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Calculus Archive: Questions from October 26, 2022

• find all first-order partial derivatives.
  8. \( f(x, y)=\int_{x}^{x+y} e^{y^{2}-t^{2}} d t \) 9. \( f(x, y, z)=3 x \ln \left(x^{2} y z\right)+x^{y / z} \) 10. \( f(x, y, z)=\frac{2}{\sqrt{x^{2}+y^{2}+z^{2}}}-x^{2} e^{x y / z} \)
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  5.Differentiate (a) \( y=\sec \theta \tan \theta \) (b) \( y=\frac{\cot t}{e^{t}} \) (c) \( y=\frac{\sin t}{1+\tan t} \)
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  Find \( d y / d x \) by implicit differentiation. (a) \( x^{4}+x^{2} y^{2}+y^{3}=5 \) (b) \( x e^{y}=x-y \) (c) \( x \sin y+y \sin x=1 \)
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  3. Ealuate the first detrative of the following functors. a) \( y=x \sin (x) \) b) \( y=\frac{2 x+1}{x-2} \) c) \( y=\frac{\sqrt{x}}{1+3 x^{2}} \) d) \( y=3 x^{4} \tan 3 x \) \( y=\frac{e^{x}}{x} \) f
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  5. Differentiate (a) \( y=\sec \theta \tan \theta \) (b) \( y=\frac{\cot t}{e^{t}} \) (c) \( y=\frac{\sin t}{1+\tan t} \)
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  5. Dillerentiate the following functions with respect to \( x \). a) \( y=\left(x^{2}+8 q\right)^{2} \) b) \( y=\ln (3 x+1) \) c) \( y=\sin ^{2} x \) d) \( y=i^{x^{2}} \) e) \( y=\sqrt{\sin \left(x^{2
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  \( \int_{0}^{7} \int_{-\sqrt{49-x^{2}}}^{\sqrt{49-x^{2}}} x^{2} e^{\left(x^{2}+y^{2}\right)} d y d x \)
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  Find \( f_{x} \) and \( f_{y} \) \[ f(x, y)=\int_{y}^{x} 7 e^{t^{10}} d t \] \[ f_{x}(x, y)= \]
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  4. Solve the following separable ODEs: (i) \( y^{\prime}+y=1, y(0)=2.5 \) (ii) \( \quad y^{\prime}=2 x y, \quad y(1)=4 \)
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  Find \( d y / d x \) if \( y=\ln \left((x-1)^{3}\left(x^{2}+1\right)^{4}\right) \) Answer:
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• Determine the solution using Laplace Transforms
  \( y^{\prime \prime}+2 y^{\prime}+5 y=0, y(0)=2, \quad y^{\prime}(0)=-1 \)
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  Differentiate \( y=3 \sin (\sin (\sin x)) \). \[ y^{\prime}= \]
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  Sketch the surface \( z=f(x, y) \). \[ f(x, y)=-\sqrt{x^{2}+y^{2}} \]
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  Find the limit. \[ \begin{array}{ll} \lim _{(x, y) \rightarrow(0,0)} & \frac{4 x^{2}+6 y^{2}+5}{4 x^{2}-6 y^{2}+4} \\ 1 & \\ \frac{5}{4} \\ -1 \\ \text { No limit } \end{array} \]
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• I need #27!!, Find local maxima, local minima,saddle points!!
  23. \( f(x, y)=y \sin x \) 24. \( f(x, y)=e^{2 x} \cos y \) 25. \( f(x, y)=e^{x^{2}+y^{2}-4 x} \) 26. \( f(x, y)=e^{y}-y e^{x} \) 27. \( f(x, y)=e^{-y}\left(x^{2}+y^{2}\right) \) 28. \( f(x, y)=e^{x}\
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• 4. Find the absolute extreme values of h(x, y) = x 2 + 2xy when − p 5/2 ≤ x ≤ p 5/2, x 2 (5) − 2 ≤ y ≤ 3 − x 2
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• Rewrite the integral using the order dxdzdy to coordinates to cylindrical or spherical. Then comment on whether or not it is convenient to change the coordinates to cylindrical or spherical ones and w
  III. Reescriba el integral utilizando el orden dxdzdy. Luego comente si es o no es conveniente hacer cambio de coordenadas a cilíndricas o esféricas y trabaje su evalaución. \[ \int_{0}^{4} \int_{0
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• Determine mass and center of mass of the solid with given density bounded by the graphs of the equations. Clearly state and evaluate the triple integral that allows determine it
  Ejercicios: 1) Determine masa y el centro de masa del sólido con densidad dada acotado por las gráficas de las ecuaciones. Establezca y evalúe claramente el integral triple que permite determinarlo
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• 1. which of these integrals represents the area of the surface generated by turning the curve r=e 20, 2.reason why the others are not correct.
  1. ¿Cuál de estas integrales representa el área de la superficie generada al girar la curva \( r=e^{2 \theta}, \quad 0 \leq \theta \leq \frac{\pi}{2} \), alrededor de la linea \( \theta=\frac{\pi}{
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• Find the area of the region outside: a. Draw the region using the Wplotsp function grapher. b. Determine limits of integration. c.evaluate the appropriate integral. translation:
  Hallar el área de la región que queda fuera de: \( r=1+\cos \theta \quad y \) dentro de \( r=3 \cos \theta \) a. Dibujar la región utilizar el graficador de funciones Wplotsp. (4 pts.) b. Determina
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  1. \( \int\left(\frac{1}{\sqrt{6 x-4}}\right) d x \) 2. \( \int\left(\frac{2}{(4-x)^{2}}\right) d x \) 3. \( \int\left(5 x \sqrt{8-5 x^{2}}\right) d x \) 4. \( \int\left(\frac{\sin \sqrt{x}}{\sqrt{x}\
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• Resolves: 1. Classify whether the following pairs of vectors are parallel, perpendicular or neither. 2. Given vectors A= i + 2j + 3k, B= 2i + j - 5k Determines: 3.The vector component of vector A perp
  1. Clasificar si los siguientes pares de vectores son paralelos, perpendiculareso ninguno de los dos. a. \( \vec{A}=i+2 j-k \quad \vec{B}=2 i+2 j+6 k \) b. \( \overrightarrow{\mathrm{C}}=\mathrm{i}+2
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• 19 and 21
  17-28 Use the guidelines of Section \( 3.5 \) to sketch the curve. 17. \( y=2-2 x-x^{3} \) 18. \( y=-2 x^{3}-3 x^{2}+12 x+5 \) 19. \( y=3 x^{4}-4 x^{3}+2 \) 20. \( y=\frac{x}{1-x^{2}} \) 21. \( y=\fra
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• 1. Define the scalar product of 2 vectors. 2. Define what is meant by orthogonal vectors. If 2 vectors are not parallel or parallel or orthogonal, how can you calculate the angle they form?
  1. Definir el producto escalar de 2 vectores 2. Definir qué se entiende por vectores ortogonales. Si 2 vectores no son paralelos ni paralelos ni ortogonales, ¿Cómo se puede calcular el ángulo que
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• Resolves: 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 points P(3,0,2) and Q(1,-4
  1. Encuentre la ecuación del plano determinado por los puntos \( P(1,2,3) \) \( Q(2,3,1) R(0,-2,-1) \) 2. Encuentre las ecuaciones paramétricas y simétricas para la línea que pasa por los puntos \
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• Given the coordinates of points P(-1,-2,3) Q(-2,3,5) and R(4,5,6) find: a. the vectors PQ and PR b. PQ x PR c. What is the relationship between the components of the vector product and the coefficient
  Dadas las coordenadas de los puntos \( P(-1,-2,3) Q(-2,3,5) \) y \( R(4,5,6) \) hallar: a. Los vectores \( \overrightarrow{P Q} \) y \( \overrightarrow{P R} \) b. \( \overrightarrow{P Q} \times \overr
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• Situation: Explain the procedure for graphing equations in space. Use the following exercises to supplement your answer: 1. x 2/4 + z 2/25 =1 2. 9y 2 +z 2 = 9 - x
  Explique el procedimiento para graficar ecuaciones en el espacio. Utilice los siguientes ejercicios para complementar su contestación: 1. \( \frac{x^{2}}{4}+\frac{z^{2}}{25}=1 \) 2. \( 9 y^{2}+z^{2}=
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• Identify the graphic in space a. Hyperboloid of a mantle parallel to the"x" axis b. Two-mantle hyperboloid parallel to the"x" axis c. Elliptic paraboloid parallel to "y" axis D. Elliptical cylinder pa
  En los ejercicios (1-3) Seleccione la respuesta correcta 1. Identifique la gráfica en el espacio \( \frac{x^{2}}{4}-\frac{y^{2}}{25}-\frac{z^{2}}{49}=1 \) a. Hiperboloide de un manto paralelo al eje
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• The surface represented by the equation p=2csco describes: a. A straight b. A point c. A plan d. A cylinder
  3. La superficie representada por la ecuación \( \rho=2 \csc \phi \) describe: a. Una recta b. Un punto c. Un plano d. un cilindro
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• Identification of the superfice cusesecca a. cono elliptico b. paraboloide elliptico c. parabolone hipperbolico d. ninguna de las anteriores
  Escoge la mejor contestación 1. Identifique la superficie cuadrica \( z^{2}=\frac{x^{2}}{4}+\frac{y^{2}}{16} \) a. Cono elíptico b. Paraboloide elíptico c. Paraboloide hiperbólico d. Ninguna de la
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• Write the standard equation for 4x 2 - 9y 2 - 36z =0. Identify the surface: a. b. c. d.
  2. Escriba la ecuación en forma estándar para \( 4 x^{2}-9 y^{2}-36 z=0 \). Identifique la superficie. a. \( 36 z=4 x^{2}-9 y^{2} \); Cono b. \( z=\frac{x^{2}}{9}-\frac{y^{2}}{4} \); Paraboloide Hip
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• Express the cylindrical point in rectangular coordinates (4, 5pi/6, 3)
  3. Exprese en coordenadas rectangulares el punto en cilindricas \( \left(4, \frac{5 \pi}{6}, 3\right) \) a. \( (2,2 \sqrt{3}, 3) \) b. \( (2 \sqrt{3},-2,3) \) c. \( (-2 \sqrt{3}, 2,3) \) d. \( (-2,2 \
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• Express in cylindrical coordinates the point (1,sqrs 3, 2)
  4. Exprese en coordenadas cilindricas el punto \( (-1, \sqrt{3}, 2) \) a. \( (2,5 \pi / 6,2) \) b. \( (2, \pi / 6,2) \) c. \( (2,2 \pi / 3,2) \) d. \( (2,-\pi / 6,2) \)
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• Find the equation p=9 csc0csc0 in rectangular twines: to. b. c. d.
  5. Encuentre la ecuación \( \rho=9 \csc \phi \csc \theta \) en coordenadas rectangulares a. \( x=9 \) b. \( y=9 \) c. \( x y=9 \) d. \( y=1 / 9 \)
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  Define \( g(x, y, z)=z^{x y z}, z>0 \). Find \( \nabla g(x, y, z) \).
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  4. Explain whether the vector field \( \vec{F}(x, y)=\frac{-y}{x^{2}+y^{2}} \widehat{\boldsymbol{\imath}}+\frac{x}{x^{2}+y^{2}} \widehat{\jmath} \) is conservative in \( D_{1}=\left\{(x, y) \in \mathb
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• show all steps to each question
  5. Calculate \( y^{\prime} \) (a) \( y=\left(x^{4}-3 x^{2}+5\right)^{3} \) (b) \( y=\sqrt{x}+\frac{1}{\sqrt[3]{x^{4}}} \) (c) \( y=\frac{e^{\frac{1}{x}}}{x^{2}} \) (d) \( y=\frac{\sec 2 \theta}{1+\tan
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  Calculate \( y^{\prime} \) if \( y=e^{\sin 2 x}+\sin \left(e^{2 x}\right) \)
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  4. Explain whether the vector field \( \vec{F}(x, y)=\frac{-y}{x^{2}+y^{2}} \hat{\imath}+\frac{x}{x^{2}+y^{2}} \widehat{\jmath} \) is conservative in \[ \begin{array}{ll} D_{1}=\left\{(x, y) \in \math
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  Differentiate each function (5 marks each \( -25 \) marks total) a. \( y=\left(\sqrt{x}+\frac{1}{\sqrt[3]{x}}\right)^{2} \) b. \( F(y)=\left(\frac{1}{y^{2}}-\frac{3}{y^{4}}\right)\left(y+5 y^{3}\right
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• Explain how to find.
  Problema: Explique cómo hallar \[ L^{-1}\left\{\frac{1}{s^{2}(s+1)}\right\} \]
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• Answer all and show work please
  39. \( y=\left[x^{2}+(1-3 x)^{5}\right]^{3} \) 41. \( y=\sqrt{x+\sqrt{x}} \) 43. \( g(x)=(2 r \sin r x+n)^{p} \) 45. \( y=\cos \sqrt{\sin (\tan \pi x)} \)
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  2. Find \( d y / d x \) by implicit or logarithmic differentiation. (a) \( y=\frac{(x-2)^{3} e^{x^{2}}}{2^{x} \sqrt[3]{1-x}} \) (b) \( 3 x y^{2}-\sqrt{x y}=3 x-y \). (c) \( y^{y}=x^{x} \).
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  Differentiate each function (5 marks each - 25 marks total) a. \( y=\left(\sqrt{x}+\frac{1}{\sqrt[3]{x}}\right)^{2} \) b. \( F(y)=\left(\frac{1}{y^{2}}-\frac{3}{y^{4}}\right)\left(y+5 y^{3}\right) \)
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  6. \( \left[9=2+3+4\right. \) marks] Find \( \frac{d y}{d x} \) for the following functions (a) \( y=\sin (\sqrt{x+1}) \) (b) \( y=\frac{1}{\tan ^{-1}\left(x^{2}\right)} \) (c) \( y=(x+1)^{x^{2}} \)
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  Sketch the domain of the function \( f(x, y)=\sqrt{16-x^{2}-y} \)
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  4. Find \( y^{\prime \prime} \) *do not simplify* a) \( y=\left(x^{4}+x\right)^{\frac{2}{3}} \) b) \( y=\frac{3}{x^{4}}-\frac{1}{x} \) Find \( y^{(6)} \) : c) \( y=x^{4}-3 x^{3}-7 x^{2}-6 x+9 \)
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• 9. Let y(x) be the solution of y' = y − 3x such that y(2) = 1. Using a step size of 0.5, set y0 = y(2) = 1 and apply Euler's Method to approximate the values y1,y2,y3, and y4.(y1, y2, y3, y4) = PLE
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  III. Reescriba el integral utilizando el orden dxdzdy. Luego comente si es o no es conveniente hacer cambio de coordenadas a cilíndricas o esféricas y trabaje su evalaución. \[ \int_{0}^{4} \int_{0
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• Solve: y'''=0, y(0)=3, y'(1)=4, y"(2)=6
  Solve: \( y^{\prime \prime \prime}=0 \quad, y^{(0)}=3, y^{\prime}(1)=4, y^{\prime \prime}(2)=6 \)
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  Solve the initial value problem. \[ \frac{d^{2} y}{d t^{2}}=1-e^{2 t}, y(1)=-4, y^{\prime}(1)=0 \]
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  (1) USE THE CHAIN RUE TO FIND THE DERIVATIVE a) \( y=(5 x-7)^{11} \) b) \( y=7 \sin ^{9}(x) \) c) \( y=(8-3 x)^{-4} \) d) \( y=\sin ^{3}\left(x^{2}+7\right) \)
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  Use impicit Differentiation to find \( d y / d x \) \( y \sin (x y)=y^{2}+2 \) OR \[ y \sqrt{x+4}=x y+8 \]
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• Find y(t) if y"-4y'+4y=0 y(0)=1, y'(0)=3
  find \( y(\tau) \) if \( y^{\prime \prime}-y y^{\prime}+y y=0 \) \( y(0)=1, y^{\prime}(0)=3 \)
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• Differentiate each function
  b. \( F(y)=\left(\frac{1}{y^{2}}-\frac{3}{y^{4}}\right)\left(y+5 y^{3}\right) \) c. \( y=\cos \sqrt{\sin (\tan \pi x)} \)
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• Diferentiate Each function
  d. \( f(s)=\sqrt{\frac{s^{2}+1}{s^{2}+4}} \) e. \( y=\frac{\cos \pi x}{\sin \pi x+\cos \pi x} \)
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• derivate
  (1) \( f(x)=6 x-5 \) (2) \( f(x)=3 x^{2} \) (3) \( f(x)=9 x^{4}-12 x^{3} \) (4) \( f(x)=-8 x^{-3} \) (5) \( f(x)=e^{7 x-4} \)
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  Find the Jacobian of the transformation. \[ x=u / v, y=v / w, z=w / u \] \[ \frac{\partial(x, y, z)}{\partial(u, v, w)}= \]
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  2. (15 points) Let \( y=x^{2} \sin (\pi x) \). Find \( y^{\prime} \).
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• Express in rectangular coordinates the point in cylindrical
  Exprese en coordenadas rectangulares el punto en cilindricas \( \left(4, \frac{5 \pi}{6}, 3\right) \) a. \( (2,2 \sqrt{3}, 3) \) b. \( (2 \sqrt{3},-2,3) \) c. \( (-2 \sqrt{3}, 2,3) \) d. \( (-2,2 \sqr
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  Differentiate. \[ F(y)=\left(\frac{1}{y^{2}}-\frac{6}{y^{4}}\right)\left(y+2 y^{3}\right) \]
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  (1) Differentiate the following functions. (a) \( y=x^{2}-\int_{x^{3}+1}^{0}\left(t^{2}+2\right)^{1 / 3} d t \) (b) \( y=\int_{\cos x}^{\sin x}\left(u^{7}+3\right)^{11} d u \)
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• Un poste telefónico tiene tres cables que halan como se muestra desde arriba, con Halle la fuerza neta sobre el poste telefónico en forma de componentes. b) Halle la magnitud y la dirección de esta
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  4. Explain whether the vector field \( \vec{F}(x, y)=\frac{-y}{x^{2}+y^{2}} \widehat{\boldsymbol{\imath}}+\frac{x}{x^{2}+y^{2}} \widehat{\jmath} \) is conservative in \( D_{1}=\left\{(x, y) \in \mathb
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  Evaluate the triple integral.
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• Find the partial derivatives of the function on the given point
  1. Hallar las derivadas parciales de la función en el punto dado. \[ f(x, y)=x^{2} e^{2 y}, \quad P(2, O) \] 2. Hallar las derivadas parciales de la función en el punto dado. \[ f(x, y)=e^{x y} \sin
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• Find the slope in the direction of x and y axis on the given point
  6. Hallar las pendientes en la dirección del eje \( x \) y en la dirección del eje \( y \) en el punto dado: \[ \begin{array}{ll} g(x, y)=4-x^{2}-y^{2} & \\ (1,1,2) & h(x, y)=x^{2}-y^{2} \\ & (-2,1,
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• Use the proper chain rule to find the solution
  1. Usando la regla de la cadena apropiada hallar \( \frac{d W}{d t} \) a \( w=x y, x=2 \sin (t), \quad y=\cos (t) \) b. \( w=\cos (x-y), \quad x=t^{2}, y=t^{3} \) c. \( w=x y \cos (z), x=t, y=t^{2}, z
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• Find the directional derivative of each function on point P in the direction of V
  1. Hallar la derivada direccional de cada función en el punto \( P \) en la dirección de \( v \) a. \( g(x, y)=\sqrt{x^{2}+y^{2}}, \quad P(3,4), \quad \mathbf{v}=3 \mathbf{i}-4 \mathbf{j} \) b. \( f
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• Find the tangent plane equation
  1. Hallar una ecuación del plano tangente a la superficie en el punto dado \[ f(x, y)=\frac{y}{x} \text { en el punto }(1,2,2) \] 2. Hallar una ecuación del plano tangente a la superficie en el punt
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• Solve the function
  \( \frac{\partial}{\partial x}\left(\frac{\partial f}{\partial y}\right)=\frac{\partial^{2} f}{\partial x y} \) y \( \frac{\partial}{\partial y}\left(\frac{\partial f}{\partial x}\right)=\frac{\partia
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• find the domain and graf it
  \( \operatorname{Sif}(\mathrm{x}, \mathrm{y})=\frac{\sqrt{x+y+1}}{x-1} \) a. Halle Dominio de \( \mathrm{f} \) Grafique la región del dominio de f
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  Calculate all four second-order partial derivatives of \( f(x, y)=\sin \left(\frac{2 x}{y}\right) \). \[ f_{x x}(x, y)= \] \[ f_{x y}(x, y)= \] \[ f_{y x}(x, y)= \] \[ f_{y y}(x, y)= \]
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• find the domain and graff the region
  6. \( \operatorname{Si~} f(x, y)=x \ln \left(y^{2}-x\right) \), c. Halle Dominio de \( f \) d. Grafique la región del dominio de \( \mathrm{f} \)
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  Solve the initial value problem: \[ \left\{\begin{array}{c} y^{\prime \prime}-7 y^{\prime}+12 y=0 \\ y(0)=1 \\ y^{\prime}(0)=5 \end{array}\right. \]
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  Given \( f(x, y)=-3 x^{6}+5 x^{2} y^{5}+6 y^{3} \) \[ f_{x}(x, y)= \] \( f_{y}(x, y)= \) \[ f_{x x}(x, y)= \] \[ f_{x y}(x, y)= \]
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  Given \( f(x, y)=2 x^{7} \cos \left(y^{9}\right) \) \[ f_{x y}(x, y)= \] \( f_{y y}(x, y)= \)
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  Let \( f(x, y)=6 e^{3 x} \sin (2 y) \) \[ \frac{\partial f}{\partial x}= \] \( \frac{\partial f}{\partial y}= \) Given \( f(x, y)=-4 x^{3}+3 x y^{6}+2 y^{2} \) \( f_{x}(x, y)= \) \[ f_{y}(x, y)= \]
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  Calculate \( \frac{d^{2} y}{d x^{2}} \) \[ y=4 x^{0.8}-2 x \] \[ \frac{d^{2} y}{d x^{2}}= \]
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  Let \( y \) be the solution of the initial value problem \[ y^{\prime \prime}+y=-\sin (2 x), y(0)=0, y^{\prime}(0)=0 . \] The maximum value of \( y \) is
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  Determine \( f_{x} \) and \( f_{y} \) if (A) \( f(x, y)=(\sin (\sqrt{x})) \ln \left(y^{2}\right) \) \( f_{x}= \) \( f_{y}= \) (B) \( f(x, y)=\sin \left(\sqrt{x} \ln \left(y^{2}\right)\right) \) \( f_{
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  ind \( \frac{d y}{d x} \) if \( y=(x+2)^{x+2} \) \[ y=x(\arctan x)-\frac{1}{2} \ln x \]
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  Find \( \frac{d y}{d x} \) if \( y=(x+2)^{x+2} \) \( y=x(\arctan x)-\frac{1}{2} \ln x \)
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  \( \frac{2}{s}+\frac{1}{s^{2}} \) \( \frac{2}{s}+\frac{e^{-s}}{s^{2}}-\frac{e^{-s}}{s} \) \( -\frac{2}{s} e^{-s}+\frac{2}{s}-\frac{1}{s^{2}} \) \( 2+\frac{e^{-s}}{s^{2}}-2 \frac{e^{-s}}{s} \) \( \frac
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  Determine \( f_{x} \) and \( f_{y} \) if (A) \( f(x, y)=(\sin (\sqrt{x})) \ln \left(y^{2}\right) \) \( f_{x}= \) \( f_{y}= \) (B) \( f(x, y)=\sin \left(\sqrt{x} \ln \left(y^{2}\right)\right) \) \( f_{
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  4. Find the horizontal asymptotes, if any for (a) \( y=\frac{2 x-7}{x^{2}-4 x} \) (b) \( y=\frac{x^{3}-x^{2}+10}{3 x^{2}-4 x} \) (c) \( y=\frac{2 x^{2}-6}{x^{2}+5 x} \).
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  3) Find the length of each curve. a) \( y=4(x-1)^{3 / 2}, \quad 1 \leq x \leq 4 \) b) \( y=2 \ln \left(\sin \frac{1}{2} x\right), \quad \frac{\pi}{3} \leq x \leq \pi \) c) \( 12 x=4 y^{3}+3 y^{-1}, \q
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  Find \( \int_{0}^{2} f(x, y) d x \) and \( \int_{0}^{3} f(x, y) d y \) \[ f(x, y)=9 x+3 x^{2} y^{2} \] \( \int_{0}^{2} f(x, y) d x= \) \( \int_{0}^{3} f(x, y) d y= \)
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