Final answer:
To find f given f '(t) = sec(t)(sec(t) tan(t)), you need to integrate the given expression. The integral of sec(t)(sec(t) tan(t)) can be found by using the substitution method. The function is f(t) = (sec²(t))/2 - 9/2.
Step-by-step explanation:
To find f given f '(t) = sec(t)(sec(t) tan(t)), we need to integrate the given expression.
The integral of sec(t)(sec(t) tan(t)) can be found by using the substitution method.
Let u = sec(t). Then, du = sec(t) tan(t) dt.
The integral becomes:
∫ sec(t)(sec(t) tan(t)) dt = ∫ u du
Simplifying the integral:
= ∫ u du
= (u²)/2 + C
Replacing u with sec(t):
= (sec²(t))/2 + C
So, f(t) = (sec²(t))/2 + C, where C is the constant of integration.
Given that f(4) = -4, we can substitute t = 4 into the equation and solve for C:
-4 = (sec²(4))/2 + C
Substituting the value of sec(4) into the equation and solving for C:
(sec(4))² = (1/cos(4))² = 1 + tan²(4) = 1 + (tan(4))² = 1 + (sin(4)/cos(4))² = 1 + (sin²(4))/(cos²(4))
Substituting this expression into the equation and solving for C:
-4 = (1 + (sin²(4))/(cos²(4)))/2 + C
Simplifying the equation:
-8 = 1 + (sin²(4))/(cos²(4)) + 2C
Subtracting 1 + (sin²(4))/(cos²(4)) from both sides of the equation:
-9 = 2C
Solving for C:
C = -9/2
Therefore, the function is f(t) = (sec²(t))/2 - 9/2.