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1 For mor important qustions visit : www4onocom CHAPTER 5 CONTINUITY AND DIFFERENTIATION POINTS TO REMEMBER A function f() is said to b continuous at = c iff lim f f c c i, lim f lim f f c c c f() is continuous in (a, b) iff it is continuous at c c a b f() is continuous in [a, b] iff (i) f() is continuous in (a, b) (ii) lim f f a, a (iii) lim f f b b Trigonomtric functions ar continuous in thir rspctiv domains Evr polnomial function is continuous on R If f () and g () ar two continuous functions and c R thn at = a (i) f () ± g () ar also continuous functions at = a (ii) g () f (), f () + c, cf (), f () ar also continuous at = a, (iii) f g is continuous at = a providd g(a) 0 f () is drivabl at = c in its domain iff 39 XII Maths

2 f f c f f c lim lim, and is finit c c c c Th valu of abov limit is dnotd b f (c) and is calld th drivativ of f() at = c d u v u dv v du d d d du dv v u d v v d u d d du If = f(u) and u = g(t) thn f u g t Chain Rul If = f(u), = g(u) thn, dt du dt d d du f u d du d g u d sin, cos d d d d d d tan, cot d d sc, cosc d d d, log d d f () = [] is discontinuous at all intgral points and continuous for all R Z Roll s thorm : If f () is continuous in [ a, b ], drivabl in (a, b) and f (a) = f (b) thn thr ists atlast on ral numbr c (a, b) such that f (c) = 0 XII Maths 40

3 Man Valu Thorm : If f () is continuous in [a, b] and drivabl in (a, b) thn thr ists atlast on ral numbr c (a, b) such that f b f a f c b a f () = log, ( > 0) is continuous function VERY SHORT ANSWER TYPE QUESTIONS ( MARK) For what valu of, f() = 7 is not drivabl Writ th st of points of continuit of g() = What is drivativ of 3 at = 4 What ar th points of discontinuit of f Writ th numbr of points of discontinuit of f() = [] in [3, 7] 3 if 6 Th function, f 4 if if R, find is a continuous function for all tan3, 0 7 For what valu of K, f sin K, 0 is continuous R 8 Writ drivativ of sin wrt cos 9 If f() = g() and g() = 6, g () = 3 find valu of f () 0 Writ th drivativ of th following functions : (i) log 3 (3 + 5) (ii) log (iii) 6 log, 4 XII Maths

4 (iv) sc cosc, (v) 7 sin (vi) log 5, > 0 SHORT ANSWER TYPE QUESTIONS (4 MARKS) Discuss th continuit of following functions at th indicatd points, 0 (i) f at 0, 0 sin, 0 (ii) 3 g at (iii) cos f 0 at (iv) f() = + at =, (v) f at 0 For what valu of k, f 0, 3 k 3 5, is continuous 3 For what valus of a and b a if f a b if b if is continuous at = XII Maths 4

5 4 Prov that f() = + is continuous at =, but not drivabl at = 5 For what valu of p, p f sin is drivabl at = 0 6 If tan tan, 0, find d 7 If sin tan thn? d 8 If = 5 + thn prov that 5 0 d 9 If a thn show that d 0 If a thn show that d If ( + ) m + n = m n thn prov that d Find th drivativ of tan wrt sin 3 Find th drivativ of log (sin ) wrt log a (cos ) 4 If + + = m n, thn find th valu of d 5 If = a cos 3, = a sin 3 thn find d d 43 XII Maths

6 6 If = a t (sint cos t) = a t (sint + cost) thn show that at is d 4 7 If sin thn find d thn find d log 8 If log 9 Diffrntiat wrt d 30 Find, if cos cos 3 If sin sin tan whr sin sin find d 3 If sin log a thn show that ( ) a = 0 33 Diffrntiat log log, w r t 34 If sin = sin (a + ) thn show that d sin a sina 35 If = sin, find d d in trms of 36 If a d thn show that b d, 4 b 3 a cos 37 If a d,, show that a 0 d d 38 If 3 = 3a 3 thn prov that d a 5 d 39 Vrif Roll's thorm for th function, = + in th intrval [a, b] whr a =, b = 40 Vrif Man Valu Thorm for th function, f() = in [, 4] XII Maths 44

7 ANSWERS = 7/ R 3 4 = 6, 7 5 Points of discontinuit of f() ar 4, 5, 6, 7 i four points Not : At = 3, f() = [] is continuous bcaus lim f 3 f k 4 8 cot (i) log3 3 5 log (ii) log (iii) 6 ( ) 5 (iv) 0 (v) 7 7 (vi) log 5 log (i) Discontinuous (ii) Discontinuous (iii) Continuous (iv) continuous (v) Discontinuous k = 3 a = 0, b = 5 p > cot log a 4 log log d log 45 XII Maths

8 5 d 4 cosc sc d 3a 7 d log log log log log log log log d log tan logcos d tan logcos d Hint : sin cos for, 33 log log log log, 35 sc tan For mor important qustions visit : www4onocom XII Maths 46

Calculus II (MAC )

Calculus II (MAC ) Calculus II (MAC232-2) Tst 2 (25/6/25) Nam (PRINT): Plas show your work. An answr with no work rcivs no crdit. You may us th back of a pag if you nd mor spac for a problm. You may not us any calculators.