95 上微積分甲統一教學一組期中考參考答案

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Aubrey Briggs
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1 95 上微積分甲統一教學一組期中考參考答案. (%) ()Given x lim 9, find p nd q ; x px + q (b)evlute (c)find x tn tdt x lim L ; x π n lim sin L. n i n n x iπ Ans ()p, q (b) L (c) L () lim px + q p + q p + q 9 x (b) L x ( x )( px + q +) ( x )( px + q +) px + q + lim lim lim lim x px + q x px + q 9 x p( x ) x p p p+ q p p q x tn tdt x tnx x sec x lim lim lim x x x x x x cos x sin x sin x lim lim lim x x cos x x x cos x x ( x) cos x n n π iπ π i πx cosπx π (c) L lim sin πlim sin ( ) π sin dx π dx n n n n n n i i

2 . (%) π Find the eqution of the line norml to the grph of sin( xy) x cos y t the point (, ). Ans The eqution of the line: sin( ) cos xy x y d d sin( xy) ( x cos y) dx dx dy dy xy y x x y x y dx dx dy xcos y y cos( xy) dy π dx x xy x y dx π cos( )( + ) cos sin cos( ) + sin (, ) π Norml Line: y ( x ) π. (5%) A squre pper mesures 5 cm by 5 cm. A pyrmid is creted by removing the four congruent shded tringles shown below, nd then folding long the dotted lines. The bse of the pyrmid is squre mesuring x cm by x cm. ()Let V(x) be the volume of this pyrmid. Find V(x). (b)find the mximum possible volume of the pyrmid nd the vlue of x for which it occurs. Ans () V(x), (b) Mximum volume t x. x x h ( ) ( ) 5 5 x V( x) x 5 5x dv ( x) 5x x 5 5x x or dx 6 5 5x 6 V() 5, V(). x hs the mximum volume 6 5 cm

3 . (5%) ()Stte the men vlue theorem (differentil form.) No proof is needed. Apply men vlue theorem to solve (b) nd (c). (b)evlute limsin(( x + ) ) sin( x ) L n Ans L. (c)prove the inequlities < 66 <. 5 ()Suppose f is continuous on [, b] nd differentible on (, b). Then there is t lest one point C in (, b) t which f ( b) f( ) f '( c)( b ). / (b)let f ( z) sin( z ), then f '( z) cos( z ) z Let x, bx+, the Men Vlue Theorem shows tht: c (, b) st.. f( b) f( ) f '( c)( b ) / / / / sin( x+ ) sin ( x ) cos( c ) c As x then c, / / cos( c ) nd limc c / / / / / / limsin( x+ ) sin( x ) limcos( c ) c x c By the Sndwich theorem, we hve / / limsin( x+ ) sin( x ) x (c) Let g x x ( ) x, g '( x) By Men Vlue Theorem, c (6, 66) st.. g'( c) nd g (x) is decresing on (, ) c 6 g'( c) g'(6) > < < c< 66 < ( 5/), > g'(( 5/) ) 66 >. 5 5

4 ( x ) 5. (%) Study the function y f( x) x + nd nswer the following questions. ()The domin of y f( x) is. () f '( x). () y f( x) hs criticl point(s) t x. () y f( x) is incresing on intervls. y f( x) is decresing on intervls. (5) f ''( x). (6) y f( x) is concve upwrd on intervls. y f( x) is concve down on intervls. (7)Find the ( x, y) -coordintes of the following points if exist. locl mximum point(s):. locl minimum point(s):. inflection point(s):. (8)Find the symptotes of the grph y f( x) if exist. Verticl symptotes(s):. Horizontl symptotes(s): s x. Slnted symptotes(s): s x. (9)Sketch the grph of y f( x) below. y x

$()R\{-} or (-,-) (-, ).$

5 ()R\{-} or (-,-) (-, ). ( x )( x+ ) () f '( x) ( x + ) () x, () y f( x) is incresing on intervls (-,-), (, ) y f( x) is decresing on intervls (-,-), (-,) 8 (5) f ''( x) ( x + ) (6) y f( x) is concve upwrd on intervls (-, ) y f( x) is concve downwrd on intervls (-,-) (7)locl mximum point(s):(-, -). locl minimum point(s):(, ). No inflection points! (8)Verticl symptotes(s): x- is verticl symptote. No Horizontl symptotes. Slnted symptotes(s):yx+5 s x ±. (9)

6 6. (%) Given two positive constnts, b, b> >, the region bounded by the curve x + ( y b) is revolved bout the x-xis to generte solid. Find the volume of this solid of revolution. (Certinly, you cn use without proof the formul of the re of circles.) Ans. The volume is. y b x + y b x + Disc Method: x. V π( b+ x ) π( b x ) dx π b x dx b x dx π b π Shell Method: x ± ( y b), b y b+ b+ V y π [ ( y b) ( ( y b) )] dy b b+ y y b dy b π ( ) b+ π ( ) ( ) + ( ) b y b y b b y b dy π u u b u du ( Let u y b) odd function + b u du π π b

7 7. (%) Let y f( x) x nd Lx ( ) be the lineriztion of f ( x ) t x6. Find Lx. ( ) Let A (6, f(6)), B (7, f(6)), C (7, f(7)), nd D (7, L(7)). Let u be the re of the region bounded by y f( x), AB, nd BC, v be the re of Δ ABD. Determine the reltion between u nd v. By using the pproximtion u v, one cn obtin nd estimte of the form 7 ( ) 7 K +, where K nd M re integers. Find K nd M. M Ans. L(x). u v (write <,, >). K. M. y f( x) x f '( x) x Lx ( ) f(6) + f'(6)( x 6) 6 + ( x 6) L (7) 6 + Since / f '( x) x < or f is concve down f ' is decresing we know u<v. 7 / / 7 u xdx 6 x 6 [(7) 6] 6 v 6 / (7) 6 + ( + 6) (5 + ) K 5, M

x 2 1 dx x 3 dx = ln(x) + 2e u du = 2e u + C = 2e x + C 2x dx = arcsin x + 1 x 1 x du = 2 u + C (t + 2) 50 dt x 2 4 dx

x 2 1 dx x 3 dx = ln(x) + 2e u du = 2e u + C = 2e x + C 2x dx = arcsin x + 1 x 1 x du = 2 u + C (t + 2) 50 dt x 2 4 dx . Compute the following indefinite integrls: ) sin(5 + )d b) c) d e d d) + d Solutions: ) After substituting u 5 +, we get: sin(5 + )d sin(u)du cos(u) + C cos(5 + ) + C b) We hve: d d ln() + + C c) Substitute

95 上微積分甲統一教學一組 期中考參考答案

95 上微積分甲統一教學一組期中考參考答案