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@ ss and, (uit) and, (2) and, (2) and, (wi) fous Tages and, , , , dx xv¥x?-1, , . Chain Rule: Chain rule is applied when the given function is the function of function ie.,, , évisa function of, then 2-2 48g, YY due ao, iyisa unchon of x, en de du de or dies da’ do ae, , . Parametric Form: Sometimes we come across the function when both x and y are expressed in, terms of another variable say t i.e., x = p(t) and y = w(t). This form of a function is called parametric, form and f is called the parameter., , . Rolle’s Theorem: If f(x) be a real valued function, defined in a closed interval [a, b] such that:, () it is continuous in closed interval [a, b]., (ii) it is differentiable in open interval (a, b)., (it) f(@) = f(b). Then there exists at least one value ¢ € (4, b) such that f'(c) = 0., . Lagrange’s Mean Value Theorem:, If f(x) is a real valued function defined in the closed interval [a, b] such that:, ()) itis continuous in the closed interval [a, b]., , (ii) itis differentiable in the open interval (a, b)., £O=F@), (b-a), , Then there exists at least one real value c € (2, b) such that f’(c) =, , . Some Standard Results:, , @ @), , , , “1 a>0,n€Q, ti ni, ae Ee OD, , (a) lim cosr = 1, x0, , , , (ii) Evaluation of limits of inverse Geigonromicitie functions:, , _ yo, sin” xX 1 ® tim lus ie a4, , , , (a) lim, x0, , (if) Evaluation of limits of expect art oe thmic functions:, a) lime*=1, @) x-0, , log | 1+2 |, , , , (c) lim, x0, , ee, , fb, , , , (e) lim”, x0, , = log, 4, , htinuity and Differentiability [123

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(iv) Limits at infinity: This method is applied when x > «., Procedure to solve the infinite limits:, (a) Write the given expression in the form of rational function., , (b) Divide the numerator and denominator by highest pawer of x., (c) Use the result lim a =0, where n >0., xe 2, , (d) Simplify and get the required result., , MULTIPLE CHOICE QUESTIONS, , Choose and write the correct option in the following questions., , 124, , me, , The function f: R > R given by f(x) =— |x-1| is ICBSE 2020 (65/2/1)], (a) continuous as well as differentiable at x= 1, , @® not continvas bat dilfentiailents =i, , (c) continuous but not differentiable at x = 1, , (d) neither continuous nor differentiable atx = 1, , The function f(x) = el"! is INCERT Exemplar], (a) continuous everywhere but not differentiable at x = 0, , (b) continuous and differentiable everywhere, , (c) not continuous at x = 0, , (d) none of these, , , , , , The function f(x) = [v], where [x] denotes the greatest integer function, is continuous at, @4 2 (1 (15, The number of points at which the function f(x) = <n is not continuous is, , INCERT Exemplar], (a) 1 (b) 2 (c) 3 (da) none of these, , sinx e, + if 0, The function f(x) =) * toes, heer is continuous at x = 0, then the value of k is, k , ifx=0, , INCERT Exemplar], , (a) 3 (2 (1 (a) 15, a, in, af 0, , Tievahieies WRiaes he Maaiionce Ny QS" ? " eontinnaamraey, =O k, ifx=0, (a) 8 (1 ()-1 ( nonest tess, The function f(x) = cot vis discontinuous on the set INCERT Exemplar], (a) {x=na:ine Z} (b) {x=2unin eZ}, (0) {x =(Qn+I Fn € zi (d) {x =e zi, Let f(x) =|sinx|. Then INCERT Exemplar], , (a) fis everywhere differentiable, (b) fis everywhere continuos but not differentiable at x =nz,n € Z, (c) fis everywhere continuous but not differentiable at x = (Qn +1)5,n eZ, , (d) none of these, , Mathematics—XII: Term-1

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10., , 11., , 13., , 14,, , 15., , 16., , 17., , 18., , 19., , 20., , 21,, , 1, , The function f(x) = we. is discontinuous at [CBSE 2020 (63/2/2)], a>, , (a) exactly one point (b) exactly two points, , (c) exactly three points (d) no point, , If f(x) =x*sin - where x #0, then the value of the function f at x = 0, so that the function is, J x J, , continuous at x = 0, is INCERT Exemplar], (a) 0 (b) -1 (c) 1 (d) None of these, , The function f(x) = |x| + |x-1| is, , (a) continuous at x = 0 as well as at x = 1. (b) continuous at x = 1 but not at x = 0., , (c) discontinuous at x=O0aswellasatx=1. (d) continuous at x = 0 but not atx = 1., , em, The functi a=, e function f(x) ee, , , , , , (a) discontinuous at only one point (b) discontinuous at exactly two points, , (c) discontinuous at exactly three points (d) none of these, , , , The set of points where the functions f given by f (x) = |x -3| cos x is differentiable is, (QR (0) R-(3} (0) (0,) (a) none of these, Differential coefficient of sec (tan™'x) w.rt. x is INCERT Exemplar], , ows One (© svi+8 © We, , If e= sin”, , , , , , , , , , ) and v= tan”(, , , , , , , , 14x, if, (a) a () x (a1, dy, If y= logy tan, then the value of a, 1, (a) 0 1 az (dé «, di, If y=ysinx+y, then 4 is equal to, cos x cos x sinx sinx, ) yor (ay © qoay @ a7, The function f(x) | + |x-2] is, (a) differentiable at x =0 and at x= 2 (b) differentiable at x = 0 but not at x = 2., (c) not differentiable at x = 0 and at x = 2. (d) none of these, The function given by /(x) = tan x is discontinuous on the set, (a) {xix=2nn,n © Z} (b) {xix=(n-Da,n © Z}, (c) [xix=nn,n © Z) (d) (eix=Qn+1)5,neZ], The derivative of tan x w.r.t. sin x is, (a) tan? (sec © a (a) sec!, (a) tan*x sec x ©) Sinx secx, , , , The function f(x) =, , f(3) should be defined a, (a) 1 (b) 3 (c) 5 (d) none of these, , is not defined for x = 3. In order to make f(x) continuous at x = 3,, , tinuity and Differentiability [125

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24,, , 25., , 26., , 27., , ai, , 31., , 126, , If f(x) = x-3 and g(x) = x +1, then which of the following can be a discontinuous function?, , , , 3, &@), , (a) flx) + g(x) (2) flo) . g(x) (c) Ax) - g(x) (a) f), The set of points where the function f given by |3x-2| sin x is differentiable is, , (a) R (B) (0, ~) (©) r-{F} (@) none of these, Let 70) =| C8 Mh * =O rece bet aenotes the greatest Integer’ <¥. IF lim f(A) etate'th, , et f(x) = [xl+a, x<o Wherelel denotes the greatest integer <x. jim fe exis' ena, is equal to, , (a) 1 (b) 3 (c) 0 (d) none of these, Let f(x) = AA OSE oy [x] denotes th, test integer <x th, , et f(x) = Le] Ciotly, ‘De<igradg WERE Il eenits thie greatest taleger Sa then, , (a) f(x) is continuous everywhere., , (b) f(x) is not continuous at x = 1 and x= 2, (0) f(x) is not differentiable at infinite points, (d) none of these, , f :[-2a,2a]— R is an odd function such that the left hand derivative at, , x=a is zeroand f(x) = f(2a—x)V x € (a, 2a), Then its left hand derivative at x = -a is, (a) 0 (b) a (c) 1 (d) does not exist., The function f(x) = |x-3|,x ER is, , (a) is continuous and differentiable everywhere., , (b) is not continuous but differentiable at x =3, , (c) is continuous but not differentiable at x= 3, , (d) none of these, , , , , , , , , , , If f(x) =x", then the value of, fo f-@ fa Erm., $n eee $, (a) 1 (0) 0 “2 (d) 2, ye. toe0 dy, If x=e%*" ,x >0, then —is, ax, 1 x 1-x, (a) = (0) Tip (c) a (d) none of these, 1) . 1-tanx x. x ‘ 5 : Dict faa, Let f(x) = age # a elo, z | If f(x) is continuous in [o 2 i then A . ) is, 1 1, @ > O -> (1 (ad) -1, The value of p and q for which the function, sin(y + Ix +sinx, x Ek, f= om +X=0 is continuous for all x ER, are, vx ae ,x>0, x, 1 3 5 Ze 3 1, (a) Pr 74> (b) Pr pda © pray (d) none of these, , Mathematics—XII: Term-1