MATH4 Calculus II (8 Spring) MATH 4 Tuorial Noes Tuorial Noes (Phyllis LIANG) IA: Phyllis LIANG Email: masliang@us.hk Homepage: hps://masliang.people.us.hk Office: Room 3 (Lif/Lif 3) Phone number: 3587453 Office Hours: www.mah.us.hk/~suppor Tuorial Sessions: TC Tue 8:-8:5 6573 TC Fri :3-: 5 Topics covered in uorial :. Review of definie inegrals. Physical Applicaion 3. Area beween curves. Review of definie inegrals Wha you need o know: Basic differeiaio & iegraio formulas Fudaeal Theore of Calculus Inegraion by Subsiuion Basic Differeniaion Formulas Addiion and Subracion Derivaive of a Consan k [ f ( ) g( )] f ( ) g p p ( ) Power Rule ( ) p Consan Muliplicaion [ kf( )] kf ( ) u Derivaive of a Square Roo u Produc Rule [ f ( ) g( )] f ( ) g( ) g( ) f ( ) u Eponenial Rule ( e ) e f ( ) g( ) f ( ) f ( ) g( ) Quoien Rule g( ) [ g( )] Eponenial Rule wih arbirary base ( a ) a ln a dy dy du Chain Rule d du d Logarihmic Rule (ln ), Trigonomeric Rules Inverse Trigonomeric Rules (sin ) cos (cos ) sin (an ) sec (co ) csc (sec ) sec an (csc ) csc co (sin ) (an ) (cos ) (co ) Page of 7
MATH4 Calculus II (8 Spring) Tuorial Noes (Phyllis LIANG) Addiion and Subracion [ f ( ) g( )] d f ( ) d g( ) d Consan Muliplicaion kf ( ) d k f ( ) d p p Power Rule: d C p Eponenial Rule: e d e C Eponenial Rule wih arbirary base: a a d C ln a Reciprocal Rule: d ln C Logarihmic Rule: ln d ln C Basic Inegraion Formulas Properies of definie inegral = + = Trigonomeric Rules: sin d cos C cos d sin C an d ln sec C co d ln sin C secd ln sec an C ln an C 4 cscd ln csc co C ln an C The version of Fundamenal Theorem of Calculus (FTC) version of FTC: = F F Where F is an aniderivaive of. Noe: N A = = area above -ais (Region and 3) area below -ais (Region ) Eample. Evaluae π π i Eample. Evaluae Page of 7
MATH4 Calculus II (8 Spring) The Subsiuion Rule: = Indefinie inegral: () = () = Tuorial Noes (Phyllis LIANG) Definie inegral: () = = () = Useful ools:. () =. + = () 3. () = () Eample.3 Evaluae +. Remark:. Do forge o chage he upper and lower limi.. To apply subsiuion rule for definie inegral, we need o make sure ha is a one-o-one funcion in he inerval,. Eample.4 Evaluae e. +e Eample.5 Evaluae. Eample.6 Evaluae +. Eample.7 Evaluae +. Eample.8 Evaluae +. Page 3 of 7
MATH4 Calculus II (8 Spring) Eample.9 Evaluae r +r. Tuorial Noes (Phyllis LIANG). Physical Applicaion Wha you need o know: Posiio & Velociy & Acceleraion & Displacemen Rae of chage of a quaniy & Ne change of a quaniy Disace & Speed We have he following erms abou a paricle moving along a line: Posiion: Velociy: v Acceleraion: = I = = = = I = = = Q : A quaniy changing over ime Q : The rae of change of Q Q Q Q Q = Q = = Q = + Q [,] : Posiion = : The rae of change of Dipa = = = = + [,] : Veloviy = : The rae of change of v = = = = + [,] Page 4 of 7
MATH4 Calculus II (8 Spring) Disance & Speed: The disance ravelled by he paricle from = o = is: = = Where v is he speed of he paricle a ime. Tuorial Noes (Phyllis LIANG) Eample. A paricle moves along a line so ha is velociy a ime is v = 6 (measured in meers per second). (a) Find he displacemen of he paricle during he ime period. (b) Find he disance raveled by his paricle during he ime period. Eample. Find he posiion and velociy of an objec moving along a sraigh line wih he given acceleraion, iniial velociy, and iniial posiion. a = +, v =, = Page 5 of 7
MATH4 Calculus II (8 Spring) 3. Area beween curves Tuorial Noes (Phyllis LIANG) Wha you need o know: How o calculae he 3 ypes of area beween curves (a) Type : The area A of he region bounded by he curves =, =, and he lines =, =, where and are coninuous and for all in [, ] is A = [ ] Eample. Find he area of he region enclosed by he parabolas = and =. (b) Type : When for some, and for oher values of, The area encloses by he curves =, =, and lines = and = is A = Eample.3 Find he area of he region bounded by he curves = i, =, =, and = π. Page 6 of 7
MATH4 Calculus II (8 Spring) (c) Type 3: Tuorial Noes (Phyllis LIANG) Some regions are bes reaed by regarding as a funcion of. If a region bounded by he curves =, =, and he lines =, =, where and are coninuous and for all in [, ], hen is area is A = [ ] Eample.4 Find he area enclosed by he line = and he parabola = + 6. Page 7 of 7
Era eercises Zea CHAN: hps://mazea.people.us.hk Inegraion by Subsiuion dy du dy Inegraion by Subsiuion: d du du d du. Evaluae he following inegrals by a suiable change of variable. e (a) d e (b) d e e ln (c) d (d) e ln ( ) d. Evaluae he following inegrals by spliing he fracion. (a) d 3 (b) d 4 3. Evaluae he following inegrals. d (a) / 3/ 9 d (b) 4 (c) d 4. Given ha sin and cos when (a) 5 4cos d (b) sin d d (c) 3 sin cos an, evaluae he following inegrals. Page of 3
Era eercises Zea CHAN: hps://mazea.people.us.hk Physical applicaion b As displacemen s() is he primiive funcion of velociy v(), v( ) d s( b) s( a). Also, Similarly, s( ) s() v( ) d. v( ) v() a( ) d, where a() is he acceleraion. In general, Q( ) Q() Q( ) d for any quaniy Q(). 5. Find he displacemen and he disance raveled over he given inerval. (a) v() = sin ; π (b) v() = + 5 4; 5 6. The velociy in miles/hour of a hiker walking along a sraigh rail is given by v() = 3 sin (π/) for cos 4. Given ha s() = and sin, find he hiker s posiion a = 3. 7. Find he velociy funcion and posiion funcion of an objec moving along a sraigh line wih he given acceleraion funcion, iniial velociy and iniial posiion. (a) a() = e ; v() = 6, s() = 4 (b) a() = ; v() =, s() = ( ) (c) a() = ; v() =, s() = ( ) 8. Saring wih an iniial value of P() = 55, he populaion of a dog communiy grows a a rae of P'() = /5 dogs/monh for. Find he populaion P() for. 9. The populaion of a communiy of foes is observed o flucuae on a -year cycle. A =, he populaion was measured o be 35 foes. The growh rae in foes/year was observed o be P'() = 5 + sin (π/5). Find he populaion P() a any ime.. The populaion of an endangered species changes a a rae of P'() = 3 individuals/year. If he iniial populaion of he species is 3 individuals, when will he species become einc?. A -L conainer is empy a =. A he same ime, waer begins flowing ino i a a rae of Q'() = 3 liers/minue. (a) Find he amoun of waer in he conainer Q() a any ime. (b) When will he conainer be full? a Page of 3
Era eercises Zea CHAN: hps://mazea.people.us.hk Area beween Curves b Area beween wo curves on [a, b] = ( y higher ylower ) d a. Find he area enclosed by he curves y = 3 and y =.. Consider he parabolas y = 4 and = 4y, where is a real number. (a) Epress heir poins of inersecion in erms of. (b) Find he area A bounded by hese parabolas. (c) Find he rae of change of A wih respec o when = 3. 3. Consider he curves C : y = ( ) and C : y = 3( ). (a) Find heir poins of inersecion. (b) Find he urning poins of C. (c) Skech C and C on he same graph. (d) Find he area bounded by he wo curves. 4. Using he subsiuion = π/ u, find he area under he following curves on [, π/]. (a) y = a sin + b cos, where a and b are posiive consans sin (b) y = sin cos 5. Le m and n be consans. (a) Prove ha a m a n n m ( a ) d ( a ) d. 3 (b) Find he area under he curve y = 8 on [, 8]. n 6. Le I n =sec d. (a) Find he derivaive of an sec n. n (b) Prove ha ( n ) I n an sec ( n ) I n. (c) Find he area under he curve y = sec 6 on [, π/4]. 7. Find he area of he loop of he curve y = ( ) ( + 3). 8. Find he area enclosed by he circle + y = 5 and he parabola y =. 9. Find he area bounded by he curve y + 3 = and he sraigh line y + + =.. If he area bounded by he curve y = 9 3, he -ais and he lines = and = k is 8, find he possible value(s) of k. Page 3 of 3