Consider the function f(x) whose second derivative is f''(x) = 8x + 4sin(x). If f(0) = 2 and f'(0) = 2, what is f(x)?
I got... f'(x)=4(x^2 - cos[x]) and f(x)=(4(x^3 - 3sin[x]))/3 but it's wrong. Can anyone explain to me how to solve this.
y '' = 8x + 4sin(x)
y ' 4x^2 - 4cosx + C
f'(0) = 2 --- > 2 = 0 - 4(cos0) + C
2 = 0-4 + C
C = 6
so y' = 4x^2 - 4cosx + 6
y = (4/3)x^3 - 4sinx + 6x + K
f(0) = 2 ---> 2 = 0 - 4sin0 + 0 + K
K = 2
then y = (4/3)x^3 - 4sinx + 6x + 2
check by differentiating and subbing in x = 0 at each level
To find the function f(x) given f''(x), we need to integrate f''(x) twice. Let's go step by step to solve this problem.
Step 1: Integrate f''(x)
The integral of 8x with respect to x is 4x^2. The integral of 4sin(x) with respect to x is -4cos(x). Therefore, the first antiderivative of f''(x) is 4x^2 - 4cos(x) + C, where C is an arbitrary constant.
Step 2: Solve for C using the initial condition f'(0) = 2
To find the constant C, we need to evaluate the derivative of f(x) and substitute x = 0 with the given condition.
Taking the derivative of the antiderivative from Step 1, we get:
f'(x) = d/dx [4x^2 - 4cos(x) + C]
= 8x + 4sin(x)
Now, substitute x = 0 and f'(0) = 2:
2 = 8(0) + 4sin(0) + C
2 = C
So, C = 2.
Step 3: Rewrite the antiderivative with the found constant C
Using the value of C, we rewrite the antiderivative from Step 1:
f'(x) = 4x^2 - 4cos(x) + 2
Step 4: Integrate f'(x) to find f(x)
Now we integrate f'(x) to find f(x). The integral of 4x^2 with respect to x is (4/3)x^3. The integral of -4cos(x) with respect to x is -4sin(x).
Therefore, the second antiderivative of f'(x) is (4/3)x^3 - 4sin(x) + D, where D is an arbitrary constant.
Step 5: Solve for D using the initial condition f(0) = 2
To find the constant D, we need to evaluate the function f(x) and substitute x = 0 with the given condition.
Now, substitute x = 0 and f(0) = 2:
2 = (4/3)(0)^3 - 4sin(0) + D
2 = D
So, D = 2.
Step 6: Rewrite the antiderivative with the found constant D
Using the value of D, we rewrite the antiderivative from Step 4:
f(x) = (4/3)x^3 - 4sin(x) + 2
Therefore, the function f(x) is given by:
f(x) = (4/3)x^3 - 4sin(x) + 2
It seems there may have been an error in your calculations when finding f(x). Using the explanations above, you can double-check your work to ensure accuracy.