Calculus Archive: Questions from December 13, 2023
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Find \( y^{\prime} \) if \( x^{y}=y^{x} \). \[ y^{\prime}=\frac{\ln (y)+\frac{x}{y}-\frac{y}{x}}{\ln (x)} \]1 answer -
2. Establecer, pero no evaluar, una integral triple para el volumen de la pirámide encerrada por los planos \[ \frac{x}{3}+\frac{y}{2}+\frac{z}{5}=1, \quad x=0, \quad y=0, \quad z=0 \] Nota: Los vér1 answer -
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Find the potential function \( f \) for the field \( F \). \[ F=-\left(\frac{x}{\left(x^{2}+y^{2}+z^{2}\right)^{3 / 2}}\right) i-\left(\frac{y}{\left(x^{2}+y^{2}+z^{2}\right)^{3 / 2}}\right) j-\left(\1 answer -
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number 9
In Exercises \( 1-17 \), find and classify the critical points of the given functions. 1. \( f(x, y)=x^{2}+2 y^{2}-4 x+4 y \) 2. \( f(x, y)=x y-x+y \) 3. \( f(x, y)=x^{3}+y^{3}-3 x y \) 4. \( f(x, y)=1 answer -
If \( y=y(x)=f(\sin (4 x)) \) and \( f^{\prime}(0)=-1 \), determine \( y^{\prime}\left(\frac{\pi}{4}\right) \). \( y^{\prime}\left(\frac{\pi}{4}\right)= \)1 answer -
number 11 and 13
In Exercises 1-17, find and classify the critical points of the given functions. 1. \( f(x, y)=x^{2}+2 y^{2}-4 x+4 y \) 2. \( f(x, y)=x y-x+y \) 3. \( f(x, y)=x^{3}+y^{3}-3 x y \) 4. \( f(x, y)=x^{4}+1 answer -
Evaluate \( \iiint_{E} 3 x z d V \) where \( E=\{(x, y, z) \mid 1 \leq x \leq 2, x \leq y \leq 2 x, 0 \leq z \leq x+3 y\} \)1 answer -
11. Given: \( h(x)=1-x^{2} \) a. Find \( h(\cos (\theta)) \). Simplify if possible. b. Find \( h(2+t) \). Simplify if possible.1 answer -
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substitute the parameter values into the problem BEFORE YOU START SOLVING and THEN solve the problem (THE TABLE) Let R be the region in the xy plane enclosed by the lines in the xy plane. Calcu
\begin{tabular}{|l|l|l|l|l|} \hline \( \mathrm{A} \) & \( \mathrm{B} \) & \( \mathrm{C} \) & \( \mathrm{D} \) & \( \mathrm{E} \) \\ \hline 4 & 6 & 2 & 5 & 3 \\ \hline \end{tabular} 5. Sea \( \boldsym1 answer -
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Evaluate \( \iiint_{E} 2 x z d V \) where \( E=\{(x, y, z) \mid 0 \leq x \leq 3, x \leq y \leq 2 x, 0 \leq z \leq x+3 y\} \)1 answer -
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question number 1, please. step by step. thanks.
En los siguientes problemas use la definición de la Transformada de Laplace para encontrar \( \mathcal{L}\{f(t)\} \) 1. \( f(t)=\left\{\begin{array}{rr}-1, & 0 \leq t1 answer -
question number 2 please. step by step. thanks.
En los siguientes problemas use la definición de la Transformada de Laplace para encontrar \( \mathcal{L}\{f(t)\} \) 1. \( f(t)=\left\{\begin{array}{rr}-1, & 0 \leq t1 answer -
question number 1 please. step by step. thanks.
1. Construya una función \( \boldsymbol{F}(\boldsymbol{t}) \) que sea de orden exponencial pero donde \( \boldsymbol{f}(\boldsymbol{t})=\boldsymbol{F}^{\prime}(\boldsymbol{t}) \) no sea de orden expo1 answer -
question 3 please. STEP BY STEP. Thanks.
Ocupando el teorema de la derivada de una transformada halla \( \mathcal{L}\{\boldsymbol{f}(\boldsymbol{t})\} \) si: e. \( f(t)=t \cos 3 t \) \[ \begin{array}{l} f . f(t)=t^{3} \operatorname{sen} 2 t1 answer -
question 4 please. STEP BY STEP. thanks.
Ocupando el teorema de la derivada de una transformada halla \( \mathcal{L}\{\boldsymbol{f}(\boldsymbol{t})\} \) si: e. \( f(t)=t \cos 3 t \) \[ \begin{array}{l} f . f(t)=t^{3} \operatorname{sen} 2 t1 answer -
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Given \( f(x, y)=-2 x^{5}-x^{2} y^{4}+2 y^{3} \), find \[ \begin{array}{l} f_{x}(x, y)= \\ f_{y}(x, y)= \\ f_{x x}(x, y)= \\ f_{x y}(x, y)= \end{array} \]1 answer -
\( \begin{array}{l}\text { Given } f(x, y)=-3 x^{2}+4 x^{2} y^{3}+3 y^{4} \\ f_{x}(x, y)= \\ f_{y}(x, y)= \\ f_{x x}(x, y)= \\ f_{x y}(x, y)=\end{array} \)2 answers -
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Evaluate the integral: \[ \begin{array}{l} \int \frac{\sin x \cos x}{(1+\sin x)^{5}} d x \\ \frac{1}{4(1+\sin x)^{5}}-\frac{1}{5(1+\sin x)^{5}}+C \\ \frac{1}{3(1+\sin x)^{3}}-\frac{1}{4(1+\sin x)^{4}}1 answer -
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\( \begin{array}{l}\text { Given } f(x, y)=-2 x^{3}+3 x^{2} y^{4}+2 y^{5}, \\ f_{x}(x, y)= \\ f_{y}(x, y)= \\ f_{x x}(x, y)= \\ f_{x y}(x, y)=\end{array} \)1 answer -
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Evaluate \[ \begin{array}{l} \int_{C} \mathbf{F} \cdot d \mathbf{r} \\ \mathbf{F}(x, y)=x \mathbf{i}+y \mathbf{j} \\ C: \mathbf{r}(t)=(7 t+5) \mathbf{i}+t \mathbf{j}, \quad 0 \leq t \leq 1 \end{array}1 answer -
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6. Find \( \frac{d y}{d x} \) for each of the following. You do not need to simplify. 1 (a) \( y=\left(2 x^{3}+5 x^{2}-6 x-4\right)^{5} \) (g) \( y=\left(x^{2}+4\right)^{2}\left(2 x^{3}-1\right)^{3} \1 answer -
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In Problems 1-6, find the indicated partial derivative. 1. If \( f(x, y)=x^{3} y^{5} \), then find \( f_{x}(x, y) \). 2. If \( f(x, y)=x^{3} y^{5} \), then find \( f_{y}(x, y) \). 3. If \( g(x, y)=\sq1 answer -
I don't understand how they get this answer
\( \begin{array}{c}\frac{\partial \vec{r}}{\partial \theta} \times \frac{\partial \vec{r}}{\partial \varphi}=\left|\begin{array}{ccc}\vec{i} & \vec{j} & \vec{k} \\ -2 \sin \varphi \sin \theta & 2 \sin0 answers -
13) If \( y=\frac{3}{\sin x+\cos x} \), find \( \frac{d y}{d x}= \) a) \( 3 \sin x-3 \cos x \) b) \( \frac{3}{(\sin x+\cos x)^{2}} \) c) \( \frac{-3}{(\sin x+\cos x)^{2}} \) d) \( \frac{3(\cos x-\sin1 answer