Calculus Archive: Questions from November 20, 2022
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1. \( \int \sin ^{-1}(2 \sqrt{x}) d x \) 2. \( \int \tan ^{7} x \sec ^{7} x d x \) 3. \( \int \tan ^{8} x \sec ^{8} x d x \) 4. \( \int x^{5} \cosh \mathrm{x} d x \) 5. \( \int e^{3 x} \cos (5 x) d x2 answers -
Evaluate the triple integral \( \iiint_{Q} f(x, y, z) d V \). \[ \begin{array}{l} f(x, y, z)=5 x+8 y-9 z, Q=\{(x, y, z) \mid 0 \leq x \leq 9,-4 \leq y \leq 4,1 \leq z \leq 4\} \\ \iiint_{Q} f(x, y, z)2 answers -
Evaluate the triple integral \( \iiint_{Q} f(x, y, z) d V \). \[ f(x, y, z)=5 x+8 y-9 z, Q=\{(x, y, z) \mid 0 \leq x \leq 9,-4 \leq y \leq 4,1 \leq z \leq 4\} \] \[ \iiint_{Q} f(x, y, z) d V= \]2 answers -
(40 points) Solve the initial value problem \[ y^{\prime \prime}+1 x y^{\prime}-4 y=0, y(0)=7, y^{\prime}(0)=0 \] \[ y= \]2 answers -
pleas slove13 and 15 with cleat steps thank u
11-16 Use the Chain Rule to find \( \partial z / \partial s \) and \( \partial z / \partial t \). 11. \( z=(x-y)^{5}, \quad x=s^{2} t, \quad y=s t^{2} \) 12. \( z=\tan ^{-1}\left(x^{2}+y^{2}\right), \2 answers -
Change the order of integration. \[ \int_{0}^{1} \int_{0}^{10 x} f(x, y) d y d x \] \[ \int_{0}^{10} \int_{1}^{\frac{1}{10} y} f(x, y) d x d y \] \[ \begin{array}{l} \int_{0}^{10} \int_{\frac{1}{10} y2 answers -
(iii) Evaluate the integral \( \iint_{D}\left(3 x^{2}+y^{2}\right) d A \) where \( =\left\{(x, y) \mid-2 \leq y \leq 3, y^{2}-3 \leq x \leq y+3\right\} \). Solution: (iv) Evaluate the integral \( \iin2 answers -
2 answers
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Choose the correct formula for the vector field. \[ \begin{array}{l} \vec{F}(x, y)=y \vec{j} \\ \vec{F}(x, y)=-y \vec{j} \\ \vec{F}(x, y)=y \vec{i} \\ \vec{F}(x, y)=-y \vec{i} \\ \vec{F}(x, y)=x \vec{2 answers -
Evaluate \( \iiint_{\mathcal{B}} f(x, y, z) d V \) for the specified function \( f \) and \( \mathcal{B} \) : \[ f(x, y, z)=\frac{z}{x} \quad 3 \leq x \leq 24,0 \leq y \leq 9,0 \leq z \leq 8 \] \[ \ii2 answers -
Find r(t) for the following condition
1. Halle \( r(t) \) para la siguiente condición \( r^{\prime}(t)=4 e^{2 t} i+3 e^{t} j, r(0)=2 i \) 2. Halle \( r^{\prime \prime}(t) \) de la siguiente función \( r(t)=4 \cos t i+4 \sin t j \)2 answers -
(iv) Evaluate the integral \( \iint_{D} x^{2} e^{x y} d A \) where \( D=\left\{(x, y) \mid 0 \leq x \leq 2, \frac{1}{2} x \leq y \leq 1\right\} \). Solution:2 answers -
Evaluate the triple integral. \[ \iiint_{E} y d V \text {, where } E=\{(x, y, z) \mid 0 \leq x \leq 2,0 \leq y \leq x, x-y \leq z \leq x+y\} \]2 answers -
5. Differentiate Implicitly a. \( x^{4}(x+y)=y^{2}(3 x-y) \) b. \( \tan (x-y)=\frac{y}{1+x^{2}} \)2 answers -
2) [20 pts] If \( f(x, y, z)=2 x^{2}-y z+x z^{2} \), where \( x=2 \sin t, y=t^{2}-t+1 \), and \( z=e^{-3 t} \), find \( f^{\prime} t \) )2 answers -
2 answers
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1. Find r (t) for the following condition r' (t) = 4e2ti + 3e*j, r (0) = 2i 2. Find p"'(t) of the following function r (t) = 4 cost i + 4 sin t j 3. Determine the maximum height and horizontal dis
Halle \( r(t) \) para la siguiente condición \( r^{\prime}(t)=4 e^{2 t} i+3 e^{t} j, r(0)=2 i \) 2. Halle \( r^{\prime \prime}(t) \) de la siguiente función \( r(t)=4 \cos t i+4 \sin t j \) Determin2 answers -
Knowing that (1/3)=/6, use the Mclaurin series of (x) to approximate with an error less than 10^-6
(10 puntos) Sabiendo que \( \arctan (1 / \sqrt{3})=\pi / 6 \), use la serie de Mclaurin de \( \arctan (x) \) para aproximar \( \pi \) con un error menor que \( 10^{-6} \).2 answers -
Use the Mclaurin series of e^x to approximate 01e-x2dx with an error of less than 10-6
4. (10 puntos) Use la serie de Mclaurin de \( e^{x} \) para aproximar \( \int_{0}^{1} e^{-x^{2}} d x \) con un error menor que \( 10^{-6} \)2 answers -
Find \( y^{\prime} \) by implicit differentiation. Match the equations defining \( y \) implicitly with the letters labeling the expressions for \( y^{\prime} \). 1. \( 3 x \cos y+2 \cos 2 y=7 \sin y1 answer -
Simplify: \[ \frac{3 x^{-2} \cdot y^{-4}}{x^{2} y^{-1}}+3 x^{-1} \cdot y\left(x^{-2} y^{-1}\right) \]2 answers -
2 answers
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Find the gradient vector field of the function \( f \). \[ f(x, y, z)=2 x \cos \left(\frac{y}{z}\right) \] \[ \nabla f(x, y, z)= \]2 answers -
\( \operatorname{div}(\operatorname{curl} \mathbf{F})=\nabla \cdot(\nabla \times \mathbf{F}) \) \( \mathbf{F}(x, y, z)=x y z \mathbf{i}+y \mathbf{j}+z \mathbf{k} \)2 answers -
\( \begin{aligned} \operatorname{curl}(\operatorname{curl} F) &=\nabla \times(\nabla \times F) \\ F(x, y, z) &=2 x^{3} y z \mathbf{i}+y \mathbf{j}+2 z \mathbf{k} \end{aligned} \)2 answers -
Evaluate the triple integral. \[ \iiint_{E} y d V, \text { where } E=\{(x, y, z) \mid 0 \leq x \leq 2,0 \leq y \leq x, x-y \leq z \leq x+y\} \]2 answers -
Evaluate the indefinite integral \( \int \frac{2 \gamma^{2}+3 \gamma \sqrt{t}+t^{2}}{\sqrt{t}} d t \).2 answers -
Use logarithmic differentiation to find \( y^{\prime} \). \[ y=\frac{\sqrt{5-7 x}\left(x^{2}+3\right)^{2}}{x^{2}+8 x+2} \] \[ \mathrm{y}^{\prime}= \]2 answers -
1 answer
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Let \( D=\left\{(x, y, z): x^{2}+y^{2}+z^{2} \leq 1, x \geq 0, y \geq 0, z \geq 0\right\} \). Evaluate the following integral. \[ \iiint_{D} x y z d V \]2 answers -
2 answers
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Find the inverse laplace of \[ F(s)=\frac{1}{\left(s^{2}+a^{2}\right)\left(s^{2}+b^{2}\right)} \] consider \[ a^{2} \neq b^{2}, a b \neq 0 \] \[ \begin{array}{l} f(t)=\frac{a \sin a t+b \sin b t}{a^{22 answers -
18
Use logarithmic differentiation to find \( y^{\prime} \). \[ y=\frac{\sqrt{7-2 x}\left(x^{2}+1\right)^{2}}{x^{2}+8 x+3} \] \[ y^{\prime}= \]2 answers -
2 answers
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Find the domain of the function \[ f(x, y)=\sqrt{x^{2}+6 y^{2}-4} \text {. } \] 1. \( \left\{(x, y): \frac{1}{6} x^{2}+\frac{1}{4} y^{2} \geq 1\right\} \) 2. \( \left\{(x, y): \frac{1}{4} x^{2}+\frac{2 answers -
Find the Laplace Transformation of \[ \begin{array}{l} y "+2 y^{\prime}+5 y=8 e^{-x} \\ \text { when } y(0)=0, y^{\prime}(0)=8 \end{array} \] \[ \begin{array}{l} Y(s)=\frac{2 s^{2}+8 s+18}{s^{3}+3 s^{2 answers -
2 answers
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2 answers
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Find the solution to each differential equation problem. Mensajes Respuestas enviadas \[ y^{\prime}=4 x^{3}+3 x^{2} \text { if } y(0)=4 \] \[ \frac{d y}{d x}=e^{4 x} \text { if } y(0)=2 \]2 answers -
Evaluate the following definite integrals. (Do not approximate with technology.) Mensajes Respuestas enviadas \( \int_{0}^{4}(9-4 x) d x \) \( \int_{0}^{3} x\left(8 x^{2}+9\right)^{-1 / 2} \mathrm{~d}2 answers -
\( y=\ln \sqrt[4]{2-3 x^{2}} \) \( y=e^{-x} \ln x \) \( y=\ln \sqrt[5]{1-2 x} \) \( y=\ln \frac{\sqrt{4+x^{2}}}{x} \) \( f(x)=x(3 x-9)^{3} \) \( y=\left(\frac{6-5 x}{x^{2}-1}\right)^{2} \)2 answers