Question

suppose f^n (x)= -36 sin(6x) and f'(0)=6 and f(0)= -5

what is f(π /4)= ?

Answer 1

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.