Calculus Archive: Questions from June 02, 2023
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(1 point) Find \( d y / d x \) in terms of \( t \) if \[ x=t e^{t}, \quad y=-8 t-8 e^{t} \] \[ d y / d x= \]2 answers -
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36. Find \( \partial z / \partial u \) when \( u=0, v=1 \) if \( z=\sin x y+x \sin y \), \( x=u^{2}+v^{2}, y=u v \).2 answers -
3. Evaluate the triple integral \[ \iiint_{E} e^{\frac{z}{y}} d V \text { where } E=\{(x, y, z) \mid 0 \leq y \leq 1, y \leq x \leq 1,0 \leq z \leq x y\} \]2 answers -
Calcule la integral de superficie \( \iint_{s} F \cdot d S \), donde \( S \) es un cilindro \( x^{2}+y^{2}=1,0 \leq z \leq 2 \), incluyendo la parte superior e inferior del cilindro, y \( F=\left\lang2 answers -
Calcule la integral de superficie \( \iint_{s} F \cdot d S \), donde \( S \) es el límite de la caja dada por \( 0 \leq x \leq 2,1 \leq y \leq 4,0 \leq z \leq 1 \), y \( F=\left\langle x^{2}+y z, y-z2 answers -
Solve through the LAPLACE transform
32. \( 2 \frac{d y}{d t}+y=0, \quad y(0)=-3 \) 33. \( y^{\prime}+6 y=e^{4 t}, \quad y(0)=2 \) 34. \( y^{\prime}-y=2 \cos 5 t, \quad y(0)=0 \)2 answers -
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If \( f(x)=\frac{4 \sin x}{3+\cos x} \), then \[ f^{\prime}(x)=\frac{12 \cos (x)+4}{(3+\cos (x))^{2}} \]2 answers -
3. Solve the following separable DEs. (a) \[ y^{\prime}=\frac{2 \cos (2 x)}{3+2 y}, y(0)=-1 \] (b) \[ \frac{y}{x} y^{\prime}=\sin x \sqrt{1+y^{2}}, y(0)=0 \]2 answers -
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Si F(x)=f(3f(4f(x))), donde f(0)=0 y f'(0)=2 encuentre F'(0). Por favor, muestre todo el trabajo! :D2 answers
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3. Evaluate the triple integral \[ \iiint_{E} e^{\frac{z}{y}} d V \text { where } E=\{(x, y, z) \mid 0 \leq y \leq 1, y \leq x \leq 1,0 \leq z \leq x y\} \]2 answers -
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(c) Let \( y=\frac{1}{x^{7}-2 x^{2}-x} \). \[ \frac{d y}{d x}= \] (d) For \( g(x)=\sqrt{8 x-x^{4}+1} \), \[ g^{\prime}(x)= \] (e) Let \( y=\left(2 e^{x}-x^{2}\right)^{8} \). \[ \frac{d y}{d x}= \]2 answers -
(f) Let \( y=5 \ln \left(x^{3}+25\right) \). \[ \frac{d y}{d x}= \] (g) Let \( y=e^{x^{2}+4 x+9} \). \[ \frac{d y}{d x}= \] (h) Let \( y=\frac{e^{8 x}}{7} \) \[ \frac{d y}{d x} \underset{x=1}{=} \]2 answers -
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12. \( 11 \mathrm{~min} \) (282) 14.6 Implicit Function HW \( \# 2.8 \) Given \( f(x, y, z)=\sqrt{x} \sqrt[3]{y} \sqrt[4]{z}=30 \), approximate \( x \) if \( y=29.0 \) and \( z=17.5 \)2 answers -
Evaluate the integral. \[ \int_{4}^{5}\left(t^{3} \mathbf{i}+t \sqrt{t-4} \mathbf{j}+t \sin \pi t \mathbf{k}\right) d t \]2 answers -
4. Halle el límite (si existe) de \( f(x, y)=\frac{x^{3} y^{2}}{x^{2}+y^{2}} \) según \( (x, y) \rightarrow 0 \). Si el límite no existe, explique por qué.1 answer -
iii. \( \iint_{D} e^{-y^{2}} d A, \iint_{D} \sin \left(-y^{2}\right) d A \), where \( D=\left\{(x, y) \in \mathbb{R}^{2} \mid 0 \leq y \leq 2,0 \leq x \leq y\right\} \) iv. \( \int_{0}^{1} \int_{\sqrt2 answers -
\( \begin{array}{l}\iint_{R} \frac{x y^{2}}{x^{2}+1} d A, R=[0,1] \times[-3,3] \\ \iint_{R} x \sin (x+y) d A, R=[0, \pi / 6] \times[0, \pi / 3] \\ \iint_{R} y e^{-x y} d A, R=[0,2] \times[0,3]\end{arr2 answers -
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