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Find the following derivatives. Express your answer in terms of the independent variables. \( w_{s} \) and \( w_{t} \), where \( w=\frac{x-z}{3 y+z}, x=3 s+4 t, y=s t \), and \( z=3 s-4 t \) (Type an expression using \( s \) and \( t \) as the variables.) \( \frac{\partial x}{\partial t}=4 \) (Type an expression using \( s \) and \( t \) as the variables.) \( \frac{\partial y}{\partial t}=s \) (Type an expression using \( s \) and \( t \) as the variables.) \( \frac{\partial z}{\partial t}=-4 \) (Type an expression using \( s \) and \( t \) as the variables.) \( w_{s}=-\frac{24 t(t+1)}{(3 s t+3 s-4 t)^{2}} \) (Type an expression using \( s \) and \( t \) as the variables.) \( w_{t}=\square \) (Type an expression using \( s \) and \( t \) as the variables.)

Ask by Campbell Burton. in the United States
Mar 17,2025

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\( w_t = \frac{24s}{(3st + 3s - 4t)^2} \)

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To find \( w_t \), we can use the chain rule to differentiate \( w \) with respect to \( t \): \[ w_t = \frac{\partial w}{\partial x} \cdot \frac{\partial x}{\partial t} + \frac{\partial w}{\partial y} \cdot \frac{\partial y}{\partial t} + \frac{\partial w}{\partial z} \cdot \frac{\partial z}{\partial t} \] Now we need the partial derivatives \( \frac{\partial w}{\partial x}, \frac{\partial w}{\partial y}, \) and \( \frac{\partial w}{\partial z} \). Using the quotient rule, we can compute them: \[ \frac{\partial w}{\partial x} = \frac{(3y + z)}{(3y + z)^2} \] \[ \frac{\partial w}{\partial y} = \frac{-(x - z)}{(3y + z)^2} \cdot 3 = -\frac{3(x - z)}{(3y + z)^2} \] \[ \frac{\partial w}{\partial z} = \frac{(x - z)}{(3y + z)^2} \cdot 1 = \frac{x - z}{(3y + z)^2} \] Now substituting the given partial derivatives and the expressions for derivatives of \( x, y, \) and \( z \) with respect to \( t \): \[ w_t = \left(\frac{3y + z}{(3y + z)^2}\right)(4) + \left(-\frac{3(x - z)}{(3y + z)^2}\right)(s) + \left(\frac{x - z}{(3y + z)^2}\right)(-4) \] Combining these terms, we have: \[ w_t = \frac{4(3y + z) - 3s(x - z) - 4(x - z)}{(3y + z)^2} \] To sum it all up, now we compute the values: 1. \( 3y + z = 3(s t) + (3s - 4t) = 3st + 3s - 4t \) 2. \( x - z = (3s + 4t) - (3s - 4t) = 8t \) Finally, plugging these back into the expression for \( w_t \): \[ w_t = \frac{4(3st + 3s - 4t) - 3s(8t) - 4(8t)}{(3st + 3s - 4t)^2} \] \[ = \frac{12st + 12s - 16t - 24st - 32t}{(3st + 3s - 4t)^2} \] \[ = \frac{-12st + 12s - 48t}{(3st + 3s - 4t)^2} \] Thus, the answer for \( w_t \) expressed in terms of \( s \) and \( t \) is: \[ w_t = \frac{12s - 12st - 48t}{(3st + 3s - 4t)^2} \]

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If you invest \( \$ 1000 \) for \( t \) years at \( 6.71 \% \) compounded at the frequencies below, find the following. (a) Suppose you compound at \( 6.71 \% \) monthly. i) Report an expression equivalent to the value of \( \$ 1000 \) invested for \( t \) years at \( 6.71 \% \) compounded monthly by completing the box with the growth factor if compounded annually. 1000 \( \square \) Number \( t \) (Round to \( \underline{4} \) decimal places.) ii) Report the effective annual rate: \( \square \) Number \% (Round to \( \underline{2} \) decimal places.) (b) Suppose you compound at \( 6.71 \% \) continuously. i) You would expect \( 6.71 \% \) compounded continuously to give a \( \square \) Click for List yield than what is given in part (a). ii) Complete the boxes below to report the expression for the value of \( \$ 1000 \) invested for \( t \) years at \( 6.71 \% \) compounded continuously and the equivalent growth factor if compounded annually. \[ \begin{array}{l} 1000 e^{(\text {Number } t)} \\ \approx 1000(\text { Number })^{t} \end{array} \] (Round to \( \underline{4} \) decimal places.) iii) Report the effective annual rate: \( \square \) Number \% (Round to \( \underline{2} \) decimal places.) (c) Complete the boxes to summarize: i) From part (a) we have that 6.71 \% compounded monthly is equivalent to \( \square \) Number \( \% \) compounded annually. ii) From part (b) we have that 6.71 \% compounded continuously is equivalent to \( \square \) Number \( \% \) compounded annually.

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If you invest \( \$ 1000 \) for \( t \) years at \( 6.71 \% \) compounded at the frequencies below, find the following. (a) Suppose you compound at \( 6.71 \% \) monthly. i) Report an expression equivalent to the value of \( \$ 1000 \) invested for \( t \) years at \( 6.71 \% \) compounded monthly by completing the box with the growth factor if compounded annually. 1000 \( \square \) Number \( t \) (Round to \( \underline{4} \) decimal places.) ii) Report the effective annual rate: \( \square \) Number \% (Round to \( \underline{2} \) decimal places.) (b) Suppose you compound at \( 6.71 \% \) continuously. i) You would expect \( 6.71 \% \) compounded continuously to give a \( \square \) Click for List yield than what is given in part (a). ii) Complete the boxes below to report the expression for the value of \( \$ 1000 \) invested for \( t \) years at \( 6.71 \% \) compounded continuously and the equivalent growth factor if compounded annually. \[ \begin{array}{l} 1000 e^{(\text {Number } t)} \\ \approx 1000(\text { Number })^{t} \end{array} \] (Round to \( \underline{4} \) decimal places.) iii) Report the effective annual rate: \( \square \) Number \% (Round to \( \underline{2} \) decimal places.) (c) Complete the boxes to summarize: i) From part (a) we have that 6.71 \% compounded monthly is equivalent to \( \square \) Number \( \% \) compounded annually. ii) From part (b) we have that 6.71 \% compounded continuously is equivalent to \( \square \) Number \( \% \) compounded annually.
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