suppose f^n (x)= -36 sin(6x) and f'(0)=6 and f(0)= -5
what is f(π /4)= ?
To find f(π/4), we need to use the given information and apply it to the function f^n(x) = -36 sin(6x).
Since f'(0) = 6, we know that the derivative of f(x) at x = 0 is 6. Thus, we can take the derivative of f^n(x) to find f^(n-1)(x):
f^(n-1)(x) = (d/dx)[f^n(x)] = (d/dx)[-36 sin(6x)] = -36 * (d/dx)[sin(6x)]
= -36 * (6 cos(6x)) = -216 cos(6x)
Now, let's find f^(n-2)(x) by taking the derivative of f^(n-1)(x):
f^(n-2)(x) = (d/dx)[f^(n-1)(x)] = (d/dx)[-216 cos(6x)] = -216 * (d/dx)[cos(6x)]
= -216 * (-6 sin(6x)) = 1296 sin(6x)
We can continue this process until we find f(x):
f^(n-3)(x) = 7776 cos(6x)
f^(n-4)(x) = -46656 sin(6x)
Now, we know that f(0) = -5. From there, we can start integrating to find f(x):
f(x) = ∫[-46656 sin(6x)] dx = 7776 cos(6x) + C
Applying the condition f(0) = -5, we can substitute x = 0 into the equation f(x) = 7776 cos(6x) + C:
-5 = 7776 cos(6*0) + C
-5 = 7776 cos(0) + C
-5 = 7776 + C
C = -7776 - 5
C = -7781
Now that we have the value of C, we can find f(x):
f(x) = 7776 cos(6x) - 7781
To find f(π/4), we can substitute x = π/4 into the equation:
f(π/4) = 7776 cos(6*π/4) - 7781
= 7776 cos(3π/2) - 7781
= 7776 * 0 - 7781
= -7781
Therefore, f(π/4) = -7781.
To find f(π/4), we need to use the given information and apply it step-by-step.
Step 1: Start with the given equation f^n (x) = -36 sin(6x), where f^n (x) denotes the nth derivative of f(x).
Step 2: Take the derivative of both sides of the equation. Since f^n (x) is the nth derivative, the derivative will reduce the power of n by 1. So, the first derivative is denoted as f'(x).
f'(x) = d/dx(-36 sin(6x))
Step 3: Evaluate the derivative. The derivative of sin(x) is cos(x), and using the chain rule, we have:
f'(x) = -36 * cos(6x) * 6
f'(x) = -216 * cos(6x)
Step 4: Substitute x =0 into the equation to find f'(0).
f'(0) = -216 * cos(6*0)
f'(0) = -216 * cos(0)
f'(0) = -216 * 1
f'(0) = -216
Step 5: Use the given information that f'(0) = 6 to solve for a constant.
-216 = 6
Step 6: Solve the equation for a constant.
-216 = 6
Divide both sides by -6:
36 = 1
This result is not possible, so there is an error in the given information or calculations. Please double-check the values provided for f'(0) = 6.
If there was an error in the provided information or calculations, please correct it and provide the correct value for f'(0) so that we can continue solving for f(π/4).