Question 2 a. State the Refinement \( Q \) of a partition \( P \) b. Explain the fundamental theorem of calculus. c. Prove that a bounded function \( f:[a, b] \rightarrow \mathbb{R} \) is Riemann integrable if and only if for every \( E>0 \) there exists a partition \( P \) of \( [a, b] \), which may depend on \( E \), such that \( U(f ; P)-L(f ; P)<\varepsilon \). d. Use the Fundamental theorem of Calculus to compute the derivative of \( F(x)= \) \( \left(\int_{0}^{x} \cos (t) d t\right)^{3} \).

a. The refinement \( Q \) of a partition \( P \) is a partition that is finer than \( P \). This means that the subintervals of \( Q \) are either the same as the subintervals of \( P \) or have smaller lengths. b. The Fundamental Theorem of Calculus states that if a function \( f \) is continuous on the interval \([a, b]\) and \( F(x) \) is an antiderivative of \( f(x) \) on \([a, b]\), then the definite integral of \( f(x) \) from \( a \) to \( b \) is equal to the difference between the values of \( F(x) \) at \( b \) and \( a \). Mathematically, this can be expressed as: \[ \int_{a}^{b} f(x) dx = F(b) - F(a) \] c. To prove that a bounded function \( f:[a, b] \rightarrow \mathbb{R} \) is Riemann integrable if and only if for every \( E>0 \) there exists a partition \( P \) of \( [a, b] \), which may depend on \( E \), such that \( U(f ; P)-L(f ; P)<\varepsilon \), we need to show two parts: 1. If \( f \) is Riemann integrable, then for every \( E>0 \), there exists a partition \( P \) of \( [a, b] \) such that \( U(f ; P)-L(f ; P)<\varepsilon \). 2. If for every \( E>0 \), there exists a partition \( P \) of \( [a, b] \) such that \( U(f ; P)-L(f ; P)<\varepsilon \), then \( f \) is Riemann integrable. d. To compute the derivative of \( F(x)= \left(\int_{0}^{x} \cos (t) d t\right)^{3} \) using the Fundamental Theorem of Calculus, we first need to find the antiderivative of \( \cos(t) \), which is \( \sin(t) \). Then, we can apply the chain rule to find the derivative of \( F(x) \). The derivative of \( F(x) \) is given by: \[ F'(x) = 3 \left(\int_{0}^{x} \cos (t) d t\right)^{2} \cdot \cos(x) \]