Given f"(x)=-25sin(5x)and f'(0)=0 and f(0)=4.
Find f(pi/2)
f''(x) = -25sin5x
f'(x) = 5 cos5x + c
f(x) = sin5x + cx + d
f'(0) = 0 so c = -5
f(0) = 4 so d = 4
f(x) = sin5x - 5x + 4
f(pi/2) = sin(5pi/2) - 5pi/2 + 4
= 5 - 5pi/2
= 5(1 - pi/2)
To find f(pi/2), we can use the given information and integrate the function f"(x) = -25sin(5x) twice to obtain the function f(x).
Step 1: Integrate f"(x) to find f'(x):
∫ -25sin(5x) dx = -5cos(5x) + C1
where C1 is the constant of integration.
Step 2: Use the given condition f'(0) = 0 to determine the value of C1:
f'(0) = -5cos(5(0)) + C1 = -5cos(0) + C1 = -5 + C1 = 0
From this, we can conclude that C1 = 5.
Step 3: Integrate f'(x) = -5cos(5x) + 5 to find f(x):
∫ (-5cos(5x) + 5) dx = -5/5sin(5x) + 5x + C2
where C2 is the constant of integration.
Step 4: Use the given condition f(0) = 4 to determine the value of C2:
f(0) = -5/5sin(5(0)) + 5(0) + C2 = 0 + C2 = 4
Therefore, C2 = 4.
Step 5: Substitute x = pi/2 into the obtained function f(x) = -5/5sin(5x) + 5x + 4:
f(pi/2) = -5/5sin(5(pi/2)) + 5(pi/2) + 4
= -sin(5(pi/2)) + (5/2)pi + 4
= -sin((5/2)pi) + (5/2)pi + 4
= -1 + (5/2)pi + 4
= (5/2)pi + 3
Therefore, f(pi/2) = (5/2)pi + 3.
To find f(pi/2), we need to integrate the given second derivative of the function to find the expression for the function itself.
Step 1: Integrate the second derivative
Given f"(x) = -25sin(5x), we can integrate it twice to find the expression for the function f(x).
First, integrate f"(x) to get the first derivative:
f'(x) = ∫(-25sin(5x)) dx
To integrate -25sin(5x), we use the integration formula for sin(ax):
∫sin(ax) dx = (-1/a) * cos(ax)
Applying this formula, we have:
f'(x) = (-25/5) * cos(5x)
Simplifying, we get:
f'(x) = -5cos(5x)
Step 2: Use the given initial conditions
We are given f'(0) = 0 and f(0) = 4.
Since f'(0) = -5cos(5*0) = 0, this means that cos(0) = 0.
Using f(0) = 4, we can find the constant of integration when integrating f'(x) to get f(x).
f(x) = ∫(-5cos(5x)) dx
Using the integration formula for cos(ax):
∫cos(ax) dx = (1/a) * sin(ax)
Applying this formula, we have:
f(x) = (-5/5) * sin(5x) + C
Simplifying, we get:
f(x) = -sin(5x) + C
Plugging in f(0) = 4, we can solve for the constant C:
-4 = -sin(5*0) + C
-4 = -sin(0) + C
-4 = 0 + C
C = -4
Step 3: Evaluate f(pi/2)
Now that we have the expression for f(x) = -sin(5x) - 4, we can find f(pi/2) by substituting x = pi/2 into the expression:
f(pi/2) = -sin(5(pi/2)) - 4
simplifying further:
f(pi/2) = -sin(5pi/2) - 4
Since sin(5pi/2) = sin(pi/2) = 1, we have:
f(pi/2) = -1 - 4
f(pi/2) = -5
Therefore, f(pi/2) = -5.