Let g(x)= integraton (top 3x, bottom 2) e^t^4 dt. Find the value of g'(0).
To find the value of g'(0), we need to differentiate the function g(x) with respect to x and then evaluate the result at x = 0.
Let's start by finding the antiderivative of e^t^4 with respect to t. We can use a change of variables to simplify the integral.
Let u = t^4, then du = 4t^3 dt, so dt = (1/4t^3) du.
Now, we have:
g(x) = ∫[3x, 2] e^t^4 dt
= ∫[3x, 2] e^u (1/4t^3) du
= (1/4) ∫[3x, 2] e^u t^-3 du.
To evaluate this integral, we need to integrate e^u with respect to u and then substitute back u = t^4.
∫ e^u du = e^u + C .
Using the substitution u = t^4, the integral becomes:
(1/4) ∫[3x, 2] e^u t^-3 du
= (1/4) ∫[3x, 2] e^(t^4) t^-3 dt.
Now, we are ready to take the derivative of g(x).
g'(x) = [(1/4) e^(t^4) t^-3]'
= (1/4) [(e^(t^4))' (t^-3) + (e^(t^4)) (t^-3)']
= (1/4) [(4t^3 e^(t^4)) (t^-3) + e^(t^4) (-3t^-4)]
= t e^(t^4) + (1/4) e^(t^4) (-3/t^4).
Now, we can evaluate g'(0) by substituting x = 0 in the result:
g'(0) = 0 e^(0^4) + (1/4) e^(0^4) (-3/0^4)
= 0 + (1/4)(-3/0)
= -3/0.
The expression -3/0 is undefined, which means the derivative g'(x) does not exist at x = 0.
Therefore, g'(0) is undefined.