Calculus Archive: Questions from November 11, 2022
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( 40 points) Solve the initial value problem \[ y^{\prime \prime}+5 x y^{\prime}-20 y=0, y(0)=3, y^{\prime}(0)=0 \]2 answers -
Evaluate the integral. \[ \int_{0}^{4} \int_{-\sqrt{1} 6-x^{2}}^{\sqrt{ } 16-x^{2}} \int_{-\sqrt{1} 6-x^{2}-z^{2}}^{\sqrt{1} 16-x^{2}-z^{2}} \frac{1}{\left(x^{2}+y^{2}+z^{2}\right)^{1 / 2}} d y d z d2 answers -
Evaluate the double integral. \[ \iint_{D} 9 y^{2} d A, \quad D=\{(x, y) \mid-1 \leq y \leq 1,-y-2 \leq x \leq y\} \]3 answers -
Match each function with one of the graphs below. A D 1. \( f(x, y)=e^{-y} \) 2. \( f(x, y)=\sqrt{4 x^{2}+y^{2}} \) 3. \( f(x, y)=y^{2}+1 \) 4. \( f(x, y)=1+y \)2 answers -
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Find the derivative of the function. \[ \begin{array}{c} G(y)=\frac{(y-1)^{4}}{\left(y^{2}+2 y\right)^{8}} \\ G^{\prime}(y)=\frac{\left(\left(y^{2}+2 y\right)^{8} \cdot 4(y-1)^{3}-(y-1)^{4} \cdot 8\le2 answers -
Find the gradient vector field \( (\vec{F}(x, y, z)) \) of \( f(x, y, z)=x^{3} y^{4} z^{6} \). \[ \vec{F}(x, y, z)=\langle \]2 answers -
Find the gradient vector field \( (\vec{F}(x, y, z)) \) of \( f(x, y, z)=y^{2} \sin (4 z x) \). \[ \vec{F}(x, y, z)=\langle \]2 answers -
Find the divergence of the vector field \( \mathbf{F} \). \[ F(x, y, z)=\ln \left(3 x^{2}+4 y^{2}\right) \mathbf{i}+12 x y \mathbf{j}+\ln \left(4 y^{2}+7 z^{2}\right) \mathbf{k} \] \( \operatorname{di2 answers -
\( \mathbf{F}(x, y, z)=\sin y \mathbf{i}+(x \cos y+\cos z) \mathbf{j}-y \sin z \mathbf{k} \) \( C: \mathbf{r}(t)=\sin t \mathbf{i}+t \mathbf{j}+2 t \mathbf{k}, \quad 0 \leqslant t \leqslant \pi / 2 \)0 answers -
Solve the initial value problem \[ y^{\prime \prime}+3 x y^{\prime}-12 y=0, y(0)=4, y^{\prime}(0)=0 \] \[ y= \]2 answers -
( 1 point) Find \( y \) as a function of \( x \) if \[ y^{\prime \prime \prime}-11 y^{\prime \prime}+30 y^{\prime}=80 e^{x}, \] \( y(0)=20, y^{\prime}(0)=10, y^{\prime \prime}(0)=24 \). \( y(x) \)2 answers -
Solve the following differential equation.
(1) \( 3 y^{\prime}+4 y=0 \) (2) \( y^{\prime \prime}-5 y^{\prime}+6 y=0 \) (3) \( y^{\prime \prime \prime}-5 y^{\prime \prime}-y^{\prime}+5 y=0 \) (4) \( y^{\prime \prime}+9 y=0 \)2 answers -
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\( \int_{C} \mathbf{F} \cdot d \mathbf{r} \) \( \mathbf{F}(x, y)=x \mathbf{i}+y \mathbf{j} \) \( \quad C: \mathbf{r}(t)=(3 t+1) \mathbf{i}+t \mathbf{j}, \quad 0 \leq t \leq 1 \)3 answers -
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