北京邮电大学高等数学第一册答案

12. (1) y = u 3 , u = sin v , v = w and w = 1 − 2 x .(2) y = arccos u , u =
x−2 1 .(3) y = , u = 1 + v , v = arctan w , w = 2 x . u 2
(4) y = u10 , u = 1 + 2 x .(5) y = u 2 , u = arcsin v , v = x 2 .(6) y = ln (1 + u ) , u = 1 + v , v = x 2 .(7) y = 2u , u = v 3 , v = sin x . 13.
(4)
( 0, +∞ ) .(5) ( −4, −2 ) .(6) ( −3, −2] . (1, +∞ ) .
⎡ 2 ⎤ (7) (1, 2 ) ∪ ( 2, 4] . (8) ⎢ − ,1⎥ . (9) ⎣ 2 ⎦
源自文库
( 0, +∞ ) .
(10) [ 0, 2 ) .
(11)
6.
1 ⎧ ⎪ a ≤ x ≤ 1 − a, 0 < a ≤ 2 ⎪ . (1) [ −1, 0] .(2) [ 0,1] .(3) ⎡ 2kπ , ( 2k + 1) π ⎤ , k ∈ Z .(4) ⎨ ⎣ ⎦ ⎪∅ , a > 1 ⎪ 2 ⎩
17. (1) x = − 1 − y 2 ,
2
( 0 ≤ y ≤ 1) .(2)
( −2 ≤ y ≤ 2 ) .(4)
x = log 3
(1 − y )
y
,
( 0 < y < 1) .
⎛ y +1⎞ −∞ < y < 1 ⎧ y, ⎜ ⎟ −1 ⎪ ⎝ 1− y ⎠ 1≤ y ≤ 2 . (5) x = y ∈ [ −1,1) .(6) x = ⎨ y , 2 ⎪log y, 2 < y < +∞ ⎩ 2
(2) x = 1 is an essential discontinuous point. (3) x = 0 is a jump discontinuous point or discontinuous point of first type. (4) x = ±1 are both jump discontinuous point or discontinuous point of the first type. 6. (1) x = 0 is an essential discontinuous point or discontinuous point of second type. (2) x = 1 is an essential discontinuous point or discontinuous point of second type. (3) x = 0 is a continuous point. (4) x = 0 is a jump discontinuous or a discontinuous of the first type. (5) x = −1 x = 2k + 1 , ( k ∈ N + )are essential discontinuous points or discontinuous points of second type. x = 0 is a jump discontinuous or a discontinuous point of the first kind.. x = 1 is a continuous point. 7. (1)
Oct. 2011
北京邮电大学双语高等数学教学组 2011 年第一版
⎧0, x >1 ⎪ 15. f ( x ) = ⎨h ( x + 1) , −1 ≤ x < 0 ⎪− h ( x − 1) , 0 ≤ x ≤ 1 ⎩
⎧ ⎪0, x < −1 ⎪ ⎪ g ( x ) = ⎨ 1 − x2 , −1 ≤ x ≤ 1 . ⎪ ⎪ 3 ( x − 1) , x > 1 ⎪ 3 ⎩ 1 y x = e y −1 − 2 .(3) x = arcsin , 3 2
2.
f ( x) =
( −∞ < x < 0 ) .
3.
⎧ − x + 1, x ∈ ( −1, 0] ⎪ f −1 ( x ) = ⎨ x ∈ [1, 2] ⎪ x − 1, ⎩
6. 8.
f ( x) = x +1 .
1⎞ ⎛ 1⎞ 1 ⎛ f ( x ) = x2 − 2 . f ⎜ x − ⎟ = ⎜ x − ⎟ − 2 = x 2 + 2 − 4 . x⎠ ⎝ x⎠ x ⎝
2
4.
eπ
1.4 Part A
1. 2.
∀ε > 0 , there exists a δ > 0 , such that α ( x ) < ε holds for all x ∈ U ( x0 , δ ) . ∀M > 0
, there exists a X > 0
, such that f ( x ) > M
2
1.2 Part A
1. (1) No. (2) Yes. (3) Yes,.(4) No. 2. (1) wrong.(2) wrong(3) wrong.. 5. wrong 6. wrong
1 9. (1) . 2
1 (2) . 3
(3) 2 . (4) 2 .
(5)
1 . 3
(6)
1 . e
π
2
(2) 1
(3) −2
(4) e
−
1 2
(5) 0
3 2
9. (1) a = 0
(2) a = 3
(3) a = 2 b = −
Advanced Mathematics
School of Science, BUPT
Oct. 2011
北京邮电大学双语高等数学教学组 2011 年第一版
Part B
1. (1)
(f
g )( x ) = 0,
( x = 0) , ( g
f )( x ) = 2 − x 2 ,
(1 ≤ x ≤ 2 ) .
⎛x ⎞ (2) ( f g )( x ) = arcsin ⎜ − 1⎟ , ⎝2 ⎠ 1 − 2 x 2 + 2 x3 , −1 + x
⎧1 0 ≤ x ≤1 ⎪ 2 arcsin ( x − 1) , ⎪ . ( 0 ≤ x ≤ 4 ) , ( g f )( x ) = ⎨ ⎛ 2 ⎞ ⎪ 1 arcsin ⎜ x − 1 ⎟ , 1 < x ≤ 2 ⎪2 ⎝ 2 ⎠ ⎩
1.5 Part A
2. Wrong 5. (1) x = 2 is a removable discontinuous point or discontinuous point of the first type.
x = −2 is an essential discontinuous point or discontinuous point of the second type.
(6) e −2
(7) π
(8) e 2
8. (1) a = −1 b = −2
(2) a =
Part B
1. (1)
4 3
(2) e −2
(3) e −1/ 2
2
(4) e π
f ( x ) − A > ε holds for all x ∈ U ( x0 , δ ) .
3. some ε > 0 , there exists a δ > 0 , such that
7. (1) No. (2) No.(3) 8. (1) Yes.(2) Yes.(3)
No. (4)Yes.(5) No. (6) Yes.(7) No. (8)Yes.(9) No. (10) Yes. Yes.
⎧5 − 3x, x < 1 ⎪ 11. f ( x ) = ⎨3 − x, 1 ≤ x < 2 . ⎪3x − 5 x ≥ 2 ⎩
北京邮电大学双语高等数学教学组 2011 年第一版
1.1 Part A
1. (1) A ∪ B = {1, 2,3, 4,5, 6, 7,8} , A ∩ B = {8} , A \ B = {1,3,5, 7} , B \ A = {2, 4, 6} . (2) A ∪ B = {all parallelograms} , A ∩ B = {all rectangles} , A \ B = {all parallelograms except rectangles} , B \ A = ∅ . (3) A ∪ B = {1, 2,3, 2. . ∩ Aic = {5, 9} .
i =1 5
},
A ∩ B = {2, 4, 6,
},
A \ B = {1,3,5,
},
B \ A = ∅.
3.
A ∪ B = {1 < x ≤ 3} A ∩ B = ∅ .
1⎤ ⎛ ⎜ −∞, ⎥ . 2⎦ ⎝
5. (1)
(2) (α , β ) ∪ ( γ , +∞ ) .
π 2π ⎤ ⎡ (3) ⎢ 2kπ + , 2kπ + . 3 3 ⎥ ⎣ ⎦
f ( x ) − A < ε holds for all x0 − x < δ .
(3) ∀M < 0 , ∃δ > 0 , f ( x ) < M holds for all x − 2 < δ . 2. (1) wrong. (2) right. 3. (1) wrong. (2) right. (2) right. (2) wrong. (2)wrong. (2) wrong. (7) Yes. (8) No.
5. (1) No. (2) No. (3) No. (4) No. (5) No. (6) No. 6. (1) −3 7. (1)
1 2
(2) 1 (2)
1 4
(3) 1 (3) −1
(4) 3 (4)
1 2
(5)
33 4 (6) cos x (7) − sin x (8) 0 2
2
π
(5) e −6
⎧1/ e, | x |< 1 ⎪ ( g f )( x ) = ⎨1, | x |= 1 ⎪e, | x |> 1 ⎩
⎧−1, x < 0 ⎪ 14. ( f g )( x ) = ⎨0, x = 0 ⎪1, x>0 ⎩
Advanced Mathematics
School of Science, BUPT
Advanced Mathematics
School of Science, BUPT
Oct. 2011
北京邮电大学双语高等数学教学组 2011 年第一版
⎧0, l<m ⎪ al n + al −1n + + a1n + a0 ⎪ al 12. lim =⎨ , l=m. m −1 n →∞ b n m + b + + b1n + b0 ⎪ bm m m −1 n ⎪∞, l > m ⎩
l l −1
13. (1) convergent. (2) divergent. (3) convergent.
1.3 Part A
1. (1) ∀ε >0 , ∃X > 0 ,
f ( x ) − A < ε holds for all x > X . (2) ∀ε > 0 , ∃δ > 0 ,
4. (1)wrong. (2) wrong. 5. (1) x (2) x (3) x (4) x 4 / 3 7. (1) 2 (2) 1 3 (3) −
1 2
Part B
1. (2) y = x − 1
⎧a = ±1 ⎪ 2. (1) ⎨b = ± 1 ⎪ ⎩ 2
(2) a =
3 1 , b= c=2 16 2
( f φ )( x ) = sin 3 2 x − sin 2 x,
x ∈ ( −∞, +∞ ) , (φ f )( x ) = sin 2 ( x 3 − x ) ,
x ∈ ( −∞, +∞ ) ,
(f
f )( x ) = x − 2 x3 + 3x5 − 3x 7 + x9 , x ∈ ( −∞, +∞ ) .
holds for all x > X .
3. (1)wrong. (2) wrong. (3)wrong. (4) wrong. (5) wrong.
Advanced Mathematics School of Science, BUPT Oct. 2011
北京邮电大学双语高等数学教学组 2011 年第一版