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Calculus Archive: Questions from October 11, 2022

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  Differentiate the function. \[ y=\frac{5}{x^{2}+x+7} \]
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  Differentiate the function. \[ y=\frac{7}{\pi}+\frac{6}{x^{2}+9} \]
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  In exercises \( 1[\sqrt[[0]-6]{[\subseteq]} \), describe and sketch the dor 1. \( f(x, y)=\frac{1}{x+y}\{\mathrm{~A}: \) 2. \( f(x, y)=\frac{3 x y}{y-x^{2}} \) 3. \( f(x, y)=\ln \left(x^{2}+y^{2}-1\ri
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• Solve the non-exact differential equation xydx+ (1 + x^2)dy = 0, y(1) = 1, determining its factor integrating and then solve it.
  Resuelva la ecuación diferencial no exacta \( x y d x+\left(1+x^{2}\right) d y=0 \), \( y(1)=1 \), determinando su factor integrando y luego resuélvala.
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• Solve the differential equation y' + (3/x)(y) = x^4 y^1/3
  Resuelva la ecuación diferencial \( y^{\prime}+\frac{3}{x} y=x^{4} y^{1 / 3} \).
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• a. Find the differential equation whose general solution is y = c1 cos2x + c2 sin2x b.Determine if it exists and write in the plane (𝑥, 𝑦) a region in which the existence and uniqueness theorem
  Halle la ecuación diferencial cuya solución general es \[ y=c_{1} \cos 2 x+c_{2} \sin 2 x \] Determine si existe \( \mathrm{y} \) escribala en el plano \( (x, y) \) una región en la cual el teorema
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  6. Find the general solution of the following separable ODEs. (a) \[ \left(1+x^{2}\right) \frac{d y}{d x}=1+y^{2} \] Hint: You may use \[ \tan (x+y)=\frac{\tan x+\tan y}{1-\tan x \tan y} \] (b) \[ \fr
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  7. Solve the following initial-value problems. (a) \[ \frac{d y}{d x}=(1-y)(2-y), \quad y(0)=0 . \] (b) \[ \cos y \frac{d y}{d x}=\frac{-x \sin y}{1+x^{2}}, \quad y(1)=\pi / 2 \]
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  Find \( \partial y / \partial x_{1} \) and \( \partial y / \partial x_{2} \) for each of the following functions: (a) \( y=2 x_{1}^{3}-11 x_{1}^{2} x_{2}+3 x_{2}^{2} \) (c) \( y=\left(2 x_{1}+3\right)
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• Calculate y' (first derivative): a) b)
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• Determine if it exists and write in the plane (𝑥, 𝑦) a region in which the existence and uniqueness theorem holds for the initial value problem dy/dx = tany, y(π/4) =1
  Determine si existe y escribala en el plano \( (x, y) \) una región en la cual el teorema de existencia y unicidad se cumpla para el problema de valor inicial \( \frac{d y}{d x}=\tan y, \mathrm{y}\le
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  Let \( f(x, y, z)=\frac{x^{2}-4 y^{2}}{y^{2}+4 z^{2}} \). Then \[ \begin{array}{l} f_{x}(x, y, z)= \\ f_{y}(x, y, z)= \\ f_{z}(x, y, z)= \end{array} \]
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  5. Find \( y^{\prime} \) if \( y=\sqrt{\frac{x^{2}-5}{x^{2}+4}} \) two different ways.
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• Find the volume of the smallest region obtained by cutting a sphere of radius a with a plane at a distance h from its center.
  Encontrar el volumen de la región más pequeña que se obtiene al cortar a una esfera de radio \( a \) con un plano a una distancia \( h \) de su centro.
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  Use the rules to differentiate. DO NOT SIMPLIFY 1. \( y=\tan ^{-1}\left(x^{2}\right) \) 2. \( y=\frac{1}{x-\sin x} \) 3. \( y=\ln (\cos (2 x)) \)
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  \[ \frac{d}{d x}\left(\sin \left(7 x^{3}\right)\right)=\cos \left(21 x^{2}\right) \] True False
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  5. La integral que representa el volumen del que se obtiene al rotar la región acotada por \( y=\sqrt{x} ; y=0 \); \( x=0 ; x=1 \) con respecto el eje de \( y \) es a. \( \int_{0}^{1} 2 \pi x^{3} d x
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  6. El promedio de \( g(x)=\sqrt[3]{x} \) en el intervalo \( [1,125] \) es
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• 1. La siguiente serie se puede usar para estimar la tangente inversa de x: arctan(x) = x -x^3/3+x^5/3-x^7/7+... -1<x<1 a. [5%| Verifique a mano que la serie realmente converge a la tangente inve
  1. La siguiente serie se puede usar para estimar la tangente inversa de \( \mathrm{x} \) : \[ \arctan (x)=x-\frac{x^{3}}{3}+\frac{x^{5}}{5}-\frac{x^{7}}{7}+\cdots \quad-1 \leq x \leq 1 \] a. [5\%] Ver
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  ( 1 point) Find \( y \) as a function of \( x \) if \[ y^{\prime \prime \prime}+4 y^{\prime}=0 \] \[ \begin{array}{l} y(0)=4, \quad y^{\prime}(0)=-4, \quad y^{\prime \prime}(0)=12 \\ y(x): \end{array}
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• Find y': y=(3x-5)^sinx
  Question 2: Find \( y^{\prime} \) : \[ y=(3 x-5)^{\sin x} \]
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  Find the general solution of the differential equation \[ \frac{d^{3} y}{d x^{3}}+\frac{d^{2} y}{d x^{2}}-\frac{d y}{d x}-y=0 \] \[ y=C_{1} e^{-x}+C_{2} e^{-2 x}+C_{3} e^{2 x} \] \[ y=C_{1} e^{-x}+\le
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  Solve \[ y^{\prime \prime}+4 y=0, y\left(\frac{\pi}{4}\right)=-1, y^{\prime}\left(\frac{\pi}{4}\right)=-2 \]
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• evaluate the given limit or state that it doesn't exist
  En los problemas 1-4, evalúe el límite dado o enuncie que éste no existe. 3. \( \lim _{t \rightarrow 1}\left\langle\frac{t^{2}-1}{t-1}, \frac{5 t-1}{t+1}, \frac{2 e^{t-1}-2}{t-1}\right\rangle \)
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• with the given info. calculate the limit given
  En los problemas 5 y 6 , suponga que \[ \lim _{t \rightarrow a} \mathbf{r}_{1}(t)=\mathbf{i}-2 \mathbf{j}+\mathbf{k} \quad \mathrm{y} \quad \lim _{t \rightarrow a} \mathbf{r}_{2}(t)=2 \mathbf{i}+5 \ma
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• determine if the vectorial function indicated is continous at t=1
  En los problemas 7 y 8 , determine si la función vectorial indicada es continua en \( t=1 \). 7. \( \mathbf{r}(t)=\left(t^{2}-2 t\right) \mathbf{i}+\frac{1}{t+1} \mathbf{j}+\ln (t-1) \mathbf{k} \)
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• find 2 indicated vectors for the given vectorial function
  En los problemas 9 y 10 , encuentre los dos vectores indicados para la función vectorial dada. 9. \( \mathbf{r}(t)=(3 t-1) \mathbf{i}+4 t^{2} \mathbf{j}+\left(5 t^{2}-t\right) \mathbf{k} ; \quad \mat
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• determine r'(t) and r"(t) for the given vectorial function do only 11
  En los problemas \( 11-14 \), determine \( \mathbf{r}^{\prime}(t) \) y \( \mathbf{r}^{\prime \prime}(t) \) para la función vectorial dada. 11. \( \mathbf{r}(t)=\ln t \mathbf{i}+\frac{1}{t} \mathbf{j}
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• determine r'(t) and r"(t) for the given vectorial function in ex.13
  En los problemas 11-14, determine \( \mathbf{r}^{\prime}(t) \) y \( \mathbf{r}^{\prime \prime}(t) \) para la función vectorial dada. 11. \( \mathbf{r}(t)=\ln t \mathbf{i}+\frac{1}{t} \mathbf{j}, \qua
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• plot the curve C that is described by and plot at the point corresponding to the value r(t) r'(t) indicated of t. do only 15
  En los problemas \( 15-18 \), grafique la curva \( C \) que es descrita por \( \mathbf{r}(t) \) y grafique \( \mathbf{r}^{\prime}(t) \) en el punto correspondiente al valor indicado de \( t \). 15. \(
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• plot the curve C that is described by and plot at the point corresponding to the value r(t) r'(t) indicated of t. do only 17
  En los problemas \( 15-18 \), grafique la curva \( C \) que es descrita por \( \mathbf{r}(t) \) y grafique \( \mathbf{r}^{\prime}(t) \) en el punto correspondiente al valor indicado de \( t \). 15. \(
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• Find parametric equations of the tangent line to the given curve at the point corresponding to the indicated value of t. do only 19
  En los problemas 19 y 20 , encuentre ecuaciones paramétricas de la recta tangente a la curva dada en el punto correspondiente al valor que se indica de \( t \). 19. \( x=t, y=\frac{1}{2} t^{2}, z=\fr
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• Determine a unit tangent vector to the given curve at the point corresponding to the indicated value of t. Find parametric equations of the tangent line at this point. do only 21
  En los problemas 21 y 22 , determine un vector tangente unitario para la curva dada en el punto correspondiente al valor que se indica de \( t \). Encuentre ecuaciones paramétricas de la recta tangen
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• just 18,22, and 28
  \[ \begin{array}{l} f(t)=t \sin x t \\ f(t)=e^{\text {at }} \sin b t \\ A(r)=\sqrt{r} \cdot e^{r^{3}+1} \\ F(x)=(4 x+5)^{3}\left(x^{2}-2 x+5\right)^{4} \end{array} \] \[ G(z)=(1-4 z)^{2} \sqrt{z^{2}+1
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  \( f(x)=x^{5}+23 x-3750 \) \( f(x)=7 x^{4}-6 x^{3}+140 x \) \( y=\sqrt[4]{x}+85 x \)
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• Use Euler's method with step size 0.5 to compute the approximate y-values y1, y2, y3, and y4 of the solution of the initial-value problem y' = y − 3x, y(4) = 2 y1 = ______ y2= ______ y3 = ______ y4=
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  Let \( x^{3}+y^{3}=126 \). Find \( y^{\prime \prime}(x) \) at the point \( (5,1) \). \[ y^{\prime \prime}(5)= \] If \( \sqrt{x}+\sqrt{y}=12 \) and \( y(49)=25 \), find \( y^{\prime}(49) \) by implici
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• 6.12) Un profesor de sexto año de primaria acostumbra hacer al inicio de cada año escolar un intervalo de confianza del 95% con los resultados de una prueba estandarizada para medir el coeficiente d
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  Evaluate the integral. \[ \int 19 \sin ^{2} x \cos ^{2} x d x \] \[ \int 19 \sin ^{2} x \cos ^{2} x d x= \]
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  Given \( f(x, y)=-x^{6}+2 x^{2} y^{3}-2 y^{4} \) \( f_{x}(x, y)= \) \( f_{y}(x, y)= \) \( f_{x x}(x, y)= \) \( f_{x y}(x, y)= \)
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  Find the first partial derivatives of the function. \[ \begin{array}{l} f(x, y, z, t)=\frac{x y^{4}}{t+6 z} \\ f_{x}(x, y, z, t)= \\ f_{y}(x, y, z, t)= \\ f_{z}(x, y, z, t)= \\ f_{t}(x, y, z, t)= \end
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  Let \( f(x, y, z)=y^{2} e^{x z} \), find \( \partial f /\left.\partial z\right|_{(2,1,1)} \).
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• Solve the following integral
  10 Integre lo siguiente paso a paso \( f(x)=15 \int_{0}^{10} e^{-15(x-20)} d x \)
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• NEED HELP ASAP Find d^2y/dx^2 for y = 3x+ 8x^2+5x^3
  Find \( \frac{d^{2} y}{d x^{2}} \) for \( y=3 x+8 x^{2}+5 x^{3} \) A) \( 16 x^{2}+5 x^{3} \) B) \( 16+30 \mathrm{x} \) C) \( 31 \mathrm{x} \) D) \( 3+5 x \) E) \( 8 x+5 x^{2} \) F) \( 3+31 \mathrm{x}
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• find the center of mass of the solid Que of uniform density find center of mass moment of inertia in X moment of inertia in Y radius of rotation in X radius of rotation in Y
  9) Halle el centro de masa del sólido \( Q \) de densidad uniforme, limitado por abajo por la hoja superior del cono \( z^{2}=x^{2}+y^{2} \) y por arriba por la esfera \( x^{2}+y^{2}+z^{2}=16 \) (uti
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  4. (Section 16.3) Evaluate \( \iint_{R} e^{-3 x^{2}-3 y^{2}} d A ; \quad R=\left\{(x, y) \mid x^{2}+y^{2} \leq 4, x \leq 0, y \geq 0\right\} \)
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  1. Evaluate the integral, if possible: \( \quad \int_{0}^{\infty} x e^{-x^{2}} d x \) 2. Evaluate the integral, if possible: \( \int_{0}^{\infty} x e^{-x} d x \)
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• 1. Find the equation of the plane determined by the points P(1,2,3) Q(2,3,1) R(0,-2,-1) 2. Find the parametric and symmetric equations for the line passing through the points P(3,0,2) and Q(1,-4,0)
  1. Encuentre la ecuación del plano determinado por los puntos \( \mathrm{P}(1,2,3) \) \[ Q(2,3,1) \quad R(0,-2,-1) \] 2. Encuentre las ecuaciones paramétricas y simétricas para la linea que pasa po
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• find y' of 3x^2y^4 - 5xy^2 = 4xy + y^3 and x^3 - 5x^2y^3 = x + y
  \( \left(3 x^{2}\right)^{\prime} y^{4}+3 x^{2}\left(y^{4}\right)^{\prime}-(5 x \) \( \left(3 x^{2}\right)^{\prime} y^{4}+3 x^{2}\left(y^{4}\right)^{\prime}-(5 x \) \( - \) \( x^{3}-5 x^{2} y^{3}=x+y
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  Differentiate \( y=\frac{-8 \sin ^{2} x}{\cos x} \) \( y^{\prime} \)
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  1. Find \( \mathrm{dy} / \mathrm{dx} \) a) \( y=\cos ^{2} x-\sin (\tan x)+\cot \left(e^{x}\right) \) \( y=\frac{\left(2 x^{3}+3\right)^{4}}{\left(6-5^{x}\right)^{5}} \) c) \( y=\csc \left(5 x^{2}\righ
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• 5. let s(u) = 5e^2u^3 +2u +7/sqrt u^3
  \( s(u)=5 e^{2 u^{3}+2 u}+\frac{7}{\sqrt{u^{3}}} \)
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