MTH -26 Assigmet : Real Numbers, Sequeces. Fid the supremum of the set { m m+ : N, m Z}. 2. Let A be a o-empty subset of R ad α R. Show that α = supa if ad oly if α is ot a upper boud of A but α + is a upper boud of A for every N. 3. Let y (, ) ad (, ). Evaluate lim (2) y. 4. For a R, let = a ad + = 4 (2 + 3) for all N. Show that ( ) coverges if ad oly if a 3. Moreover, fid the limit of the sequece whe it coverges. 5. Show that the sequece ( ) defied by = 2 ad + = ( 7 3 + 2 ) for N satisfies the Cauchy criterio. 6. Let = + 2 + 3 +... + for N. Show that 2 2 sequece ( ) satisfy the Cauchy criterio? for every N. Does the 7. Let ( ) be defied by =, 2 = 2 ad +2 = + + 2 for. Show that ( ) coverges. Further, by observig that +2 + + 2 = + + 2, fid the limit of ( ). Assigmet 2 : Cotiuity, Eistece of miimum, Itermediate Value Property. Let [] deote the iteger part of the real umber. Suppose f() = g()h() where g() = [ 2 ] ad h() = si 2π. Discuss the cotiuity/discotiuity of f, g ad h at = 2 ad = 2. 2. Determie the poits of cotiuity for the fuctio f : R R defied by f() = { 2 if is ratioal + 3 if is irratioal. 3. Let f : R R be a cotiuous fuctio ad let, c R. Show that if f( ) > c, the there eists a δ > such that f() > c for all ( δ, + δ). 4. Let f : [, ] (, ) be a o-to fuctio. Show that f is ot cotiuous o [, ]. 5. Let f : [a, b] R ad for every [a, b] there eists y [a, b] such that f(y) < 2 f(). Fid if{ f() : [a, b]}. Show that f is ot cotiuous o [a, b]. 6. Let f : [, 2] R be a cotiuous fuctio ad f() = f(2). Prove that there eist real umbers, 2 [, 2] such that 2 = ad f( 2 ) = f( ). 7. Let p be a odd degree polyomial ad g : R R be a bouded cotiuous fuctio. Show that there eists R such that p( ) = g( ). Further show that the equatio 3 3 + 4 + si = + cos 2 has a solutio i R. + 2 Assigmet 3 : Derivatives, Maima ad Miima, Rolle s Theorem. Show that the fuctio f() = is differetiable at. More geerally, if f is cotiuous at, the g() = f() is differetiable at.
2. Prove that if f : R R is a eve fuctio (i.e., f( ) = f() for all R) ad has a derivative at every poit, the the derivative f is a odd fuctio (i.e.,f( ) = f() for all R). 3. Show that amog all triagles with give base ad the correspodig verte agle, the isosceles triagle has the maimum area. 4. Show that eactly two real values of satisfy the equatio 2 = si + cos. 5. Suppose f is cotiuous o [a, b], differetiable o (a, b) ad satisfies f 2 (a) f 2 (b) = a 2 b 2. The show that the equatio f ()f() = has at least oe root i (a, b). 6. Let f : (, ) R be twice differetiable. Suppose f( ) = for all N. Show that f () = f () =. 7. Let f : (, ) R be a twice differetiable fuctio such that f () >. Show that there eists N such that f( ). Assigmet 4 : Mea Value Theorem, Taylor s Theorem, Curve Sketchig. Show that y ( y) y ( y) if < y, N. 2. Let f : [, ] R be differetiable, f( 2 ) = 2 ad < α <. Suppose f () α for all [, ]. Show that f() < for all [, ]. 3. Let f : [a, b] R be cotiuous o [a, b] ad differetiable o (a, b). Suppose that f(a) = a ad f(b) = b. Show that there is c (a, b) such that f (c) =. Further, show that there are distict c, c 2 (a, b) such that f (c ) + f (c 2 ) = 2. 4. Usig Cauchy Mea Value Theorem, show that (a) 2 2! < cos for. (b) 3 3! < si for >. 5. Fid lim (6 ) 5 ad lim 5 +( + ). 6. Sketch the graphs of f() = 3 6 2 + 9 + ad f() = 2 2. 7. (a) Let f : [a, b] R be such that f () for all [a, b]. Suppose [a, b]. Show that for ay [a, b] f() f( ) + f ( )( ) i.e., the graph of f lies above the taget lie to the graph at (, f( )). (b) Show that cos y cos ( y) si for all, y [ π 2, 3π 2 ]. 8. Suppose f is a three times differetiable fuctio o [, ] such that f( ) =, f() = ad f () =. Usig Taylor s theorem show that f (c) 3 for some c (, ). Assigmet 5 : Series, Power Series, Taylor Series. Let f : [, ] R ad a = f( ) f( + ). Show that if f is cotiuous the = a coverges ad if f is differetiable ad f () < for all [, ] the = a coverges. 2. I each of the followig cases, discuss the covergece/divergece of the series = a where a equals:
(a) + (e) (b) cos (c) 2 ( ) (d) l 2 (f) log, (p > ) p (g) e (cos ) 2 si 3. Let a ad = that if = b be series of positive terms satisfyig a + a b + b = b coverges the a also coverges. Test the series = 4. Show that the series 4 + 5 2 + 3 4 3 + 5 4 + 5 4 5 + 5 6 + 7 4 7 + coverges. 5. Show that the series = ( ) si 6. Determie the values of for which the series 7. Show that cos = ( ) = (2)! 2, R. coverges but ot absolutely. = ( ) 2 2 3 = coverges. ( ) + (+) for all N. Show 2 e! for covergece. Assigmet 6: Itegratio. Usig Riema s criterio for the itegrability, show that f() = is itegrable o [, 2]. 2. If f ad g are cotiuous fuctios o [a, b] ad if g() for a b, the show the mea value theorem for itegrals : there eists c [a, b] such that (a) Show that there is o cotiuous fuctio f o [, ] such that N. b a f()g()d = f(c) b a f()d = g()d. for all (b) If f is cotiuuous o [a, b] the show that there eists c [a, b] such that b a f()d = f(c)(b a). (c) If f ad g are cotiuous o [a, b] ad b a f()d = b a g()d the show that there eists c [a, b] such that f(c) = g(c). 3. Let f : [, 2] R be a cotiuous fuctio such that 2 f()d = 2. Fid the value of 2 [f() + f(t)dt]d. u 4. Show that ( f(t)dt)du = f(u)( u)du, assumig f to be cotiuous. 5. Let f : [, ] R be a positive cotiuous fuctio. Show that lim (f( )f( 2 ) f( )) = e lf(). Assigmet 7: Improper Itegrals. Test the covergece/divergece of the followig improper itegrals: (a) (e) d log(+ ) (b) si(/) d (f) d log(+) e 2 d (g) (c) log si 2 d, 2. Determie all those values of p for which the improper itegral (d) (h) si(/)d. π/2 cot d. e d coverges. p
3. Show that the itegrals si d. 4. Show that log (+ 2 ) 2 d =. 5. Prove the followig statemets. si 2 d ad 2 si d coverge. Further, prove that si 2 2 d = (a) Let f be a icreasig fuctio o (,) ad the improper itegral f() eist. The i. f()d f( )+f( 2 ii. lim f( )+f( 2 (b) lim l +l 2 (c) lim! = e. )+ +f( ) f()d. )+ +f( ) = f()d. + +l =. Assigmet 8: Applicatios of Itegratio, Pappus Theorem. Sketch the graphs of r = cos(2θ) ad r = si(2θ). Also, fid their poits of itersectio. 2. A curved wedge is cut from a cylider of radius 3 by two plaes. Oe plae is perpedicular to the ais of the cylider. The secod plae crosses the first plae at a 45 agle at the ceter of the cylider. Fid the volume of the wedge. 3. Let C deote the circular disc of radius b cetered at (a, ) where < b < a. Fid the volume of the torus that is geerated by revolvig C aroud the y-ais usig (a) the Washer Method (b) the Shell Method. 4. Cosider the curve C defied by (t) = cos 3 (t), y(t) = si 3 t, t π 2. (a) Fid the legth of the curve. (b) Fid the area of the surface geerated by revolvig C about the -ais. (c) If (, y) is the cetroid of C the fid y. 5. A square is rotated about a ais lyig i the plae of the square, which itersects the square oly at oe of its vertices. For what positio of the ais, is the volume of the resultig solid of revolutio the largest? 6. Fid the cetroid of the semicircular arc ( r) 2 + y 2 = r 2, r > described i the first quadrat. If this arc is rotated about the lie y + m =, m >, determie the geerated surface area A ad show that A is maimum whe m = π/2. Assigmet 9: Vectors, Curves, Surfaces, Vector Fuctios. Cosider the plaes y + z =, + ay 2z + = ad 2 3y + z + b =, where a ad b are parameters. Determie the values of a ad b such that the three plaes (a) itersect at a sigle poit,
(b) itersect i a lie, (c) itersect (take two at a time) i three distict parallel lies. 2. Determie the equatio of a coe with verte (, a, ) geerated by a lie passig through the curve 2 = 2y, z = h. 3. The velocity of a particle movig i space is d dt c(t) = (cos t) i (si t) j + k. Fid the particle s positio as a fuctio of t if c() = 2 i + k. Also fid the agle betwee its positio vector ad the velocity vector. 4. Show that c(t) = si t 2 i + cos t 2 j + 5 k has costat magitude ad is orthogoal to its derivative. Is the velocity vector of costat magitude? 5. Fid the poit o the curve c(t) = (5 si t) i + (5 cos t) j + 2t k at a distace 26π uits alog the curve from (, 5, ) i the directio of icreasig arc legth. 6. Reparametrize the curves (a) c(t) = t2 2 i + t3 3 k, t 2, (b) c(t) = 2 cos t i + 2 si t j, t 2π i terms of arc legth. 7. Show that the parabola y = a 2, a has its largest curvature at its verte ad has o miimum curvature. Assigmet : Fuctios of several variables (Cotiuity ad Differetiability). Idetify the poits, if ay, where the followig fuctios fail to be cotiuous: (i) f(, y) = { y if y y if y < { y if y is ratioal (ii) f(, y) = y if y is irratioal. 2. Cosider the fuctio f : R 2 R defied by { 2 y 2 if (, y) (, ) f(, y) = 2 y 2 +( y) 2 if(, y) = (, ) Show that the fuctio satisfy the followig: [ ] [ ] (a) The iterated limits lim lim f(, y) ad lim lim f(, y) eist ad equals ; y y (b) lim f(, y) does ot eist; (,y) (,) (c) f(, y) is ot cotiuous at (, ); (d) the partial derivatives eist at (, ). 3. Let f(, y) = ( 2 +y 2 ) si 2 +y 2 if (, y) (, ) ad, otherwise. Show that f is differetiable at every poit of R 2 but the partial derivatives are ot cotiuous at (, ). 4. Let f(, y) = y for all (, y) R 2. Show that (a) f is differetiable at (,.)
(b) f (, y ) does ot eist if y. 5. Suppose f is a fuctio with f (, y) = f y (, y) = for all (, y). The show that f(, y) = c, a costat. Assigmet : Directioal derivatives, Maima, Miima, Lagrage Multipliers. Let f(, y) = 2 ( y y ). Is f cotiuous at (, )? Which directioal derivatives of f eist at (, )? Is f differetiable at (, )? 2. Let f(, y) = 2 y for (, y) (, ) ad f(, ) =. Show that the directioal derivative 2 +y 2 of f at (, ) i all directios eist but f is ot differetiable at (, ). 3. Let f(, y) = 2 e y + cos(y). Fid the directioal derivative of f at (, 2) i the directio ( 3 5, 4 5 ). 4. Fid the equatio of the surface geerated by the ormals to the surface + 2yz + yz 2 = at all poits o the z-ais. 5. Eamie the followig fuctios for local maima, local miima ad saddle poits: i) 4y 4 y 4 ii) 3 3y 2 6. Fid the absolute maima of f(, y) = y o the uit disc {(, y) : 2 + y 2 }.. Evaluate the followig itegrals: Assigmet 2 : Double Itegrals i) 2 y 2 dyd ii) π π si y y dyd iii) y 2 ep y ddy. 2. Evaluate ddy where R is the regio ( y) 2 ad y 2. R 3. Usig double itegral, fid the area eclosed by the curve r = si3θ give i polar cordiates. 4. Compute lim ep (2 +y 2) ddy, where a D(a) i) D(a) = {(, y) : 2 + y 2 a 2 } ad ii) D(a) = {(, y) : a, y a}. Hece prove that (i) e 2 d = π 2 (ii) 2 e 2 d = π 4. 5. Fid the volume of the solid which is commo to the cylider 2 + y 2 = ad 2 + z 2 =. Assigmet 3 : Triple Itegrals, Surface Itegrals, Lie itegrals. Evaluate the itegral W dzdyd + 2 +y 2 +z 2 ; where W is the ball 2 + y 2 + z 2.
2. What is the itegral of the fuctio 2 z take over the etire surface of a right circular cylider of height h which stads o the circle 2 + y 2 = a 2. What is the itegral of the give fuctio take throughout the volume of the cylider. 3. Fid the lie itegral of the vector field F (, y, z) = y i j + k alog the path c(t) = t (cos t, si t, 2π ), t 2π joiig (,, ) to (,, ). 4. Evaluate T dr, where C is the circle 2 + y 2 = ad T is the uit taget vector. C 5. Show that the itegral yzd+(z+)dy+ydz is idepedet of the path C joiig (,, ) C ad (2,, 4). Assigmet 4 : Gree s /Stokes /Gauss Theorems. Use Gree s Theorem to compute C (2 2 y 2 ) d + ( 2 + y 2 ) dy where C is the boudary of the regio {(, y) :, y & 2 + y 2 }. 2. Use Stokes Theorem to evaluate the lie itegral C y 3 d + 3 dy z 3 dz, where C is the itersectio of the cylider 2 + y 2 = ad the plae + y + z = ad the orietatio of C correspods to couterclockwise motio i the y-plae. 3. Let F = r r 3 where r = i + y j + z k ad let S be ay surface that surrouds the origi. Prove that S F. dσ = 4π. 4. Let D be the domai iside the cylider 2 +y 2 = cut off by the plaes z = ad z = +2. If F = ( 2 + ye z, y 2 + ze, z + e y ), use the divergece theorem to evaluate F dσ. D