answer:Applying the four-step process to find the derivative, we have: {eq}begin{align*} f'(x) &= lim _{hrightarrow0}frac{f(x+h) - f(x) }{h} &= lim _{hrightarrow0}frac{(3 + x + h - (x + h)^2) - (3 + x - x^2)}{h} &= lim _{hrightarrow0}frac{3 + x + h - x^2 - 2xh - h^2 - 3 - x + x^2}{h} &= lim _{hrightarrow0}frac{h - 2xh - h^2}{h} &= lim _{hrightarrow0}frac{h(1 - 2x - h)}{h} &= lim _{hrightarrow0} (1 - 2x - h) quad (text{since } h neq 0) &= 1 - 2x end{align*} {/eq} Thus, the derivative {eq}f'(x) {/eq} is given by {eq}1 - 2x. {/eq}

answer:Given: - Volume of hot water, ( V_1 = 50 ) mL - Temperature of hot water, ( T_1 = 65^{circ}C ) - Volume of cold water, ( V_2 = 25 ) mL - Temperature of cold water, ( T_2 = 25^{circ}C ) Since the system is thermally isolated, the heat lost by the hot water equals the heat gained by the cold water: [ -Q_{hot} = Q_{cold} ] Using the specific heat capacity equation, we can express this as: [ -m_1 c (T_f - T_1) = m_2 c (T_f - T_2) ] Here, ( m_i ) represents the mass of each volume, ( rho ) is the density of water, and ( c ) is its specific heat capacity. We can write the mass of water in terms of volume: [ m_i = rho V_i ] Substituting and simplifying, we get: [ V_1 (T_1 - T_f) = V_2 (T_f - T_2) ] Rearranging for the final temperature ( T_f ): [ T_f (V_2 + V_1) = V_1 T_1 + V_2 T_2 ] [ T_f = frac{V_1 T_1 + V_2 T_2}{V_2 + V_1} ] Now, applying the given values: [ T_f = frac{(50 , text{mL}) (65^{circ}C) + (25 , text{mL}) (25^{circ}C)}{50 , text{mL} + 25 , text{mL}} ] Evaluating the expression, we find: [ boxed{T_f = 51.67^{circ}C} ]

answer:To multiply a scalar with a matrix, you multiply each element of the matrix by the scalar. Thus, we have: -frac{3}{2} left( begin{array}{cc} 6 & -2 end{array} right) = left( begin{array}{cc} -frac{3}{2} cdot 6 & -frac{3}{2} cdot (-2) end{array} right) = left( begin{array}{cc} -9 & 3 end{array} right) The multiplication results in the matrix: left( begin{array}{cc} -9 & 3 end{array} right)

answer:To rewrite a fraction as a mixed number, we need to divide the numerator by the denominator and express the remainder as a fraction. In this case, 31 divided by 9 is 3 with a remainder of 4. Therefore, 31/9 can be written as 3 4/9. The mixed number 3 4/9 represents 3 whole units and 4/9 of another unit. The answer is 3 4/9