Answer the following True or False. Since (d)/(dx)3x^((1)/(3))=x^(-(2)/(3)), thefundamental theorem of calculus tells us that int_(-1)^(1)x^(-(2)/(3))dx=3(1^((1)/(3)))-3(-1)^((1)/(3))=6. True False

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Answer the following True or False. Since  (d)/(dx)3x^((1)/(3))=x^(-(2)/(3)), thefundamental theorem of calculus tells us that  int_(-1)^(1)x^(-(2)/(3))dx=3(1^((1)/(3)))-3(-1)^((1)/(3))=6. True False

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Verify Derivative: Let's first verify the derivative given in the statement. Given the function f(x) = 3x^(1/3), we need to find its derivative f'(x). Using the power rule for derivatives, we have: f'(x) = d/dx [3x^(1/3)] = 3 * (1/3) * x^((1/3) - 1) = x^(-2/3).Apply Fundamental Theorem: Now, let's apply the Fundamental Theorem of Calculus, which states that if F is an antiderivative of f on an interval [a, b], then: ∫_a^b f(x) dx = F(b) - F(a). Here, f(x) = x^(-2/3) and F(x) = 3x^(1/3) is its antiderivative.Evaluate Integral: We will now evaluate the integral using the antiderivative: ∫_(-1)^(1) x^(-2/3) dx = F(1) - F(-1) = 3(1^(1/3)) - 3((-1)^(1/3)).Be Careful with Evaluation: We need to be careful with the evaluation of F(-1) because (-1)^(1/3) is not equal to -1. The cube root of -1 is -1, so we have: F(1) - F(-1) = 3(1) - 3(-1) = 3 + 3 = 6.Statement Verification: The statement given is ＂Since (d)/(dx)3x^((1)/(3))=x^(-(2)/(3)), the fundamental theorem of calculus tells us that ∫_(-1)^(1)x^(-(2)/(3))dx=3(1^((1)/(3)))-3(-1)^((1)/(3))=6.＂ Based on our calculations, this statement is true.

Answer the following True or False. $\newline$ Since $\frac{d}{dx}3x^{\frac{1}{3}}=x^{-\frac{2}{3}}$ , the fundamental theorem of calculus tells us that $\int_{-1}^{1}x^{-\frac{2}{3}}dx=3(1^{\frac{1}{3}})-3(-1)^{\frac{1}{3}}=6$ . $\newline$ True $\newline$ False

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