Consider the function f(t) = 2 sec^2 (t) − 10t^3 . Let F (t) be the antiderivative of f(t) with F(0) = 0 .
Then F(3)=
f(t) = 2 sec^2 (t) − 10t^3
f(t) = 2tan (t) - 5/2 t^4 + c
F(0) = = 2tan 0 - 0 + c = 0 , so c = 0
then
F(t) = 2tan t - (5/2) t^4
F(3) = 2tan 3 - (5/2)(3^4) = 2tan3 - 405/2
To find the value of F(3), we need to find the antiderivative of f(t) and evaluate it at t = 3.
Given the function f(t) = 2sec^2(t) - 10t^3, let's calculate its antiderivative step by step.
Step 1: Find the antiderivative of 2sec^2(t):
The derivative of sec(t) is sec(t)tan(t), so the antiderivative of sec^2(t) is tan(t). Since the coefficient is 2, the antiderivative of 2sec^2(t) is 2tan(t).
Step 2: Find the antiderivative of -10t^3:
The power rule for integration gives us (1/n+1)t^(n+1). Applying this rule, we get -(10/4)t^4 = -5/2t^4.
Step 3: Combine the antiderivatives:
The antiderivative of 2sec^2(t) - 10t^3 is 2tan(t) - 5/2t^4.
Step 4: Evaluate the antiderivative at t = 3:
Substitute t = 3 into the antiderivative to find F(3):
F(3) = 2tan(3) - 5/2(3)^4
= 2tan(3) - 5/2(81)
= 2tan(3) - 405/2
Therefore, F(3) = 2tan(3) - 405/2.
To find the value of F(3), we need to evaluate the antiderivative of f(t) at t = 3.
The function f(t) = 2 sec^2(t) - 10t^3 is a combination of a trigonometric term and a polynomial term.
To find the antiderivative of f(t), we first need to identify the individual antiderivatives of each term.
The antiderivative of 2 sec^2(t) with respect to t is 2 tan(t).
The antiderivative of -10t^3 with respect to t is -10 * (1/4) * t^4 = - (5/2) * t^4.
Now, let's determine the antiderivative F(t) by combining the antiderivatives of each term:
F(t) = 2 tan(t) - (5/2) * t^4 + C,
where C is the constant of integration.
Given that F(0) = 0, we can substitute t = 0 into the equation:
F(0) = 2 tan(0) - (5/2) * 0^4 + C = 0.
Simplifying this equation, we find that C = 0.
Now, we can evaluate F(3) by substituting t = 3 into the equation:
F(3) = 2 tan(3) - (5/2) * 3^4.
Using a scientific calculator, you can find the value of tangent (tan) of 3, and then calculate the remaining term using the exponentiation operator (^).
After substituting the values, you will find the value of F(3).