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Calculus Archive: Questions from December 18, 2022

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  6. Find the derivative of the following functions. a. \( y=x^{2} \sin ^{4} x+x \cos ^{-2} x \), b. \( y=\sin ^{2}\left(3 x^{2}\right) \), c. \( g(t)=\left(\frac{1+\sin 3 t}{3-2 t}\right)^{-1} \), d. \
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  \begin{tabular}{|c|c|} \hline 1 & \( x^{2} y^{\prime \prime}+2 x y^{\prime}-6 y=0 ; \quad y_{1}=x^{2} \) \\ \hline 2 & \( y^{\prime \prime}+y=\sec ^{2} x \) \\ \hline 3 & \( x^{3} y^{\prime \prime \pr
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  (6 points) Find the partial derivatives of the function
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  \( y^{\prime \prime \prime}-6 y^{\prime \prime}+11 y^{\prime}-6 y=3 x \) \( y^{\prime \prime}-4 y^{\prime}+4 y=2 e^{2 x}+4 x-12 \) \( y^{(4)}+y^{\prime \prime}=0 \)
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  (6 points) Find the partial derivatives of the function \[ f(x, y)=x y e^{5 y} \] \[ f_{x}(x, y)= \] \[ f_{u}(2 \] \[ f_{x u}( \] \[ f_{y x}( \]
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  Evaluate the triple integral. \[ \iiint_{E} y d V, \text { where } E=\{(x, y, z) \mid 0 \leq x \leq 5,0 \leq y \leq x, x-y \leq z \leq x+y\} \]
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  Solve \( F_{y y}+3 F_{x y}+2 F_{x x}=2 x, F(x, 0)=-x, \partial_{y} F(x, 0)=0 \) \( F(x, y)= \)
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  Evaluate the following integral \( \int x^{2} \cos 2 x d x \) \[ \begin{array}{l} \frac{1}{2}(\sin -1 x)^{2}+C \\ x \sin ^{-1} x+\sqrt{1-x^{2}}+C \\ x^{2} \sin x+2 x \cos x-2 \sin x+C \\ \frac{1}{3} x
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  \( \frac{d y}{d x} \) \( \left(x^{2}+3 y^{2}\right)^{5}=2 x+y \) \( y^{2}+2 x y^{2}-3 x+1=0 \) \( y=\left(x^{2}+1\right)^{2 x} \)
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  Which of the following is a solution of the differential equation \( \frac{\mathrm{d}^{2} y}{\mathrm{~d}^{2}}+4 y=0 \) ? \[ \begin{array}{l} y=\frac{1}{4 x+1} \\ y=e^{2 x} \\ y=\sin 2 x \\ y=e^{-4 x}
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  Which of the following is a solution of the differential equation \( \frac{d y}{d x}+4 y^{2}=0 \) ? \[ \begin{array}{l} y=e^{2 x^{2}} \\ y=e^{-4 x} \\ y=4 x \\ y=e^{4 x} \\ y=e^{2 x} \\ y=2 x^{2} \\ y
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  \( x^{2} y^{\prime \prime}+2 x y^{\prime}-6 y=0 ; \quad y_{1}=x^{2} \) \( y^{\prime \prime}+y=\sec ^{2} x \) \( x^{3} y^{\prime \prime \prime}+x y^{\prime}-y=0 \)
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  Solve using Laplace transform \[ \begin{array}{l} y^{\prime \prime}+2 y^{\prime}+y=d(t) \\ t=0: y=1 \\ y^{\prime}=0 \end{array} \]
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• please answer number 2, 4, 6, 8, 14, 12, 16, 18
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  \( \int_{0}^{\frac{\pi}{6}} \int_{0}^{1} \int_{0}^{\pi}(x y \sin (18 y z)) d x d y d z \)
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  Evaluate \( \int_{0}^{4} \int_{0}^{27} \int_{\sqrt[3]{x}}^{3} e^{y^{4}} d y d x d z \)
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  \( \int_{16}^{91} \int_{0}^{33} \int_{y}^{33} \frac{x \cos z}{z} d z d y d x \)
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  Express \( x=t, y=\tan ^{-1}\left(t^{0}+e^{t}\right) \) in the form \( y=f(x) \) by eliminating the parameter \[ y= \]
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  \( \int_{0}^{\pi} \int_{\sin (x)}^{3 \sin (x)}[x(1+y)] d y d x \)
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  Given the spherical coordinates \( x=r \cos \theta \sin \gamma, y=r \sin \theta \sin \gamma, z=r \cos \gamma \) show that the Jacobian \[ J=\frac{\partial(x, y, z)}{\partial(r, \theta, \gamma)}=r^{2}
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• will leave upvote☺️
  For the given parametric equations, find the points \( (x, y) \) corresponding to the parameter values \( t=-2,-1,0,1,2 \). \[ \begin{array}{ll} & x=9 t^{2}+9 t, \quad y=3^{t+1} \\ t=-2 & (x, y)=( \\
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  Solve the following differential equations, then give the particular solution with \( y(0)=1 \). \[ y^{\prime}=-2 y^{2} x^{2}, \quad y^{\prime}=\frac{e^{-x}}{y^{2}}, \quad y^{\prime}=y \sin x, \quad y
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• Find the general solution of y′′ + y′ + 2y = sin x.
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• Find the general solution of y′′ + y′ + y = ex
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  20] \#7 Find a particular solution of the D.E. \[ y^{\prime \prime}+y=(8-4 x) \cos x-(8+8 x) \sin x \]
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  Solve using Laplace transform \[ \begin{aligned} y^{\prime \prime}+2 y^{\prime}+y=\delta(t) & \\ t=0: \quad & y=1 \\ & y^{\prime}=0 \end{aligned} \]
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