Calculus Archive: Questions from June 06, 2023
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Convert \( \int_{0}^{4} \int_{0}^{\sqrt{16-y^{2}}} \int_{\sqrt{x^{2}+y^{2}}}^{\sqrt{32-x^{2}-y^{2}}}\left(x^{2}+y^{2}+z^{2}\right) d z d x d y \) into spherical coordinates. \[ \begin{array}{l} \int_{2 answers -
Calcular la integral doble \( \iint_{R} \quad y \operatorname{sen}(x y) D a \) Usando cambio de variables, siendo \( R \) la región entre \[ \begin{array}{l} x y=5 \\ x y=1 \\ y=5 \\ y=1 \end{array}2 answers -
solve the following exercises
(a) Sea \( f(x)=6 x^{2}+x-5 \). Encuentre \( F(x) \) si se sabe que \( F(0)=2 \). (b) Sea \( f(x)=12 x^{2}-6 x+1 \). Encuentre \( F(x) \) si se sabe que \( F(1)=5 \). (c) Sea \( f^{\prime}(x)=9 x^{2}+2 answers -
2. (5 points) Evaluate \( \iint_{R}(3 x+2 y) d A \), where \( R=\left\{(x, y): \frac{x^{2}}{36}+\frac{y^{2}}{16} \leq 1, x \geq\right. \) \( 0, y \geq 0\} \)2 answers -
2. The plane \( x+y+z=1 \) intercepts the eliptic paraboloid \( z=x^{2}+2 y 2 \) in an ellipse. Using Lagrange's Method, find the closest and furthest points to the origin on the ellipse. 3.Using Lagr2 answers -
(3) Evaluate the Jacobian \( \partial(x, y, z) / \partial(\rho, \phi, \theta) \), where \[ x=\rho \sin \phi \cos \theta, y=\rho \sin \phi \sin \theta, \text { and } z=\rho \cos \phi \]2 answers -
Solve the following ODEs: (a) \( y^{\prime}=e^{2 x-1} \cdot y^{2} \) (b) \( x y^{\prime}=y+3 x^{4} \cos ^{2}(y / x), y(1)=0 \). (c) \( \left(x^{2}+y^{2}\right) d x-2 x y d y=0 \) (d) \( y d x+(y+\tan2 answers -
Solve the following IVPs: (a) \( y^{\prime \prime}+4 y^{\prime}+4 y=e^{-2 x} \sin 2 x, y(0)=1, y^{\prime}(0)=-1.5 \). (b) \( y^{\prime \prime}+9 y=\sec 3 x \). (c) \( y^{\prime \prime}+6 y^{\prime}+92 answers -
Given the function \( f(x, y)=\frac{x^{2}+y^{2}}{24 y-6 x y+90 x-360} \), the domain of \( f(x, y) \) is \( \left\{(x, y) \in \mathbb{R}^{2}, x \neq A, y \neq B\right\} \). Find \( A= \) and \( B= \)2 answers -
\[ y^{(4)}+2 y^{(3)}+2 y^{\prime \prime}=3.5 t+2 \sin t \] The general solution is \[ \begin{array}{l} y_{g}=c_{1}+c_{2} t+c_{3} e^{-t} \cos t+c_{4} e^{-t} \sin t+A t^{3}+B t^{2}+D t \cos t+E t \sin t2 answers -
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\( \begin{array}{l}\frac{d y}{d x} \text { if } y=\sqrt[7]{x^{3}-5 x^{4}+7 x} \\ \frac{d y}{d x}=\frac{1}{7}\left(3 x^{2}-20 x^{3}+7\right)^{(-6 / 7)} \\ \frac{d y}{d x}=\frac{1}{7}\left(3 x^{2}-20 x^2 answers -
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Sea f dos veces diferenciable con f(0)=2, f(1)=5 y f'(1)=8. Evalúa la integral xf"(x)dx entre 0 y 1.2 answers
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Q3. Find \( \nabla \cdot(\nabla \times \boldsymbol{F}) \) and \( \nabla \times(\nabla \times \boldsymbol{F}) \) if \( \boldsymbol{F}(x, y, z)=e^{y z} x \boldsymbol{i}+\cos (x y- \) \( 3 z) \boldsymbol2 answers