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Calculus Archive: Questions from August 04, 2022

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  Solve the ODE by Laplace Transform (1) \( y^{\prime \prime}-y=t, y(0)=1 \& y^{\prime}(0)=-1 \) (2) \( y^{\prime \prime}+y^{\prime}-6 y=0, y(0)=1 \& y^{\prime}(0)=1 \) (3) \( y^{\prime \prime}+0.04 y=0
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  LARCALCET7 \( 3.4 .008 . \) Complete the table. \[ \begin{array}{c|c|c} y=f(g(x)) & u=g(x) & y=f(u) \\ \hline y=\frac{5}{\sqrt{x^{2}+19}} & u= & y=\frac{5}{\sqrt{u}} \end{array} \]
  3 answers
• Use Picard's Method to approx y1,y2 for y' = x2 + y2, y(0) = 1
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  Find the first partial derivatives of the function. \[ f(x, y, z)=4 x \sin (y-z) \] \[ f_{x}(x, y, z)= \] \[ f_{y}(x, y, z)= \] \[ f_{z}(x, y, z)= \]
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  Evaluate \( \int_{-1}^{2} \int_{x^{2}-2}^{x} d y d x \). \( \frac{9}{2} \) \( \frac{1}{2} \) 2 1
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  Given \( f(x, y)=-2 x^{6}-x y^{4}+2 y^{5} \) \[ f_{x}(x, y)= \] \( f_{y}(x, y)= \)
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  Find \( \frac{d y}{d x} \) for \( y=\frac{6}{x}+7 \) si \[ \frac{d}{d x}\left(\frac{6}{x}+7 \sin x\right)= \]
  1 answer

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  (7 pts) For \[ g(x, y, z)=5 y^{2} \sin (4 x-2 z)-7 y z^{3} \] \( \frac{\partial^{2} g}{\partial x \partial z} \)
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  \( \int_{0}^{70} \int_{x}^{\sqrt{140 x-x^{2}}} 9 \sqrt{x^{2}+y^{2}} d y d x= \)
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  Find the divergence of \( F(x, y, z)=x \hat{i}+y^{3} z^{2} \widehat{j}+x z^{3} \hat{k} \). \[ \begin{array}{l} 1+3 y^{2} z^{2}+3 x z \\ 1+3 y^{2} z^{2}+3 x z^{2} \\ 1+3 y^{2} z^{2} \end{array} \] 1
  1 answer

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  \( y^{\prime \prime}+4 y^{\prime}+4 y=x+e^{x} \)
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  Find the divergence of \( F(x, y, z)=x \hat{i}+y^{3} z^{2} \widehat{j}+x z^{3} \hat{k} \). \[ \begin{array}{l} 1+3 y^{2} z^{2}+3 x z \\ 1+3 y^{2} z^{2}+3 x z^{2} \\ 1+3 y^{2} z^{2} \\ 1 \end{array} \]
  1 answer
• Help please.
  Find \( g^{\prime}(x) \) \[ g(x)=\int_{\frac{\pi}{6}}^{\sin (2 x)}\left(t^{2}-3 t\right) d t \] A) \( x^{2}-3 x \) B) \( (\sin 2 x)^{2}-3 \sin (2 x) \) C) \( \left((\sin 2 x)^{2}-3 \sin (2 x)\right) \
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  \( \iint_{R}\left(6 x^{2} y^{3}-10 x^{4}\right) d A, R=\{(x, y) \mid 0 \leq x \leq 1,0 \leq y \leq 4\} \)
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  If \( y=f(x)+g(x) \), then \( y^{\prime}=f^{\prime}(x) g(x)+g^{\prime}(x) f(x) \) True False
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• Find the derivative of the function: y = 3^2 + (x^2+8x+3)/sqrt(x) A: y' = (2x+8)/2x^3/2 B: y' =(3x^2+8x-3)/2x^3/2 C: y' = 6 + (2x+8)/x D: y' = (3x^2+8x-3)/x
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  Find the divergence of \( F(x, y, z)=x \widehat{i}+y^{3} z^{2} \widehat{j}+x z^{3} \hat{k} \). \[ \begin{array}{l} 1+3 y^{2} z^{2}+3 x z \\ 1+3 y^{2} z^{2}+3 x z^{2} \\ 1+3 y^{2} z^{2} \\ 1 \end{array
  1 answer

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  6. Evaluate the indefinite integral. a. \[ \int x^{2}\left(x^{3}+4\right)^{7} d x \] b. \[ \int \frac{y}{(y+3)^{\frac{2}{3}}} d y \]
  3 answers
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  \[ \frac{d y}{d x}-\frac{2 y}{x^{2}}=0 \] \[ y=C e^{2 x} \] None of these \[ y=e^{\frac{2}{x}}+2 \] \[ y=2 e^{\frac{-2}{x}}+C \] \[ y=C e^{\frac{-2}{x}} \]
  1 answer
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  The solution of the differential equation \[ \frac{d y}{d x}=\frac{3 x^{2}}{y}, y(1)=-2 \] \[ y=\left(x^{3}+1\right) \] \[ y=\sqrt{2}\left(2 x^{3}+1\right)^{\frac{1}{2}} \] \( y=\sqrt{2}\left(x^{3}+1\
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  (a) (C) \[ y^{\prime}=-3 y \] (u) \[ y^{\prime}=-3 x \] \[ y^{\prime}=e^{x} \] \[ y^{\prime}=\frac{1}{2} y+x \] \[ y^{\prime}=-x y \]
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  Find the curl \( \mathbf{F} \) in the vector field. \[ \mathbf{F}(x, y, z)=\mathbf{i}+\sin z \mathbf{j}+y \cos z \mathbf{k} \] A. \( \cos z \mathbf{j}+\sin z \mathbf{k} \) B. 0 C. \( y \sin z \mathbf{
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  3. Find \( y^{\prime} \) and \( y^{\prime \prime} \) \[ y=\left(-x^{5}+3 \sqrt{x}\right)\left(2 x^{7}+17\right) \]
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  If \( R=\{(x, y): 0 \leq x \leq 1,0 \leq y \leq 1\} \), then find \( \iint_{R}\left(\frac{x^{2}+1}{y^{2}+2}\right) d A \). \( \frac{1}{3} \arctan \left(\frac{3}{\sqrt{2}}\right) \) \( \frac{2 \sqrt{2}
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  Evaluate \( \int_{0}^{1} \int_{x^{2}}^{\sqrt{x}} 160 x y^{3} d y d x \) 2 6 3 10
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  Find the divergence of \( F(x, y, z)=x \hat{i}+y^{3} z^{2} \widehat{j}+x z^{3} \hat{k} \). \[ \begin{array}{l} 1+3 y^{2} z^{2}+3 x z \\ 1+3 y^{2} z^{2}+3 x z^{2} \\ 1+3 y^{2} z^{2} \\ 1 \end{array} \]
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  If \( f(x, y)=e^{x} \cos y \), then find \( \frac{\partial^{2} f}{\partial x^{2}}+\frac{\partial^{2} f}{\partial y^{2}} \). 0 1 2 3 3
  1 answer
• help asap pls!!!
  Find \( \frac{d y}{d x} \). \[ y=\tan ^{-1}\left(\ln 5 x^{3}\right) \]
  3 answers
• Solve the DE
  \( y^{\prime \prime}-2 y^{\prime}+2 y=e^{x} \tan \)
  3 answers
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  \( f(x)=\left\{\begin{array}{ll}\frac{4 x-\sin (4 x)}{2 x^{3}} & \text { if } x \neq 0 \\ \frac{16}{3} & \text { if } y \quad 0\end{array}\right. \) \( \sum_{n=0}^{\infty}\left(\frac{16}{3} \cdot \fra
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  If \( R=\{(x, y): 0 \leq x \leq 1,0 \leq y \leq 1\} \), then find \( \iint_{R}\left(\frac{x^{2}+1}{y^{2}+2}\right) d A \).
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  Find the gradient \( \nabla \mathrm{f} \). \[ f(x, y, z)=x y^{3} e^{x+z} \]
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  Given \( f(x, y)=5 x^{2}+2 x y^{4}-2 y^{5} \) \[ f_{x x}(x, y)= \] \( f_{x y}(x, y)= \)
  3 answers