Help with Calculus?? If f'(x) = cos x and g'(x) = 1 for all x, and if f(0) = g(0) = 0….?

4 Answers

• Hi Julia,

  df(x)/dx = cos(x).

  dg(x)/dx = 1.

  Integrate:

  f(x) = sin(x) + constant1.

  g(x) = x + constant2.

  Use initial conditions:

  f(0) = 0:

  f(0) = sin(0) + constant1 = 0 + constant1 = 0.

  So, constant1 = 0.

  Thus, f(x) = sin(x).

  g(0) = 0:

  g(0) = 0 + constant2 = 0.

  So, constant2 = 0.

  Thus, g(x) = x.

  Substitute in functions:

  lim [x –> 0] (f(x) / g(x)) = lim [x –> 0] (sin(x) / x)

  This is 0 / 0. Therefore, you can use L’Hopital’s rule:

  = lim [x –> 0] (f'(x) / g'(x)) = lim [x –> 0] (cos(x) / 1) = lim [x –> 0] cos(x) = cos(0) = 1.

  Solution:

  1.

• “If f'(x) = cos x”

  So f(x) = sin(x) + C, where C = some constant.

  “and g'(x) = 1”

  So g(x) = x + D where D = some other constant.

  “f(0) = g(0) = 0”

  Plug in x = 0. That tells you what C and D need to be.

  Then take the limit as x->0.

• The anti-derivative of cosx = sinx which is f(x)

  The anti-derivative of g'(x) = 1 is x

  So f(0): sin(0) = 0

  and g(0) = 0

  Therefore lim x->0 of f(x)/g(x) = 0/0 which is indeterminate

• im taking calculas next semester, thanks for scaring me now lol