Consider the function f(t)=2sec^2(t)–6t^2 . Let F(t) be the antiderivative of f(t) with F(0)=0 .
Then F(5)=?
I am confused on how I would find the antiderivative, after I do that do I plug in five?
You might want to use a table of integrals for the first term. Most people prefer to call the "antiderivative" the (indefinite) integral.
In your case the integral is
F(t) = 2 tan t -2t^3 + C
where C is an arbitary constant.
Since F(0) = 0, C = 0
Now solve for F(5)
when i plug this answer in I get -500 is this correct?
I get 2 tan 5 - 2*125 = -249.8
The "5" is understood to be in radians. The -2t^3 term dominates
To find the antiderivative F(t) of the function f(t), you need to integrate f(t) with respect to t. Let's break it down step by step.
Step 1: Integrate f(t)
Start by integrating each term of the function f(t). The antiderivative of 2sec^2(t) can be found using the identity integral(sec^2(t)) dt = tan(t) + C, where C is the constant of integration. The antiderivative of -6t^2 is -2t^3 + C.
So, F(t) = 2tan(t) - 2t^3 + C.
Step 2: Use F(0) = 0 to find C
Since F(0) = 0, you can substitute t = 0 into the expression for F(t).
0 = 2tan(0) - 2(0)^3 + C
C = 0.
Hence, the constant of integration C is 0.
Step 3: Substitute t = 5 into F(t)
Now that you have the expression for F(t) and the value of C, you can substitute t = 5 to find F(5).
F(5) = 2tan(5) - 2(5)^3
F(5) = 2tan(5) - 250.
So, to find F(5), you need to evaluate 2tan(5) - 250. Note that this assumes the given function and antiderivative are correct.