Question

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

f(π /4)= ?

Answer 1

To find f(π/4), we need to use the given information and apply the properties of derivatives and integrals.

Given:
f^n (x) = -36 sin(6x)
f'(0) = 6
f(0) = -5

First, let's find the first derivative of f(x) by differentiating f^n (x):
f'(x) = d/dx (-36 sin(6x))
Using the chain rule, we get:
f'(x) = -36 * 6 cos(6x)
f'(x) = -216 cos(6x)

Since we are given f'(0) = 6, we can substitute x = 0 into the equation:
-216 cos(6(0)) = 6
-216 cos(0) = 6
-216 * 1 = 6
-216 = 6
This equation is not true, so let's check if there's a mistake in the given information.

Now, let's find the integral of f'(x) to find f(x):
∫f'(x) dx = ∫(-216 cos(6x)) dx
Using the integral of cos(x), we get:
∫(-216 cos(6x)) dx = -216 * (1/6) * sin(6x) + C
= -36 sin(6x) + C

Since we are given f(0) = -5, we can substitute x = 0 into the equation:
-36 sin(6(0)) + C = -5
-36 sin(0) + C = -5
-36 * 0 + C = -5
0 + C = -5
C = -5

Therefore, the equation for f(x) is: f(x) = -36 sin(6x) - 5

To find f(π/4), substitute x = π/4 into the equation:
f(π/4) = -36 sin(6(π/4)) - 5
= -36 sin(3π/2) - 5
= -36 * (-1) - 5
= 36 - 5
= 31

So, f(π/4) = 31.