโ^2 x^ + c 2 xeโ^2 x
(b) y(x) = c 1 eโ^4 x^ cos 2x + c 2 eโ^4 x^ sin 2x
(c) y(x) = c 1 eโ^2 x^ + c 2 e^2 x^ + c 3 xe^2 x
(d) y(x) = c 1 eโx^ + c 2 e^2 x^ + 53 e^2 x
(e) y(t) = c 1 cos 2t + c 2 sin 2t +
32 t^ โ^
1 12 t
3 )^ cos 2t + (^7 4 t^ +^
13 16 t
2 )^ sin 2t
(f) y(t) = c 1 eโ^2 t^ + c 2 teโ^2 t^ + 16 t^3 eโ^2 t^ โ 18 e^2 t
- (a) y(t) = (2 โ t)e^4 t
(b) y(t) = et^ โ cos t
- (a) y(t) = eโt^ (cos 2t + 2 sin 2t) =
5 eโt^ cos(2t โ arctan 2). This system is underdamped.
(b) y(t) = โ 14 eโt/^2 + 54 eโ^5 t/^2 = eโ^3 t/^2
โ 14 et^ + 54 eโt
. This system is overdamped.
(c) y(t) = eโ^2 t^ โ 2 eโ^3 t^ = eโ^5 t/^2
et/^2 โ 2 eโt/^2
. This system is overdamped.
- y(t) = 16 cos 34 t + 12 sin 34 t โ 16 cos 2t
- (a) y(x) = (c 1 + c 2 t)e^3 t^ + 4t^5 /^2 e^3 t
(b) y(x) = c 1 ex^ + c 2 xex^ โ ex^ ln x
