Assigmet : Real Numbers, Sequeces. 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. 2. Let y (, ) ad x (, ). Evaluate lim (2) y x. 3. For a R, let x = a ad x + = 4 (x2 + 3) for all N. Show that (x ) coverges if ad oly if a 3. Moreover, fid the limit of the sequece whe it coverges. 4. Show that the sequece (x ) defied by x = 2 ad x + = ( 7 x 3 + 2 ) for N satisfies the Cauchy criterio. 5. Let x = + 2 + 3 +... + for N. Show that x 2 x 2 sequece (x ) satisfy the Cauchy criterio? for every N. Does the 6. Let (x ) be defied by x =, x 2 = 2 ad x +2 = x+x + 2 for. Show that (x ) coverges. Further, by observig that x +2 + x + 2 = x + + x 2, fid the limit of (x ). Assigmet 2 : Cotiuity, Existece of miimum, Itermediate Value Property. Determie the poits of cotiuity for the fuctio f : R R defied by f(x) = { 2x if x is ratioal x + 3 if x is irratioal. 2. Let f : R R be a cotiuous fuctio ad let x, c R. Show that if f(x ) > c, the there exists a δ > such that f(x) > c for all x (x δ, x + δ). 3. Let f : [a, b] R ad for every x [a, b] there exists y [a, b] such that f(y) < 2 f(x). What is the miimum value of the fuctio x f(x). Show that f is ot cotiuous o [a, b]. 4. Let f : [, 2] R be a cotiuous fuctio ad f() = f(2). Prove that there exist real umbers x, x 2 [, 2] such that x 2 x = ad f(x 2 ) = f(x ). 5. Let p be a odd degree polyomial ad g : R R be a bouded cotiuous fuctio. Show that there exists x R such that p(x ) = g(x ). Further show that the equatio x 3 3x + 4x + six = + cos 2 x has a solutio i R. +x 2 Assigmet 3 : Derivatives, Maxima ad Miima, Rolle s Theorem. Show that the fuctio f(x) = x x is differetiable at. More geerally, if f is cotiuous at, the g(x) = xf(x) is differetiable at. 2. Prove that if f : R R is a eve fuctio (i.e., f( x) = f(x) for all x R) ad has a derivative at every poit, the the derivative f is a odd fuctio (i.e.,f( x) = f(x) for all x R). 3. Show that amog all triagles with give base ad the correspodig vertex agle, the isosceles triagle has the maximum area. 4. Show that exactly two real values of x satisfy the equatio x 2 = xsix + cosx.
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 (x)f(x) = x has at least oe root i (a, b). 6. Let f : (, ) R be twice differetiable. Suppose f( ) = for all N. Show that f () = f () =. Assigmet 4 : Mea Value Theorem, Taylor s Theorem, Curve Sketchig. Show that y (x y) x y x (x y) if < y x, N. 2. 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. 3. Usig Cauchy Mea Value Theorem, show that (a) x2 2! < cos x for x. (b) x x3 3! < si x for x >. 4. Fid lim (6 x) x 5 ad lim x 5 x +( + x )x. 5. Sketch the graphs of f(x) = x 3 6x 2 + 9x + ad f(x) = x2 x 2. 6. 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 (x) < for all x [, ] the = a coverges. 2. I each of the followig cases, discuss the covergece/divergece of the series = a where a equals: (a) + (b) cos ( ) (c) 2 ( ) (d) + (+) l (e) 2 (f) log, (p > ) p 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. Determie the values of x for which the series 5. Show that cos x = ( ) = (2)! x2, x R. = (x ) 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(x) = x is itegrable o [, 2].
2. If f ad g are cotiuous fuctios o [a, b] ad if g(x) for a x b, the show the mea value theorem for itegrals : there exists c [a, b] such that b a f(x)g(x)dx = f(c) Usig this result show that there is o cotiuous fuctio f o [, ] such that for all N. b a g(x)dx. x f(x)dx = 3. Let f : [, 2] R be a cotiuous fuctio such that 2 f(x)dx = 2. Fid the value of 2 [xf(x) + x f(t)dt]dx. x u x 4. Show that ( f(t)dt)du = f(u)(x 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(x). Assigmet 7: Improper Itegrals. Test the covergece/divergece of the followig improper itegrals: (a) (e) dx log(+ x) (b) si(/x) x dx (f) 2. Show that the itegrals si x x dx. 3. Show that x log x (+x 2 ) 2 dx =. 4. Prove the followig statemets. dx x log(+x) e x2 dx si 2 xdx ad x 2 (g) (c) log x x si x 2 dx, (d) (h) si(/x)dx. π/2 cot xdx. si x x dx coverge. Further, prove that si 2 x x 2 dx = (a) Let f be a icreasig fuctio o (,) ad the improper itegral f(x) exist. The i. f(x)dx f( )+f( 2 ii. lim f( )+f( 2 (b) lim l +l 2 (c) lim! = e. )+ +f( ) f(x)dx. )+ +f( ) = f(x)dx. + +l =. Assigmet 8: Applicatios of Itegratio, Pappus Theorem. Sketch the graphs 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 axis 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 f be a cotiuous fuctio o R. A solid is geerated by rotatig about the X-axis, the regio bouded by the curve y = f(x), the X-axis ad the lies x = a ad x = b. For fixed a, the volume of this solid betwee a ad b is b 3 + b 2 ab a 3 for each b > a. Fid f(x). 4. A square is rotated about a axis lyig i the plae of the square, which itersects the square oly at oe of its vertices. For what positio of the axis, is the volume of the resultig solid of revolutio the largest? 5. Fid the cetroid of the semicircular arc (x r) 2 + y 2 = r 2, r > described i the first quadrat. If this arc is rotated about the lie y + mx =, m >, determie the geerated surface area A ad show that A is maximum whe m = π/2. Assigmet 9: Vectors, Curves, Surfaces, Vector Fuctios. Cosider the plaes x y + z =, x + ay 2z + = ad 2x 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 vertex (, a, ) geerated by a lie passig through the curve x 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 legth 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 the origi 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 = ax 2, a has its largest curvature at its vertex 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(x, y) = { xy if xy xy if xy < { xy if xy is ratioal (ii) f(x, y) = xy if xy is irratioal.
2. Cosider the fuctio f : R 2 R defied by { x 2 y 2 if (x, y) (, ) f(x, y) = x 2 y 2 +(x y) 2 if(x, y) = (, ) Show that the fuctio satisfy the followig: [ ] [ ] (a) The iterated limits lim lim f(x, y) ad lim lim f(x, y) exist ad equals ; x y y x (b) lim f(x, y) does ot exist; (x,y) (,) (c) f(x, y) is ot cotiuous at (, ); (d) the partial derivatives exist at (, ). 3. Let f(x, y) = (x 2 +y 2 ) si x 2 +y 2 if (x, y) (, ) ad, otherwise. Show that f is differetiable at every poit of R 2 but the partial derivatives are ot cotiuous at (, ). 4. Suppose f is a fuctio with f x (x, y) = f y (x, y) = for all (x, y). The show that f(x, y) = c, a costat. Assigmet : Directioal derivatives, Maxima, Miima, Lagrage Multipliers. Let f(x, y) = 2( x y ) x y. Is f cotiuous at (, )? Which directioal derivatives of f exist at (, )? Is f differetiable at (, )? 2. Fid the equatio of the surface geerated by the ormals to the surface x + 2yz + xyz 2 = at all poits o the z-axis. 3. Examie the followig fuctios for local maxima, local miima ad saddle poits: i) 4xy x 4 y 4 ii) x 3 3xy 4. Fid the absolute maxima of f(x, y) = xy o the uit disc {(x, y) : x 2 + y 2 }. 5. Miimize the quatity x 2 +y 2 +z 2 subject to the costraits x+2y+3z = 6 ad x+3y+9z = 9.. Evaluate the followig itegrals: Assigmet 2 : Double Itegrals i) x 2 y 2 dydx ii) π π x si y y dydx iii) y x 2 exp xy dxdy. 2. Evaluate xdxdy where R is the regio x( y) 2 ad xy 2. R 3. Usig double itegral, fid the area eclosed by the curve r = si3θ give i polar cordiates. 4. Compute lim exp (x2 +y 2) dxdy, where a D(a) i) D(a) = {(x, y) : x 2 + y 2 a 2 } ad ii) D(a) = {(x, y) : x a, y a}.
Hece prove that e x2 dx = π 2. Assigmet 3 : Triple Itegrals, Surface Itegrals, Lie itegrals. Evaluate the itegral W dzdydx +x 2 +y 2 +z 2 ; where W is the ball x2 + y 2 + z 2. 2. What is the itegral of the fuctio x 2 z take over the etire surface of a right circular cylider of height h which stads o the circle x 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 (x, y, z) = y i x 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 x 2 + y 2 = ad T is the uit taget vector. C 5. Show that the itegral yzdx+(xz+)dy+xydz 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 (2x 2 y 2 ) dx + (x 2 + y 2 ) dy where C is the boudary of the regio {(x, y) : x, y & x 2 + y 2 }. 2. Use Stokes Theorem to evaluate the lie itegral C y 3 dx + x 3 dy z 3 dz, where C is the itersectio of the cylider x 2 + y 2 = ad the plae x + y + z = ad the orietatio of C correspods to couterclockwise motio i the xy-plae. 3. Let F = r r 3 where r = x 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 x 2 +y 2 = cut off by the plaes z = ad z = x+2. If F = (x 2 + ye z, y 2 + ze x, z + xe y ), use the divergece theorem to evaluate F dσ. D