Question 3: Given f'(x) = -25 sin(5x) and f'(0) = -2 and f(0) = -5, find f(75).
Solution:
To find f(75), we need to integrate f'(x) with respect to x.
∫f'(x) dx = ∫(-25 sin(5x)) dx
Using the integral of sin(x) = -cos(x), we can rewrite the integral as:
∫(-25 sin(5x)) dx = -25/5 ∫sin(5x) dx = -5 ∫sin(5x) dx
Integrating sin(5x) with respect to x, we get:
-5 ∫sin(5x) dx = -5 (-1/5 cos(5x)) + C = cos(5x) + C
Now, we can find the value of C by using the given information f'(0) = -2 and f(0) = -5.
f'(0) = cos(5(0)) + C = cos(0) + C = 1 + C = -2
Solving for C, we get:
C = -2 - 1 = -3
Therefore, the equation for f(x) is:
f(x) = cos(5x) - 3
To find f(75), we substitute x = 75 into the equation:
f(75) = cos(5(75)) - 3
Simplifying further:
f(75) = cos(375) - 3
Note: The square root symbol was not mentioned in the given information, so there is no need to correct any errors related to it.