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Limits, Continuity and Differentiability

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  1. 151.Let R be such that the function [f(x)=cases^-1(1-x^2)^-1(1-x)x-x^3, & x0, & x=0cases] is c…
  2. 152.Let f:S S, where S=(0,), be a twice differentiable function such that f(x+1)=xf(x).
  3. 153.Let f:[0,3]R be defined by f(x)=x-[x],1+[x]-x where [x] is the greatest integer x.
  4. 154.Let f:(-4,4)R be defined as [f(x)=cases(1+| x|)^(3a)/(| x|), & -4<x<0, & x=0^4x/2x, & 0<x<…
  5. 155.Let f:[0,)[0,3] be defined by f(x)=cases t:0 t x, & 0 x<2+ x, & xcases.
  6. 156.Let f:(a,b)R be a twice differentiable function such that f(x)=_a^xg(t),dt for a different…
  7. 157.Let f:RR be defined as f(x)=cases|x^2-5x+6|(5x-x^2-6), & x<2, & x=2e^(x-2)x-[x], & x>2case…
  8. 158.Let f:[0,)[0,) be defined as f(x)=_0^x[y],dy, where [x] is the greatest integer x.
  9. 159.If f(x)=cases_0^x(5+|1-t|),dt, & x>25x+1, & x2cases, then:
  10. 160.Consider the function f(x)=casesP(x)(x-2), & x27, & x=2cases, where P(x) is a polynomial s…
  11. 161.Let f:RR be defined as f(x)=casesx^3(1-2x)^2_e(1+2xe^-2x(1-xe^-x)^2), & x0, & x=0cases.
  12. 162.Let f:RR be defined by f(x)=cases3(1-|x|2), & |x|20, & |x|>2cases.
  13. 163.Let a function f:RR be defined as f(x)=cases x-e^x, & x0+[-x], & 0<x<12x-b, & x1cases, whe…
  14. 164.Let a function g:[0,4]R be defined as g(x)=cases_0 t x(t^3-6t^2+9t-3), & 0 x34-x, & 3<x4ca…
  15. 165.The function f(x)=(x^2-2x-3) e^|9x^2-12x+4| is not differentiable at exactly
  16. 166.If the function f(x)=cases1x_e(1+x/a1-x/b), & x<0, & x=02x-2x-1x^2+1-1, & x>0cases is cont…
  17. 167.Let a,bR, b0.
  18. 168.Let [t] denote the greatest integer t.
  19. 169.Let x^k+y^k=a^k, (a,k>0) and dydx+( yx)^1/3=0, then k is:
  20. 170.The value of c in the Lagrange's Mean Value Theorem for the function f(x)=x^3-4x^2+8x+11, …
  21. 171.Let [t] denote the greatest integer t and _x0x[4x]=A.
  22. 172.If the function f defined on (-13,13) by f(x)=cases1x_e(1+3x1-2x), & x0, & x=0cases is con…
  23. 173.If c is a point at which Rolle's theorem holds for the function f(x)=_e(x^2+7x) in the int…
  24. 174.If f(x)=cases(a+2)x+ xx, & x<0, & x=0(x+3x^2)^1/3-x^1/3x^4/3, & x>0cases is continuous at …
  25. 175.Let f and g be differentiable functions on R such that f g is the identity function.
  26. 176.Let S be the set of all functions f:[0,1]R, which are continuous on [0,1] and differentiab…
  27. 177.Let the function f:[-7,0]R be continuous on [-7,0] and differentiable on (-7,0).
  28. 178.Let S be the set of points where the function f(x)=|2-|x-3||, xR, is not differentiable.
  29. 179.If x=2-2 and y=2-2, [0,2], then d^2ydx^2 at = is:
  30. 180.If y()=2(+)1+^2+1^2, (34,), then dyd at =56 is:
  31. 181.Let y=y(x) be a function of x satisfying y√1-x^2=k-x√1-y^2 where k is a constant and y(12)…
  32. 182.For all twice differentiable functions f:RR, with f(0)=f(1)=f'(0)=0:
  33. 183.If y^2+_e(^2x)=y, x(-2,2), then:
  34. 184.Let f(x)=x[x^2], for -10<x<10, where [t] denotes the greatest integer function.
  35. 185.If a function f(x) defined by f(x)=casesae^x+be^-x, & -1 x<1^2, & 1 x3^2+2cx, & 3<x4cases …
  36. 186.Let f:RR be a function defined by f(x)=x,x^2.
  37. 187.If the function f(x)=casesk_1(x-)^2-1, & x_2 x, & x>cases is twice differentiable, then th…
  38. 188.Let f be a twice differentiable function on (1,6).
  39. 189.Suppose a differentiable function f(x) satisfies the identity f(x+y)=f(x)+f(y)+xy^2+x^2y f…
  40. 190.The function f(x)=cases4+^-1x, & |x|112(|x|-1), & |x|>1cases is:
  41. 191.The derivative of ^-1(√1+x^2-1x) with respect to ^-1(2x√1-x^21-2x^2) at x=12 is:
  42. 192.If (a+√2b x)(a-√2b y)=a^2-b^2, where a>b>0, then dxdy at (4,4) is
  43. 193.If y=_k=1^6k^-1(35 kx-45 kx), then dydx at x=0 is _______.
  44. 194.Let f:RR be a function defined as f(x)=cases5, & x1+bx, & 1<x<3+5x, & 3 x<530, & x5cases.
  45. 195.Let f be a differentiable function such that f(1)=2 and f'(x)=f(x) for all xR.
  46. 196.Let f(x)=cases-1, & -2 x<0^2-1, & 0 x2cases and g(x)=|f(x)|+f(|x|).
  47. 197.If x_e(_ex)-x^2+y^2=4 (y>0), then dydx at x=e is equal to:
  48. 198.Let K be the set of all real values of x where the function f(x)=|x|-|x|+2(x-)|x| is not d…
  49. 199.Let f(x)=cases|x|,x^2, & |x|28-2|x|, & 2<|x|4cases.
  50. 200.Let f:(-1,1)R be a function defined by f(x)=-|x|,-√1-x^2.