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Calculus Archive: Questions from December 08, 2022

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  Solve the IVP: \[ \frac{d y}{d x}+(\tan x) y=x \cos (x), \quad y(0)=1 \]
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  c) Compute the Jacobian matrix for the following system of three equations: \[ \begin{array}{l} f(x, y, w)=3 x^{2}+2 w^{3}-w y^{2} \\ g(x, y, w)=2 x y^{2}-5 y^{\frac{1}{3}}-w \\ h(x, y, w)=x^{2} w^{2}
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• ​​​​​​​ Part II: Evaluate f(2, 1) and f(2.1, 1.05) for the following functions of two variables and calculate Az. Use total differentiation dz to approximate Δz: 1. f(x, y) = 2x-3y 2. f
  Evalúe \( \mathrm{f}(2,1) \) y \( \mathrm{f}(2.1,1.05) \) para las siguientes funciones de dos variables y calcular \( \Delta z \). Utlice la diferenciación total dz para aproximar \( \Delta z \) :
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  Oeter mine whether the series converges of diverges (1) \( \sum\left(1-\frac{1}{k}\right)^{k} \) (2) \( \sum \frac{1}{\sqrt[3]{k^{2}}} \)
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  1. \( \lim _{x \rightarrow 0} \frac{\sin (5 x)}{7 x}= \) 2. \( \lim _{x \rightarrow 2} \frac{\sin (5 x)}{7 x}= \) 3. \( \lim _{x \rightarrow 0} \frac{\sin (x)}{7}= \) 4. \( \lim _{x \rightarrow 0} \fr
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  Evaluate the double integral. \[ \iint_{D}(2 x+y) d A, \quad D=\{(x, y) \mid 1 \leq y \leq 2, y-1 \leq x \leq 1\} \]
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  Find the derivative. \[ y=7 x^{2} e^{3 x} \] A. \( 14 x e^{3 x}(2 x+3) \) B. \( 7 x e^{3 x}(3 x+2) \) C. \( 14 e x^{3 x}(3 x+2) \) D. \( 7 x e^{3 x}(2 x+3) \)
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  Let \( y \) be the solution of IVP \[ y^{\prime \prime \prime}+3 y^{\prime \prime}+3 y^{\prime}+y=0, y(0)=1, y^{\prime}(0)=0, y^{\prime \prime}(0)=1 \text {. Then } y(-1)= \] a. \( -e \) b.e c. \( 2 e
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  \( f(x, y)=y \ln \)
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  \( f(x, y)=y \ln \left(\frac{1}{x}\right) \)
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• help please
  Given \( f(x, y)=3 x^{3}+2 x y^{2}-y^{6} \) \[ f_{x x}(x, y)= \] \[ f_{x y}(x, y)= \]
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  Find each limit. \[ f(x, y)=6 x^{2}+5 y^{2} \] (a) \( \lim _{\Delta x \rightarrow 0} \frac{f(x+\Delta x, y)-f(x, y)}{\Delta x} \) (b) \( \lim _{\Delta y \rightarrow 0} \frac{f(x, y+\Delta y)-f(x, y)}{
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• Discover the inverse laplace transformation of the following
  з [20 puntos] Halle la transformada inversa de Laplace de: \[ \text { a } F(s)=\frac{2 s\left(e^{-\pi s}-e^{-2 \pi s}\right)}{s^{2}+4} \] \[ \text { b } F(s)=\frac{s-1}{s^{2}+2 s+2} \]
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• Given the differential equation a region that satisfies the existence and uniqueness of the DE is:
  Dada la ecuación diferencial \( \left(x^{2}+y^{2}\right) y^{\prime}=y^{2} \), una región que satisface la existencia y unicidad de la ED es: a. Región para todo \( x y y>0 \) b. región done \( x>0
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• calculate the double integral
  \( \iint_{R} \frac{x y^{2}}{x^{2}+1} d A, \quad R=\{(x, y) \mid 0 \leqslant x \leqslant 1,-3 \leqslant y \leqslant 3\} \)
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• x=c1cost+c2sint is a solution of the differential equation x′′+x=0 with initial conditions x(π/4)=√2,x′(π/4)=2√2, the values ​​of the constants are (the answer must be entered separate
  \( x=c_{1} \operatorname{cost}+c_{2} \sin t \) es una solución de la ecuación diferencial \( x^{\prime \prime}+x=0 \) con condiciones iniciales \( x(\pi / 4)=\sqrt[2]{2}, x^{\prime}(\pi / 4)=2 \sqrt
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  15) \( \int \frac{\left(\sin ^{-1} x\right)^{3}}{\sqrt{1-x^{2}}} d x= \) a) \( \frac{1}{4}\left(\sec ^{-1} x\right)^{4}+c \) b) \( -3 \sin ^{-2} x+c \) c) \( \frac{1}{4}\left(\sin ^{-1} x\right)^{4}+c
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  Find the gradient vector field \( \nabla f \) of \( f \). \[ f(x, y, z)=7 \sqrt{x^{2}+y^{2}+z^{2}} \] \[ \nabla f(x, y, z)= \]
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• 4. 4. f(x) = 3x is increasing and decreasing. It is increasing and decreasing.
  4. [4 pts.] Determine los intervalos donde la función \( f(x)=3 x^{4}- \) \( 4 x^{3}-15 x^{2} \) es creciente \( y \) decreciente.
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• 5. Determine the concavity intervals and the inflection points of
  5- [4 pts.] Determine los intervalos de concavidad y los puntos de inflexión de \( f(x)=\frac{x^{2}}{x^{2}+27} \).
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• 6. Determine the concavity intervals and the inflection points of f(x) = 6x
  6- [4 pts.] Determine los intervalos de concavidad y los puntos de inflexión de \( f(x)=6 x^{2 / 3}-x \)
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  Use polar coordinate to compute \[ \iint_{D} f(x, y) d x d y \] dove \[ \begin{array}{l} D=\left\{(x, y): x^{2}+y^{2} \leq 1\right\} \text { and } f(x, y)=\ln \left(x^{2}+y^{2}+1\right) \\ D=\left\{(x
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  Given \( f(x, y)=-5 x^{5}-6 x y^{4}+6 y^{6} \) \[ f_{x x}(x, y)= \] \[ f_{x y}(x, y)= \]
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• which is the answer to that exercise
  fonga on cuonth que. xrmax \( = \) distaricia \( x \) maxima vre vo= velocidad inicial, use la misrra quo uso on el ejorcicio anterior thota=ingulo on grados (de los ejercicios antertores) 9 - acelora
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  \( z=f(x, y)=\frac{3 x^{2}+4 y}{y^{2}} \)
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  1. Determine whether the following functions are homogeneous. If so, of what degree? (a) \( f(x, y)=\sqrt{x y} \) (d) \( f(x, y)=2 x+y+3 \sqrt{x y} \) (b) \( f(x, y)=\left(x^{2}-y^{2}\right)^{1 / 2} \
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  Given: \[ z=x^{2}+x y^{3}, \quad x=u v^{2}+w^{4}, \quad y=u+v e^{w} \] Find \( \frac{\partial z}{\partial w} \) when \( u=1, v=-1, w=0 \)
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  Solve the initial value problem: \[ \begin{array}{l} y^{\prime}=-0.01 y-2 \\ y(0)=110 \end{array} \]
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  \( y=2 \sqrt{x} \quad y=\frac{1}{4} x^{2} \quad \) Use Greanis Th. to fird area
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  8. Extremize \( P(x, y)=60 x+20 y \) subject to \[ 12 x+12 y \geq 24,6 x+4 y \leq 36,10 x+5 y \leq 50, x \geq 0, y \geq 0 \]
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  \( f(x) \) es uree fericion linieal, con RaMin de Ciamhio Promedio en el interule \( [1,3]=7_{4} \) Belectione unat b. 14 c. No as punde ideterminuar d. \( \mathrm{e} \)
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  7. \( \mathrm{y}=\frac{(2 x) 2^{x}}{\sqrt{x^{2}+1}} \) 8. \( r=\left(\frac{1+\sin \theta}{1-\cos \theta}\right)^{2} \)
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  Compute the following Line Integral \[ \int_{C} \frac{x}{y} d s, C: x=t^{2}, y=2 t, 0 \leq t \leq 3 \]
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  Find \( \frac{d f}{d w} \) given \( f(w, x, y)=3 w x-5 x^{4} y^{3}+(2 w-5 y)^{6} \)
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  Sketch the graph of a function that satisfies all of the given conditions.
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  Find the integral. \[ \begin{array}{l} \int \sin ^{3} x \cos ^{6} x d x \\ -\frac{1}{7} \cos ^{7} x+\frac{1}{5} \cos ^{5} x+C \\ \frac{1}{7} \cos ^{7} x-\frac{1}{5} \cos ^{5} x+C \\ -\frac{1}{9} \cos
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  (1 point) Let \( F(x, y, z)=\left(-5 x z^{2},-3 x y z, 9 x y^{3} z\right) \) be a vector field and \( f(x, y, z)=x^{3} y^{2} z \). \[ \begin{array}{l} \nabla f=( \\ \nabla \times F=( \\ F \times \nabl
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  pan resoberla. c \( x^{2} \frac{d y}{d x}+x y=\sin x \) A Cretiventas himigened B Enewhic \( \frac{A}{B} \frac{d y}{d x}=\frac{x^{2}-y^{2}}{x y} \) E Bencoulle Quelian \( \# 2 \)
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  Questión \#4 Qa ecuacion diferencial \( \left(2-y^{2}\right) d x+x^{2} y d y=0 \) es lineal - Falsa - cierto
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  Question \#8 - Falso a cierto
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  Determine the global extreme values of the function \( f(x, y)=4 x^{3}+4 x^{2} y+4 y^{2}, \quad x, y \geq 0, x+y \leq 1 \) \[ \begin{array}{l} f_{\min }= \\ f_{\max }= \end{array} \]
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• determine the equation of the tangent line mas= + para= for
  Determina la ecuación de la línea tangente. \( f(x)=-6 x^{2} \) más \( 9 x+-6 \) para \( a=-4 \) \( y=\quad x+ \)
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• you get in an elevator of a fair. the height of the cabin 'y in feet then 't in seconds, is given for the equation determine the aceleration 'a of the cabin after 3 seconds
  Te monfas en un elevador de una feria. La altura de la cabina \( y \) en pies luego de \( t \) segundos, está dada por la ecuación: \[ y=t^{3}-18 t^{2}+72 t \] Determina la aceleración a de la cabi
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• determine if it satifices the hypothesis of the Rolle teorem for en= in
  Determina si se satisface la hipótesis del Teorema de Rolle para \( f(x)=9 x^{3}+-1323 x+1 \) en \( [0, \sqrt{21}] \). Cierto Falso
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• when a rock falls in a lake, form a circular wave. the velocity in which it travels is 63cm/s determine the reason of the growth in the area after 3 seconfs reason of the average change=
  Cuando una piedra cae en un lago, forma una honda circular. La velocidad en que viaja la honda es de \( 63 \mathrm{~cm} / \mathrm{s} \). Determina la razón de crecimiento del área del círculo luego
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• maximun absolute minimum absolute mas= + en= in
  Determina los puntos extremos de la función. \[ \begin{array}{l} f(x)=9 x^{3} \text { más }-216 x^{2}+1701 x+-4392 \text { en }[6,8] \\ \text { Máximo Absoluto }=( \\ \text { Minimo Absoluto }=( \e
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• select the component of the graoh of the fuction by the signs derivated
  Selecciona el componente de la gráfica de la función según los signos de las derivadas. \[ f^{\prime}(x):-y f^{\prime \prime}(x): 0 \] Select one: \( r \) - 1 \( \imath \)
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• teorem rolle
  Determina el valor \( c \) que satisface el Teorema de Rolle para \[ f(x)=-6 x^{3}+648 x+6 \text { en }[0, \sqrt{108}] \]
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  Question 15 La cugcion difarencid \( 6 x y d x+\left(4 y+9 x^{2}\right) d y=0 \) Determine tacior de integracain apropisdo
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• Find dy/dx
  \( \frac{1}{x}+\frac{1}{y}=4 \) \( \mathrm{e}^{\mathrm{y}} \cos \mathrm{x}+\mathrm{e}^{-\mathrm{x}} \sin \mathrm{y}=10 \)
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  Paree las suguientes ecuaciones diferenciales con el metado para resolverla. A coeficentes hemsyen (3) Emeta O Separmble D Lineal \( E \) Berroulli
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  Calculate \( \iint_{\mathcal{S}} f(x, y, z) d S \) For \[ y=3-z^{2}, \quad 0 \leq x, z \leq 7 ; \quad f(x, y, z)=z \] \( \iint_{\mathcal{S}} f(x, y, z) d S= \)
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  Find \( f_{x} \) and \( f_{y} \). \[ f(x, y)=5 x \ln x y \]
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