EXERCISE I JEE MAIN. x continuous at x = 0 if a equals (A) 0 (B) 4 (C) 5 (D) 6 Sol. x 1 CONTINUITY & DIFFERENTIABILITY. Page # 20

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1 Page # 0 EXERCISE I 1. A function f() is defined as below cos(sin) cos f() =, 0 and f(0) = a, f() is continuous at = 0 if a equals (A) 0 (B) 4 (C) 5 (D) 6 JEE MAIN CONTINUITY & DIFFERENTIABILITY 4. Let f() = sgn () and g() = ( 5 + 6). The function f(g()) is discontinuous at (A) infinitely many points (B) eactly one point (C) eactly three points (D) no point. f() = (1 p) (1 p), 1 0 1,0 1 i s continuous in the interval [ 1, 1], then p is equal to: (A) 1 (B) 1/ (C) 1/ (D) If y = where t =, then the number t t 1 of points of discontinuities of y = f(), R is (A) 1 (B) (C) 3 (D) infinite 1 3. Let f() = [] when. Then (where [ * ] represents greatest integer function) (A) f() is continuous at = (B) f() is continuous at = 1 (C) f() is continuous at = 1 (D) f() is discontinuous at = 0 6. The equation tan + 5 = 0 has (A) no solution (B) at least one real solution in [0, /4] (C) two real solution in [0, /4] (D) None of these

2 CONTINUITY & DIFFERENTIABILITY 7. If f() = ( 1), then indicate the correct alternative(s) (A) f() is continuous but not differentiable at = 0 (B) f() is differentiable at = 0 (C) f() is not differentiable at = 0 (D) None of these Page # The function f() = sin 1 (cos ) is (A) discontinuous at = 0 (B) continuous at = 0 (C) differentiable at = 0 (D) none of these 11. Let f() be defined in [, ] by 1/ (3e 4), 0 8. If f() = 1/ e then f() is 0, 0 (A) continuous as well differentiable at = 0 (B) continuous but not differentiable at = 0 (C) neither differentiable at = 0 not continuous at = 0 (D) none of these ma ( f() = min ( 4, 4, 1 ) 1 ), 0 then f(), 0 (A) is continuous at all points (B) is not continuous at more than one point (C) is not differentiable only at one point (D) is not differentiable at more than one point. 9. If f() = be a real valued function then 1 (A) f() is continuous, but f(0) does not eist (B) f() is differentiable at = 0 (C) f() is not continuous at = 0 (D) f() is not differentiable at = 0 1. If f() is differentiable everywhere, then (A) f is differentiable everywhere (B) f is differentiable everywhere (C) f f is not differentiable at some point (D) f + f is differentiable everywhere

3 Page # 13. Let f( + y) = f() f(y) all and y. Suppose that f(3) = 3 and f(0) = 11 then f(3) is given by (A) (B) 44 (C) 8 (D) 33 CONTINUITY & DIFFERENTIABILITY 16. Let [] denote the integral part of R and g() = []. Let f() be any continuous function with f(0) = f(1) then the function h() = f(g()) (A) has finitely many discontinuities (B) is continuous on R (C) is discontinuous at some = c (D) is a constant function. 14. If f : R R be a differentiable function, such that f( + y) = f() + f(y) + 4y, y R, then (A) f (1) = f(0) + 1 (B) f(1) = f(0) 1 (C) f(0) = f(1) + (D) f(0) = f(1) t (1 sin ) The function f defined by f()= lim t (1 sin ) 1 is t (A) everywhere continuous (B) discontinuous at all integer values of (C) continuous at = 0 (D) none of these 15. Let f() = and g() = ma f(t),0 t,0 1. sin, 1 Then in the interval [0, ) (A) g() is everywhere continuous ecept at two points (B) g() is everywhere differentiable ecept at two points (C) g() is everywhere differentiable ecept at = 1 (D) none of these 18. If f() = 1 1 sin 1 1 sin 0,,, 0 0, then f() is 0 (A) continuous as well diff. at = 0 (B) continuous at = 0, but not diff. at = 0 (C) neither continuous at =0 nor diff. at =0 (D) none of these

4 CONTINUITY & DIFFERENTIABILITY 19. The functions defined by f() = ma {, ( 1), (1 )}, 0 1 (A) is differentiable for all (B) is differentiable for all ecept at one point (C) is differentiable for all ecept at two points (D) is not differentiable at more than two points. Let f : R R be a function such that Page # 3 y f() f(y) f, f(0) = 0 and f(0) = 3, then 3 3 f() (A) is differentiable in R (B) f() is continuous but not differentiable in R (C) f() is continuous in R (D) f() is bounded in R 0. Let f() = and ma{f(t)} for 0 t for 0 1 g() = then 3 for 1 (A) g() is continuous & derivable at = 1 (B) g() is continuous but not derivable at = 1 (C) g() is neither continuous nor derivable at = 1 (D) g() is derivable but not continuous at = 1 3. Suppose that f is a differentiable function with 1 the property that f( + y) = f() + f(y) + y and lim h0 h f(h) = 3 then (A) f is a linear function (B) f() = 3 + (C) f() = 3 + (D) none of these 1. Let f() be continuous at = 0 and f(0) = 4 then f() 3f() f(4) value of lim 0 is (A) 11 (B) (C) 1 (D) none of these 4. If a differentiable function f satisfies y 4 (f() f(y)) f 3 3, y R, find f() (A) 1/7 (B) /7 (C) 8/7 (D) 4/7

5 Page # 4 5. Let f : R R be a function defined by f() = Min { + 1, + 1}. Then which of the following is true? (A) f() 1 for all R (B) f() is not differentiable at = 1 (C) f() is differentiable everywhere (D) f() is not differentiable at = 0 CONTINUITY & DIFFERENTIABILITY 8. Let f( + y) = f() f(y) for all, y, where f(0) 0. If f(0) =, then f() is equal to (A) Ae (B) e (C) (D) None of these 6. The function f : R /{0} R given by 1 f() = e 1 defining f(0) as (A) (B) 1 (C) 0 (D) 1 9. A function f : R R satisfies the equation f( + y) = f(). f(y) for all, y R, f() 0. Suppose that the function is differentiable at = 0 and f(0) = then f() = (A) f() (B) f() (C) f() (D) f() 7. Function f() = ( cos ) where [0, 4] is not continuous at number of points (A) 3 (B) (C) 1 (D) Let f() = [cos + sin ], 0 < < where [] denotes the greatest integer less than or equal to. the number of points of discontinuity of f() is (A) 6 (B) 5 (C) 4 (D) 3

6 CONTINUITY & DIFFERENTIABILITY 1 ; The function f() =, is [ ] 0 ; 0 represents the greatest integer less than or equal to (A) continuous at = 1 (B) continuous at = 1 (C) continuous at = 0 (D) continuous at = Page # Let f() be a continuous function defined for 1 3. If f() takes rational values of for all and f() = 10 then the value of f(1.5) is (A) 7.5 (B) 10 (C) 8 (D) None of these sin(ln ) 0 3. The function f() = 1 0 (A) is continuous at = 0 (B) has removable discontinuity at = 0 (C) has jump discontinuity at = 0 (D) has discontinuity of II nd type at = If f() = p sin + q. e + r 3 and f() is differentiable at = 0, then (A) p = q = r = 0 (B) p = 0, q = 0, r R (C) q = 0, r = 0, p R (D) p + q = 0, r R The set of all point for which f() = [1 ] is continuous is (where [ * ] represents greatest integer function) (A) R (B) R [ 1, 0] (C) R ({} [ 1, 0]) (D) R {( 1, 0) n, n} 36. Let f() = sin, g() = [ + 1] and g(f()) = h() then h is (where [ * ] is the greatest integer function) (A) noneistent (B) 1 (C) 1 (D) None of these

7 Page # If f() = [tan ] then (where [ * ] denotes the greatest integer function) (A) Lim f() does not eist 0 (B) f() is continuous at = 0 (C) f() is non-differentiable at = 0 (D) f(0) = 1 CONTINUITY & DIFFERENTIABILITY 40. If f() = sgn (cos sin + 3) then f() (where sgn ( ) is the signum function) (A) is continuous over its domain (B) has a missing point discontinuity (C) has isolated point discontinuity (D) has irremovable discontinuity. 38. If f() = [] + { }, then (where, [ * ] and { * } denote the greatest integer and fractional part functions respectively) (A) f() is continuous at all integral points (B) f() is continuous and differentiable at = 0 (C) f() is discontinuous {1} (D) f() is differentiable. [] 41. Let g() = tan 1 cot 1, f() = {}, [ 1] h() = g (f () ) then which of the following holds good? (where { * } denotes fractional part and [ * ] denotes the integral part) (A) h is continuous at = 0 (B) h is discontinuous at = 0 (C) h(0 ) = / (D) h(0 + ) = / f(h) f(0) 39. If f is an even function such that Lim h0 h has some finite non-zero value, then (A) f is continuous and derivable at = 0 (B) f is continuous but not derivable at = 0 (C) f may be discontinuous at = 0 (D) None of these sin 4. Consider f() = Limit n n n for > 0, 1, sin f(1) = 0 then (A) f is continuous at = 1 (B) f has a finite discontinuity at = 1 (C) f has an infinite or oscillatory discontinuity at = 1. (D) f has a removable type of discontinuity at = 1. n n

8 CONTINUITY & DIFFERENTIABILITY [{ }]e {[ {}]} for Given f()= 1/ (e 1) sgn(sin ) then, f() 0 for 0 (where {} is the fractional part function; [] is the step up function and sgn() is the signum function of ) (A) is continuous at = 0 (B) is discontinuous at = 0 (C) has a removable discontinuity at = 0 (D) has an irremovable discontinuity at = Consider f() = Page # 7 1 1, 0 ; g() = cos {} 1 f(g()) for 0, < < 0, h() = 4 1 for 0 f() for 0 then, which of the following holds good (where { * } denotes fractional part function) (A) h is continuous at = 0 (B) h is discontinuous at = 0 (C) f(g()) is an even function (D) f() is an even function [] log(1 ) for Consider f() = ln(e {} ) the for 0 1 tan (where [ * ] & { * } are the greatest integer function & fractional part function respectively) (A) f(0) = ln f is continuous at = 0 (B) f(0) = f is continuous at = 0 (C) f(0) = e f is continuous at = 0 (D) f has an irremovable discontinuity at = Consider the function defined on [0, 1] R, sin cos f() = if 0 and f(0) = 0, then the function f() (A) has a removable discontinuity at = 0 (B) has a non removable finite discontinuity at = 0 (C) has a non removable infinite discontinuity at = 0 (D) is continuous at = 0

9 Page # Let f() = for 0 & f(0) = 1 then, sin (A) f() is conti. & diff. at = 0 (B) f() is continuous & not derivable at = 0 (C) f() is discont. & not diff. at = 0 (D) None of these CONTINUITY & DIFFERENTIABILITY 49. The function f() is defined as follows if 0 f() = if 0 1 then f() is 3 1 if 1 (A) derivable & cont. at = 0 (B) derivable at = 1 but not cont. at = 1 (C) neither derivable nor cont. at = 1 (D) not derivable at = 0 but cont. at = For what triplets of real number (a, b, c) with 1 a 0 the function f() = a b c otherwise differentiable for all real? (A) {(a, 1 a, a) a R, a 0} (B) {(a, 1 a, c) a, c R, a 0} (C) {(a, b) a, b, c R, a + b + c = 1} (D) {(a, 1 a, 0) a R, a 0} is {} sin{} for If f() = then 0 for 0 (where { * } denotes the fractional part function) (A) f is cont. & diff. at = 0 (B) f is cont. but not diff. at = 0 (C) f is cont. & diff. at = (D) None of these

10 CONTINUITY & DIFFERENTIABILITY Page # 9 Answer E I JEE MAIN 1. A. B 3. D 4. C 5. C 6. B 7. B 8. B 9. B 10. B 11. D 1. B 13. D 14. D 15. C 16. B 17. B 18. B 19. C 0. C 1. C. C 3. C 4. D 5. C 6. D 7. D 8. B 9. B 30. B 31. D 3. D 33. D 34. B 35. D 36. A 37. B 38. C 39. B 40. C 41. A 4. B 43. A 44. D 45. A 46. D 47. C 48. A 49. D 50. D

WebAssign hw2.3 (Homework)

WebAssign hw2.3 (Homework) Current Score : / 98 Due : Wednesday, May 31 2017 07:25 AM PDT Michael Lee Math261(Calculus I), section 1049, Spring 2017 Instructor: Michael Lee 1. /6 pointsscalc8 1.6.001.