Calculus Archive: Questions from March 12, 2023
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1. Let \( \mathbf{r}(x, y, z)=(x, y, z) \) and \( r(x, y, z)=\sqrt{x^{2}+y^{2}+z^{2}}=\|\mathbf{r}\| \). Verify the following identities. (a) \( \nabla\left(\frac{1}{r}\right)=-\frac{\mathbf{r}}{r^{3}2 answers -
Q6. [10 marks] Let \( f(x, y)=\frac{x-y}{x+y} \). Determine if \[ \lim _{(x, y) \rightarrow(0,0)} f(x, y) \] exists or not.2 answers -
If \( z=(x+y) e^{x} \) and \( x=1+t^{2} \) and \( y=3 t \), find \( \frac{\partial z}{\partial t} \).2 answers -
if \( y=\frac{\sqrt{x^{3}+x}}{x^{2}+1} \), compute \( y^{\prime} \) (A) \( y^{\prime}=\frac{\left(x^{2}-1\right)\left(3 x^{2}+1\right)-4 x\left(x^{3}-x\right)}{\left(x^{2}+1\right)^{2}} \) (B) \( y^{\2 answers -
\( y^{\prime} \) for \( y=\frac{x^{2}-3}{\sqrt{x^{4}+2 x}} \) A) \( y^{\prime}=\frac{2 x\left(x^{4}+2 x\right)-\left(x^{2}-3\right)\left(4 x^{3}+2\right)}{\left(x^{4}+2 x\right) \sqrt{x^{4}+2 x}} \) (2 answers -
What is \( \frac{d}{d x}\left[f(x)^{g(x)}\right] \) where \( f(x)=\sin (3 x) \) and \( g(x)=3 x ? \) \[ \begin{array}{l} 3(\sin 3 x)^{x} \ln (\sin 3 x)-9 x(\sin 3 x)^{3 x}(\cot 3 x) \\ 3(\sin 3 x)^{32 answers -
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Solve the differential equation. \[ \begin{array}{l} 3 y^{\prime \prime}+17 y^{\prime}+24 y=0 \\ y=C_{1} e^{-3 x}+C_{2} e^{-8 x} \\ y=C_{1} e^{-3 x}+C_{2} e^{(-8 / 3) x} \\ y=C_{1} e^{3 x}+C_{2} e^{(82 answers -
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0) If \( y=\sec x \tan x \), then \( \frac{d y}{d x}= \) a) \( \sec x \) b) \( \sec ^{2} x(\sec x+\tan x) \) c) \( \sec x\left(\sec ^{2} x+\tan ^{2} x\right) \) d) \( -\frac{\cos ^{2} x}{\sin x} \)2 answers -
find the solution to the following differential equations
1. [20 pts.] Halla la solución de las siguientes ecuaciones diferenciales: a) \( y^{\prime}+4 x y=x e^{x^{2}} \) b) \( \left(x+y e^{\frac{y}{x}}\right) d x-x e^{\frac{y}{x}} d y=0 \)2 answers -
verify that the differential equation is exact, then find its solution.
[10 pts.] Verifica que la ecuación diferencial \( (\tan x-\sin x \sin y) d x+\cos x \cos y d y=0 \) es exacta. Luego, halla su solución.2 answers -
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Which of the following is a homogenous differential equation?
5. ¿Cuál de las siguientes ecuaciones diferenciales es una ecuación diferencial homogénea? a) \( \frac{d y}{d x}=\frac{1-x-y}{x+y} \) d) \( \frac{d y}{d x}=(x+y+1)^{2} \) b) \( \frac{d y}{d x}+\fr2 answers -
help
2. La siguiente gráfica pertenece a la función: \[ \begin{array}{l} r(t)=t \boldsymbol{i}+2 t \boldsymbol{j}+t^{2} \boldsymbol{k}-2 \leq t \leq 2 \\ r(t)=\cos (\pi t) \boldsymbol{i}+\sin (\pi t) \bo2 answers -
answer
3. La siguiente gráfica pertenece a la función: \[ \begin{array}{l} r(t)=t \boldsymbol{i}+2 t \boldsymbol{j}+t^{2} \boldsymbol{k}-2 \leq t \leq 2 \\ r(t)=\cos (\pi t) \boldsymbol{i}+\sin (\pi t) \bo2 answers -
Find the differential, dy
Let \( y=8 x^{2} e^{25 x} \) Find the differential, \( d y \). \[ d y=16 x e^{25 x}+200 x^{2} e^{25 x} \] \[ d y=\left(16 x e^{25 x}+200 x^{2} e^{25 x}\right) d x \] \[ d y=400 x e^{25 x} d x \] \[ d2 answers -
triplelntegrals: Problem 1 (1 point) Evaluate \( \iiint_{B} f(x, y, z) d V \) for the specified function \( f \) and \( \mathcal{B} \) : \[ f(x, y, z)=\frac{z}{x} \quad 3 \leq x \leq 9,0 \leq y \leq 72 answers -
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3. Find \( \partial z / \partial u \) and \( \partial z / \partial v \) if \( z=x e^{y}, x=\ln u \), and \( y=v \)2 answers -
Find the Maclaurian series for the function \( f(x)=\sin 8 x \). \[ \sin 8 x=\sum_{n=0}^{\infty} \frac{(-1)^{n}(8 x)^{2 n+1}}{(2 n+1) !} \] \[ \sin 8 x=\sum_{n=0}^{\infty} \frac{(8 x)^{2 n+1}}{(2 n+1)2 answers -
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Given \( f(x, y)=3 x^{3}-6 x^{2} y^{6}-y^{5} \) \[ f_{x}(x, y)= \] \[ f_{y}(x, y)= \] \[ f_{x x}(x, y)= \] \[ f_{x y}(x, y)= \]2 answers -
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Given \( f(x, y)=3 x^{3}-6 x^{2} y^{6}-y^{5} \) \[ f_{x}(x, y)= \] \[ f_{y}(x, y)= \] \[ f_{x x}(x, y)= \] \[ f_{x y}(x, y)= \]2 answers -
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26.) Apligu af Teorems de Rolle Sea \( f(x)=x^{4}-2 x^{2} \) Encmentro of wafur de \( c \) en el \( (-2,2) \), tal gue \( f^{\prime \prime}(e)=0 \) (Prede hnLm mes dencc)2 answers -
\( s: f(x)=\frac{6}{x^{2}+3} \) Encuentra la concrwidad de la gmfica endönde as cinvava hacia e hecia ablio2 answers -
III. Plantear y resolver la integral (ó integrales) para determinar la medida del volumen que se forma al girar la región limitada por las gráficas de las funciones \( x+y^{2}=2, \quad x+y=0 \), al2 answers -
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evaluate the integrals please
\( \int \sin ^{3} x \cos ^{5} x d x \) \( \begin{array}{c}\int \tan ^{3} x d x \\ \int \sin 3 x \sin 5 x d x\end{array} \)2 answers -
the 22 question
15-24. Domains Find the domain of the following functions. 15. \( f(x, y)=2 x y-3 x+4 y \) 16. \( f(x, y)=\cos \left(x^{2}-y^{2}\right) \) 17. \( f(x, y)=\sqrt{25-x^{2}-y^{2}} \) 18. \( f(x, y)=\frac{2 answers -
1. Let \( \mathbf{r}(x, y, z)=(x, y, z) \) and \( r(x, y, z)=\sqrt{x^{2}+y^{2}+z^{2}}=\|\mathbf{r}\| \). Verify the following identities. (a) \( \nabla\left(\frac{1}{r}\right)=-\frac{\mathbf{r}}{r^{3}2 answers -
true or false about differential equations 1. y(x)=2x is a solution for the differencial ecuation y'=2 2. the first order differential equation is not separable.
1. \( y(x)=2 x \) es una solución de la ecuación diferencial \( y^{\prime}=2 \). 2. La ecuación diferencial de primer orden \( \frac{d y}{d x}=x y+x+y+1 \) no es separable.2 answers -
41 question
35-54. Continuity At what points of \( \mathbb{R}^{2} \) are the following functions continuous? 35. \( f(x, y)=x^{2}+2 x y-y^{3} \) 36. \( f(x, y)=\frac{x y}{x^{2} y^{2}+1} \) 37. \( p(x, y)=\frac{42 answers -
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Evaluate the double integral.
\( \iint_{D} 6 y^{2} e^{x y} d A, D=\{(x, y) \mid 0 \leq y \leq 5,0 \leq x \leq y\} \)2 answers -
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Solve the following DE 1. (D^4 + 3D^3 - 4D^2)y = 0 ; where y(0) = 0, y'(0) = 0, y'(0) = 1, y'(0) = k. What is k if y = 2 + 3x + e1-4x/5 - 11ex/5 2. (D^2 - 4D +4)y = 6(e^2x - x) 3.
I. Solve the following differential equations: 1. \( \left(D^{4}+3 D^{3}-4 D^{2}\right) y=0 \) where \( y(0)=0, y^{\prime}(0)=0, y^{\prime \prime}(0)=1, y^{\prime \prime \prime}(0)=k \). What is \( k2 answers