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Use the Chain Rule to find \( \frac{d Q}{d t} \), where \( Q=\sqrt{2 x^{2}+2 y^{2}+2 z^{2}}, x=\sin t, y=\cos t \), and \( z=\sin t \). \( \frac{\partial Q}{\partial x}=\square \) (Type an expression using \( x, y \), and \( z \) as the variables.)

Ask by Moreno Rodriguez. in the United States
Mar 17,2025

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\[ \frac{\partial Q}{\partial x} = \frac{2x}{\sqrt{2x^2 + 2y^2 + 2z^2}} \]

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To find \( \frac{dQ}{dt} \) using the Chain Rule, we first need to calculate \( \frac{\partial Q}{\partial x} \). The expression for \( Q \) is: \[ Q = \sqrt{2x^2 + 2y^2 + 2z^2} \] Calculating \( \frac{\partial Q}{\partial x} \): Using the chain rule, we first express the derivative of \( Q \) with respect to its argument: \[ \frac{\partial Q}{\partial x} = \frac{1}{2\sqrt{2x^2 + 2y^2 + 2z^2}} \cdot \frac{\partial}{\partial x}(2x^2 + 2y^2 + 2z^2) \] Calculating the inner derivative: \[ \frac{\partial}{\partial x}(2x^2 + 2y^2 + 2z^2) = 4x \] Now substituting back into the expression for \( \frac{\partial Q}{\partial x} \): \[ \frac{\partial Q}{\partial x} = \frac{4x}{2\sqrt{2x^2 + 2y^2 + 2z^2}} = \frac{2x}{\sqrt{2x^2 + 2y^2 + 2z^2}} \] So the value of \( \frac{\partial Q}{\partial x} \) is: \[ \frac{\partial Q}{\partial x} = \frac{2x}{\sqrt{2(x^2 + y^2 + z^2)}} \]

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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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