What is the function derivative?

Consider the function f(x) whose second derivative is f '' (x)= 4x+2sin(x) . If f(0)=2 and f ' (0)=2, what?

• Consider the function f(x) whose second derivative is f '' (x)= 4x+2sin(x) . If f(0)=2 and f ' (0)=2 , what is f(x)?

• Answer:

  1. Take the anti derivative of f ''(x) to get f '(x) 2. Substitute the condition f '(0) = 2 into f '(x) to find C 3. Insert your new value of C back into the equation for f'(x) 4. Take the derivative of f '(x) to get f(x) 5. Insert the final condition f(0) = 2 to get the value of C in the equation for f(x) 6. Insert your new value of C back into the equation for f(x) 7. check your answer by taking the derivative twice to see if you end up with what you started with :) [1] f '(x) = 2x^2 - 2cos(x) + C [2] substitute 2 for f '(x) and 0 for 'x'.... 2 = 2(0)^2 - 2cos(0) + C 2 = 0 - 2(1) + C 2 = -2 + C 4 = C [3] f '(x) = 2x^2 - 2cos(x) + 4 [4] f(x) = (2/3)x^3 - 2sin(x) + 4x + C [5] substitute 2 for f(x) and 0 for 'x'.... 2 = (2/3)(0)^2 - 2sin(0) + 4(0) + C 2 = 0 - 2(0) + 0 + C 2 = C [6] f(x) = (2/3)x^3 -2sin(x) + 4x + 2 [7] f '(x) = 2x^2 - 2cos(x) +4 f' ''(x) = 4x -2(-sin(x)) + 0 f' ''(x) = 4x + 2sin(x)