______________________________Note: if you took answer that was display before I answered, then that answer was incorrect, since person answering forgot to multiply each term by d/dt (t³) and d/dt (4t) respectively______________________________Answer:Let G(x) = ∫ e^(x²) dxG(x) = e^(x²)g(t) = ∫[x = 4t to t³] e^(x²)g(t) = G(t³) - G(4t)g(t) = d/dt (G(t³) - G(4t))g(t) = d/dt (G(t³)) - d/dt (G(4t))Using chain rule: [G(f(t))] = G(f(t)) * f(t), we get:g(t) = G(t³) * 3t² - G(4t) * 4 -----------> recall G(x) = e^(x²)g(t) = e^((t³)²) * 3t² - e^((4t)²) * 4g(t) = 3t² e^(t⁶) - 4 e^(16t²)Mαthmφm-.........
Our teacher never thought this in class and I don't even know where to start! I don't need an answer ( I don't want anyone to feel like they're doing the work for me lol) but the steps to solving this would be appreciated. Thanks!
______________________________
Note: if you took answer that was display before I answered, then that answer was incorrect, since person answering forgot to multiply each term by d/dt (t³) and d/dt (4t) respectively
______________________________
Answer:
Let G(x) = ∫ e^(x²) dx
G'(x) = e^(x²)
g(t) = ∫[x = 4t to t³] e^(x²)
g(t) = G(t³) - G(4t)
g'(t) = d/dt (G(t³) - G(4t))
g'(t) = d/dt (G(t³)) - d/dt (G(4t))
Using chain rule: [G(f(t))]' = G'(f(t)) * f'(t), we get:
g'(t) = G'(t³) * 3t² - G'(4t) * 4 -----------> recall G'(x) = e^(x²)
g'(t) = e^((t³)²) * 3t² - e^((4t)²) * 4
g'(t) = 3t² e^(t⁶) - 4 e^(16t²)
Mαthmφm
... y = 4^(x^2 - 5x) * 6^(x^3 - 4x^2) * 7^(5x+4)
or ln(y) = (x^2 - 5x) ln(4) + (x^3 - 4x^2) ln(6) + (5x+4) ln(7)
or ln(y) = x^2 ln(4) - 5x ln(4) + x^3 ln(6) - 4x^2 ln(6) + 5x ln(7) + 4ln(7)
or ln(y) = x^3 ln(6) + x^2 ln(4) - 4x^2 ln(6) - 5x ln(4) + 5x ln(7) + 4ln(7)
or ln(y) = ln(6) x^3 + ( ln(4) - 4ln(6) ) x^2 + 5 ( ln(7) - ln(4) ) x + 4ln(7)
or ln(y) = ln(6) x^3 + ln(4/6^4) x^2 + ( ln(7/4)^5 ) x + ln(7^4)
or ln(y) = ln(6) x^3 + ln(4/6^4) x^2 + ln( (7/4)^5 ) x + ln(7^4)
or ln(y) = ln(A) x^3 + ln(B) x^2 + ln(C) x + ln(D)
where A = 6; B = 4/6^4; C = (7/4)^5; and D = 7^4
