(1)93 ç,ç‚B 2 ‚255æ 1

93 ç,ç‚B 2 ‚255æ 1. °-”M, à”Ì.æÊ, ~zp.

(a) (5 }) lim

x→0

cos x − 1 x (b) (6 }) lim

x→−∞

px²− 2x + x

2. (a) (8 }) 5?(y³+ y²x + x + y = 0. t°^dy_dx £_dx^d2y2 Êõ(0, 0) íM.

Solution.

3y²y⁰+ 2yy⁰x + y²+ 1 + y⁰= 0 ∴y⁰ = −1

6y⁰²+3y²y⁰⁰+2y⁰²x+2yy⁰+2xyy⁰⁰+2yy⁰+y⁰⁰= 0 ∴6+y⁰⁰= 0, y⁰⁰= −6 (b) (7 }) ÙlÞÊ×ò4 t 5Ú(C9 t T[Ù˚, vÙ˚J¹₃ m/sec 5

ì§ò,¯. çÙ˚®ƒËÞ16 t v, ~½Ú(C5Ù˚ JÖ×

§òs?

Solution.

y − 4 = 9 tan θ y⁰= 9 sec²dθ

dt 1

3 = 9 5 3

²

·dθ dt

∴θ⁰ = 1

75 rad/sec

¢ x = 4 cot θ, dx

dt = −4 csc²θdθ

dt = −4 5 4

²

· 1

75= −25 4 × 1

75 = − 1

12 m/sec.

3. (15 }) túf (x) = _x^2x₊₉ 5Ç$. ~nú˚4. ]Ó. ]Á. ”M. p4. ¥

õ. Úª(.

Solution. Jƒb, ¬Ÿõ. y = 0 Ѯڪ(.

f⁰(x) = 9 − x²

(x²+ 9)² = 0, x = ±3.

local min. is f (−3) = −¹6, local max. is f (3) = ¹₆. f⁰⁰(x) =2x(x²− 27)

(x²+ 9)³ = 0, x = 0, ±3√ 3.

4. (10 }) t°_dx^d Rtanx x

1 u⁴+1du.

5. (a) (5 }) „p lim

n→∞

n

X

i=1

sin iπ 2n



· π 2n = 1

1

Solution. By definition of integral,

n

X

i=1

sin iπ 2n

 π 2n−→

Z ^π2

0

sin xdx = [− cos x]

π 2

0 = 1 as n → ∞.

(b) (6 }) ° lim

n→∞

n

X

i=1

s 1 − i

n

²1 n Solution. Ñ√

1 − x² Ê [0, 1] 5 Riemann sum, ]¤”ÌÑ π 4. 6. (a) (4 }) t„ç0 < x < ^π₂ v, sin x < x.

(b) (7 })f (x) = ^tan_x^x, „pJ0 < x1< x2< ^π₂ v, f (x1) < f (x2).

Solution. f⁰(x) = x sec²x − tan x

x² .

k„: x sec²x > tan x, ¹ x > sin x cos x = ¹₂sin 2x.

I g(x) = x −¹2sin 2x, g(0) = 0, g⁰(x) = 1 − cos 2x > 0.

7. °- } (a) (6 })R x⁷

√x⁴+1dx.

(b) (6 }) Z

−π

πx³cos x 1 + x⁶ dx.

8. (15 }) t°qQkÀPÆí|×8úi$Þ .

2