MATH 152 Midterm: MATH 152 TAMU 2013c Exam 3a
▼❆❚❍ ✶✺✷✱ ❋❆▲▲ ✷✵✶✸
❈❖▼▼❖◆ ❊❳❆▼ ■■■ ✲ ❱❊❘❙■❖◆ ❆
Pr✐♥t ♥❛♠❡ ✭▲❆❙❚✱ ❋✐rst✮✿ ❙❊❈❚■❖◆ ★✿
■◆❙❚❘❯❈❚❖❘✿ ❙❊❆❚ ★✿
❚❍❊ ❆●●■❊ ❈❖❉❊ ❖❋ ❍❖◆❖❘
✧❆♥ ❆❣❣✐❡ ❞♦❡s ♥♦t ❧✐❡✱ ❝❤❡❛t✱ ♦r st❡❛❧✱ ♦r t♦❧❡r❛t❡ t❤♦s❡ ✇❤♦ ❞♦✳✧ ❇② s✐❣♥✐♥❣ ❜❡❧♦✇✱ ②♦✉ ✐♥❞✐❝❛t❡ t❤❛t ❛❧❧
✇♦r❦ ✐s ②♦✉r ♦✇♥ ❛♥❞ t❤❛t ②♦✉ ❤❛✈❡ ♥❡✐t❤❡r ❣✐✈❡♥ ♥♦r r❡❝❡✐✈❡❞ ❤❡❧♣ ❢r♦♠ ❛♥② ❡①t❡r♥❛❧ s♦✉r❝❡s✳
❙■●◆❆❚❯❘❊✿
P❆❘❚ ■✲▼❯▲❚■P▲❊ ❈❍❖■❈❊
❚❤❡ ✉s❡ ♦❢ ❛♥② ❡❧❡❝tr♦♥✐❝ ❞❡✈✐❝❡ ✐s ♣r♦❤✐❜✐t❡❞✳ ▼❛r❦ t❤❡ ❝♦rr❡❝t ❝❤♦✐❝❡ ♦♥ ②♦✉r ❙❝❛♥❚r♦♥ ✉s✐♥❣ ❛ ◆♦✳ ✷ ♣❡♥❝✐❧✳
❋♦r ②♦✉r ♦✇♥ r❡❝♦r❞s✱ ❛❧s♦ r❡❝♦r❞ ②♦✉r ❝❤♦✐❝❡s ♦♥ ②♦✉r ❡①❛♠✦ ❇❡ s✉r❡ t♦ ✇r✐t❡ ②♦✉r ♥❛♠❡✱ s❡❝t✐♦♥ ❛♥❞ ✈❡rs✐♦♥ ❧❡tt❡r
♦❢ t❤❡ ❡①❛♠ ♦♥ t❤❡ ❙❝❛♥❚r♦♥ ❢♦r♠✳ ❊❛❝❤ ♣r♦❜❧❡♠ ✐s ✇♦rt❤ ✸ ♣♦✐♥ts✳
◆❖❚❊✿ ❚❤❡ ❢♦❧❧♦✇✐♥❣ ❢♦r♠✉❧❛s ♠❛② ♦r ♠❛② ♥♦t ❜❡ ✉s❡❢✉❧ ♦♥ t❤✐s ❡①❛♠✿
✭■✮ ■❢ |f(n+1)(x)| ≤ M♦♥ t❤❡ ✐♥t❡r✈❛❧ [a, b]✱ t❤❡♥ |Rn(x)| ≤ M
(n+ 1)!|x−a|n+1 ❢♦r a≤x≤b✳
✭■■✮ s−sn≤ˆ∞
n
f(x)dx
✭■■■✮ |s−sn| ≤ |an+1|
✶✳ ❲❤✐❝❤ st❛t❡♠❡♥t ✐s tr✉❡ r❡❣❛r❞✐♥❣ t❤❡ s❡r✐❡s
∞
X
n=0
n!
(−2013)n❄
✭❛✮ ❚❤❡ s❡r✐❡s ✐s ❞✐✈❡r❣❡♥t
✭❜✮ ◆♦♥❡ ♦❢ t❤❡s❡
✭❝✮ ❚❤❡ s❡r✐❡s ✐s ❛❜s♦❧✉t❡❧② ❝♦♥✈❡r❣❡♥t
✭❞✮ ❚❤❡ s❡r✐❡s ✐s ❝♦♥✈❡r❣❡♥t✱ ❜✉t ♥♦t ❛❜s♦❧✉t❡❧② ❝♦♥✈❡r❣❡♥t
✭❡✮ ❚❤❡r❡ ✐s ✐♥s✉✣❝✐❡♥t ✐♥❢♦r♠❛t✐♦♥ t♦ ❞❡t❡r♠✐♥❡ ✐ts ❝♦♥✈❡r❣❡♥❝❡
✷✳ ❲❤✐❝❤ ♦❢ t❤❡ ❢♦❧❧♦✇✐♥❣ ✐s t❤❡ ✷♥❞ ❞❡❣r❡❡ ❚❛②❧♦r P♦❧②♥♦♠✐❛❧ ❢♦r f(x) = arctan x❝❡♥t❡r❡❞ ❛t a= 1❄
✭❛✮ T2(x) = π
4+1
2(x−1) −1
2(x−1)2
✭❜✮ T2(x) = π
4+1
2(x−1) −1
4(x−1)2
✭❝✮ T2(x) = (x−1)
✭❞✮ T2(x) = (x−1) −1
3(x−1)3
✭❡✮ ◆♦♥❡ ♦❢ t❤❡s❡
✸✳ ❯s❡ t❤❡ ▼❛❝❧❛✉r✐♥ s❡r✐❡s ❢♦r ext♦ ✜♥❞ t❤❡ t❤✐r❞✲❞❡❣r❡❡ ❚❛②❧♦r ♣♦❧②♥♦♠✐❛❧ ❢♦r f(x) = x+e−x❝❡♥t❡r❡❞ ❛t a= 0✳
✭❛✮ T3(x) = x−x2+1
2x3
✭❜✮ T3(x) = −1−1
2x2−1
6x3
✭❝✮ T3(x) = 1 −2x+1
2x2−1
6x3
✭❞✮ ◆♦♥❡ ♦❢ t❤❡s❡
✭❡✮ T3(x) = 1 + 1
2x2−1
6x3
✹✳ ●✐✈❡♥ t❤❡ ♣♦✇❡r s❡r✐❡s ❢♦r 1
1 + x✐s
∞
X
n=0
(−1)nxn❢♦r −1< x < 1✱ ✇❤✐❝❤ ♦❢ t❤❡ ❢♦❧❧♦✇✐♥❣ ✐s ❛ ♣♦✇❡r s❡r✐❡s ❢♦r
x3ln(1 + x)❄
✭❛✮ ◆♦♥❡ ♦❢ t❤❡s❡
✭❜✮
∞
X
n=0
(−1)nnxn+2
✭❝✮
∞
X
n=0
(−1)nxn+3
ln n
✭❞✮
∞
X
n=0
(−1)nxn+4
n+ 1
✭❡✮
∞
X
n=0
(−1)nxn+4
n+ 4
✺✳ ❲❤✐❝❤ ♦❢ t❤❡ ❢♦❧❧♦✇✐♥❣ ✐s t❤❡ ❡q✉❛t✐♦♥ ♦❢ t❤❡ s♣❤❡r❡ ✇❤♦s❡ ❝❡♥t❡r ✐s (0,−3,−5) ✇❤✐❝❤ ♣❛ss❡s t❤r♦✉❣❤ t❤❡ ♣♦✐♥t
(−6,0,−5)❄
✭❛✮ x2+ (y−3)2+ (z−5)2=√45
✭❜✮ x2+ (y+ 3)2+ (z+ 5)2= 45
✭❝✮ ◆♦♥❡ ♦❢ t❤❡s❡
✭❞✮ (x+ 6)2+y2+ (z+ 5)2= 45
✭❡✮ (x+ 6)2+y2+ (z+ 5)2=√45
✻✳ ❚❤❡ s❡r✐❡s
∞
X
n=1
(−1)n+1
8n3❝♦♥✈❡r❣❡s t♦ s✳ ❇❛s❡❞ ♦♥ t❤❡ ❆❧t❡r♥❛t✐♥❣ ❙❡r✐❡s ❊st✐♠❛t✐♦♥ ❚❤❡♦r❡♠✱ ❤♦✇ ♠❛♥② t❡r♠s
♦❢ t❤❡ s❡r✐❡s ✭♠✐♥✐♠✉♠✮ ❞♦ ✇❡ ♥❡❡❞ t♦ ❣✉❛r❛♥t❡❡ |s−sn|<1
4000❄ ✭◆❖❚❊✿ t❤❡ ♥✉♠❜❡r ♦❢ t❡r♠s ❞♦ ❝♦rr❡s♣♦♥❞
t♦ t❤❡ ❝❛❧❝✉❧❛t✐♦♥s ❣✐✈❡♥✮✳
✭❛✮ n > √125✱ s♦ ✇❡ ♥❡❡❞ ❛t ❧❡❛st ✶✷ t❡r♠s
✭❜✮ n > √125 −1✱ s♦ ✇❡ ♥❡❡❞ ❛t ❧❡❛st ✶✶ t❡r♠s
✭❝✮ n > 3
√500 −1✱ s♦ ✇❡ ♥❡❡❞ ❛t ❧❡❛st ✼ t❡r♠s
✭❞✮ n > 3
√500✱ s♦ ✇❡ ♥❡❡❞ ❛t ❧❡❛st ✽ t❡r♠s
✭❡✮ ◆♦♥❡ ♦❢ t❤❡s❡
✼✳ ●✐✈❡♥ t❤❡ ♣♦✐♥t P(3,4,5)✱ ✇❤❛t ✐s t❤❡ ♣r♦❥❡❝t✐♦♥ ♦❢ t❤✐s ♣♦✐♥t ♦♥t♦ t❤❡ x✲z♣❧❛♥❡❄
✭❛✮ (0,4,0)
✭❜✮ (3,0,5)
✭❝✮ (3,4,0)
✭❞✮ (0,4,5)
✭❡✮ ◆♦♥❡ ♦❢ t❤❡s❡
✽✳ ❲❤✐❝❤ ♦❢ t❤❡ ❢♦❧❧♦✇✐♥❣ ✐s ❛ ✉♥✐t ✈❡❝t♦r ✐♥ t❤❡ ❞✐r❡❝t✐♦♥ ♦❢ t❤❡ ✈❡❝t♦r −2i+ 4j+k❄
✭❛✮ −4
√21i−1
√21j+2
√21k
✭❜✮ −4
√13i−1
√13j+2
√13k
✭❝✮ ◆♦♥❡ ♦❢ t❤❡s❡
✭❞✮ −2
√21i+4
√21j+1
√21k
✭❡✮ −2
√13i+4
√13j+1
√13k
✾✳ ●✐✈❡♥ t❤❡ r❛❞✐✉s ♦❢ ❝♦♥✈❡r❣❡♥❝❡ ♦❢ t❤❡ s❡r✐❡s
∞
X
n=1
(−1)n(x−3)n
2n·n4✐s 2✱ ✇❤❛t ✐s t❤❡ ✐♥t❡r✈❛❧ ♦❢ ❝♦♥✈❡r❣❡♥❝❡❄
✭❛✮ ◆♦♥❡ ♦❢ t❤❡s❡
✭❜✮ (−2,2]
✭❝✮ (−2,2)
✭❞✮ (1,5)
✭❡✮ [1,5]
✶✵✳ ❲❤✐❝❤ ♦❢ t❤❡ ❢♦❧❧♦✇✐♥❣ ✐s t❤❡ ▼❛❝❧❛✉r✐♥ ❙❡r✐❡s ❢♦r cos(x2)❄
✭❛✮
∞
X
n=0
(−1)nx4n+2
(2n+ 1)!
✭❜✮
∞
X
n=0
(−1)nx4n+2
(4n+ 2)!
✭❝✮ ◆♦♥❡ ♦❢ t❤❡s❡
✭❞✮
∞
X
n=0
(−1)nx4n
(4n)!
✭❡✮
∞
X
n=0
(−1)nx4n
(2n)!
Document Summary
|s sn| |an+1| f (x) dx. M n (n + 1)!|x a|n+1 r a x b . T s t r r p r f (x) = arctan x t r t a = 1 . T s (x 1) (x 1) . S t r s r s r ex t t t r r r r f (x) = x + e x t r t a = 0 x3. X n=0 ( 1)nxn r 1 < x < 1 t s r s r s r. T r s r s r x3 ln(1 + x) . X n=0 ( 1)nnxn+2 ( 1)nxn+3 ln n ( 1)nxn+4 n + 1 ( 1)nxn+4 n + 4. T s t q t t s r s t r s (0, 3, 5) ss s t r t t ( 6, 0, 5) . X2 + (y 3)2 + (z 5)2 = 45.
