Given that
𝐺
(
𝑥
)
=
∫
3
𝑥
cos
⁡
(
𝜋
𝑡
/
6
)
𝑡
2
𝑑
𝑡
, what is
𝐺
′
(
2
)
?

Question

Given that 
𝐺
(
𝑥
)
=
∫
3
𝑥
cos
⁡
(
𝜋
𝑡
/
6
)
𝑡
2
𝑑
𝑡
, what is 
𝐺
′
(
2
)
?

BlackTom AI · Accepted Answer

The question as provided defines G(x) as an integral: G(x) = ∫ 3 x cos(π t/6) t^2 dt, but it does not specify the limits of integration. Without limits, the integral is not a well-defined function of x, so we cannot proceed to differentiate with respect to x or evaluate G'(2).

First, note a key principle: if you have a parameter x inside an integrand and want d/dx G(x) where G(x) = ∫ f(x, t) dt with fixed limits, you can differentiate under the integral sign, giving G'(x) = ∫ ∂f/∂x dt, provided certain regularity conditions hold. In this problem, f(x, t) = 3 x cos(π t/6) t^2. Its partial derivative with respect to x is ∂f/∂x = 3 cos(π t/6) t^2, which is independent of x.

However, to evaluate either G'(x) in general or G'(2) specifically, we still need the limits of integration. Without limits, the integral ∫ 3 cos(π t/6) t^2 dt is undefined as a numeric value, and thus G'(2) cannot be computed.

Suppose, for the sake of understanding, the limits were standard finite bounds [a, b]. Then, using the differentiation under the integral sign approach, G'(x) would be: G'(x) = ∫_{a}^{b} 3 cos(π t/6) t^2 dt. This expression would then be a constant with respect to x, since the integrand derivative with respect to x does not involve x. Consequently, G'(2) would equal that same constant ∫_{a}^{b} 3 cos(π t/6) t^2 dt, assuming the same limits. But again, the actual numeric value depends on a and b, which are not provided here.

In addition, the provided answer option displays as two lines '1' and '8', likely intending the number 18, but without limits or further context, we cannot justify any numeric result for G'(2).

If you can supply the limits of integration (the interval for t), I can show the exact steps to compute G'(2) step by step, including evaluating the definite integral ∫ 3 cos(π t/6) t^2 dt over those limits and then substituting into G'(x) at x = 2.

Until the limits are known, the critical blockers remain: the integral is not defined numerically, and thus G'(2) cannot be determined from the given information.

MATH-112-301-001 Unproctored Midcourse Exam 4 Practice Exam 1

Given that 𝐺 ( 𝑥 ) = ∫ 3 𝑥 cos ⁡ ( 𝜋 𝑡 / 6 ) 𝑡 2 𝑑 𝑡 , what is 𝐺 ′ ( 2 ) ?

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