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question:Find the coefficient of x in the expansion of p(x)q(x) where p(x) = -frac{3 x^2}{sqrt{pi }}+frac{20 x}{sqrt{pi }}-frac{1}{sqrt{pi }} and q(x) = frac{15 x}{sqrt{pi }}-frac{12}{sqrt{pi }}.

answer:begin{align*} text{Coefficient of }x&=left(15cdotfrac{20}{sqrt{pi}}right)+left(-12cdotfrac{-3}{sqrt{pi}}right) &=frac{300}{sqrt{pi}}+frac{36}{sqrt{pi}} &=boxed{frac{336}{sqrt{pi}}} end{align*}

question:Compute the binomial coefficient binom{8483}{8482}.

answer:The binomial coefficient binom{n}{k} represents the number of ways to choose k elements from a set of n elements, without regard to order. In this case, we have n = 8483 and k = 8482. Using the formula for the binomial coefficient, we have: binom{8483}{8482} = frac{8483!}{8482! cdot 1!} Simplifying this expression, we can cancel out the common factor of 8482! in the numerator and denominator: binom{8483}{8482} = frac{8483!}{8482! cdot 1!} = 8483 Therefore, the answer to the question is 8483. The answer is 8483

question:Find the least squares vector b for the system Ax = b, where the matrix A is given by A = left( begin{array}{ccccc} 1 & 3 & 2 & -1 & 0 2 & 3 & 0 & -3 & 3 3 & -3 & 3 & 0 & -2 -1 & -1 & 0 & 2 & -2 1 & -3 & 3 & 1 & -1 0 & -1 & 2 & 0 & 0 end{array} right) and the vector b is b = left( begin{array}{c} 1.91 -1.38 -1.65 2.1 2.85 0.95 end{array} right)

answer:The least squares solution for the system Ax = b is the vector x that minimizes the Euclidean norm of the residual vector r = b - Ax. For the given matrix A and vector b, the least squares vector x is x = left( begin{array}{c} 0.821 1.026 1.019 4.054 2.038 end{array} right)

question:Explain the role of compactness in proving the Extreme Value Theorem for a continuous function defined on a closed interval.

answer:Compactness (boundedness and closedness) is essential because it prevents the function from escaping to infinity or having a maximum value outside the interval. Without compactness, we can construct examples like f(x) = x on [0, 1) or f(x) = tan(x) on [-π/2, π/2) where the function is continuous but has no maximum value within the interval. Compactness ensures that the function is bounded and that any sequence within the interval has a convergent subsequence, allowing us to find a point where the function attains its maximum value.