Let F(x)= the integral from 0 to 2x of tan(t^2) dt. Use your calculator to find F″(1)
By applying the fundamental theorem of calculus, I got the derivative of the integral (F'(x)) to be 2tan(2x^2)
When I take the derivative to find F''(x) I get 8x sec^2(2x^2). When I plug 1 in for x, I don't get any of the answer choices, but I don't know where I went wrong in my evaluations. I just need to know my mistake
actually, F' = 2tan((2x)^2) = 2tan(4x^2)
F' = 2sec^2(4x^2)(8x) = 16x sec^2(4x^2)
To find F''(x), we need to differentiate F'(x) = 2tan(2x^2) with respect to x.
Using the chain rule, the derivative of tan(2x^2) with respect to x is 2x * sec^2(2x^2).
So, calculating F''(x) = d^2/dx^2 [2tan(2x^2)]:
F''(x) = d/dx [2x * sec^2(2x^2)]
Applying the product rule, we have:
F''(x) = 2 * sec^2(2x^2) + 2x * d/dx[sec^2(2x^2)]
To find d/dx[sec^2(2x^2)], we note that it is the derivative of sec^2(u) with u = 2x^2, multiplied by the derivative of u with respect to x.
The derivative of sec^2(u) is 2sec(u) * sec(u) * tan(u).
The derivative of u = 2x^2 with respect to x is du/dx = 4x.
Now, substituting these values into F''(x), we get:
F''(x) = 2 * sec^2(2x^2) + 2x * [2sec(2x^2) * sec(2x^2) * tan(2x^2)] * 4x
Simplifying further:
F''(x) = 2 * sec^2(2x^2) + 2x * 8x * sec^2(2x^2) * tan(2x^2)
Now, to find F''(1), substitute x = 1 into the equation:
F''(1) = 2 * sec^2(2*1^2) + 2*1 * 8*1 * sec^2(2*1^2) * tan(2*1^2)
Evaluating further:
F''(1) = 2 * sec^2(2) + 2(8) * sec^2(2) * tan(2)
Now, you can use your calculator to find the value of sec^2(2) and tan(2), and substitute those values into the equation to get the final answer for F''(1).
To find F''(x), you correctly differentiated F'(x) as 8x sec^2(2x^2).
When we plug in x = 1 into this expression, we get:
F''(1) = 8 * 1 * sec^2(2 * 1^2)
= 8 * 1 * sec^2(2)
= 8 * 1 * (1/cos^2(2))
= 8/cos^2(2)
The mistake is that you may have been expecting an answer choice in a different form. To evaluate the final answer, you can use your calculator to find the value of the cosine squared of 2 (cos^2(2)), and divide 8 by that value to get the numerical answer.