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Calculus Archive: Questions from February 28, 2023

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  Use various trigonometric identities to simplify the expression then integrate. \[ \begin{array}{l} \int \sin ^{2} \theta \cos 5 \theta d \theta \\ \frac{1}{2} \sin 5 \theta-\frac{1}{4} \sin 3 \theta-
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  Find \( y= \) by implicit differentiation. \[ \begin{array}{c} 3 x^{3}-4 y^{3}=7 \\ y^{\prime \prime}=\frac{3 x\left(16 y^{2}-27 x^{2}\right)}{2 y^{6}} \end{array} \]
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  2. (3 pts) Find the derivative, \( y^{\prime} \). \[ y=\left(5 x^{2}-7 x\right)^{-5}\left(x^{3}+1\right)^{3} \]
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• Use the properties of odd and even functions to find the exact value of the following expressions. Do not use a calculator.
  4. Use las propiedades de las funciones pares e impares para hallar el valor exacto de las siguientes expresiones. No use calculadora. (a) \( \operatorname{sen}\left(-\frac{\pi}{3}\right) \) (b) \( \c
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• s bar may be unehen because questions can b tion: (x+2)/(7)=(x)/(21)
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  Find the domain: \[ \begin{array}{l} f(x, y)=\frac{1}{\sqrt{x+y-4}} \\ \{(x, y): x+y \geq 4\} \\ \{(x, y): x+y \geq 0\} \\ \{(x, y): x+y>4\} \\ \{(x, y), x+y \leq 4\} \end{array} \]
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• Please solve 45. please show all work
  45-56 Use logarithmic differentiation to find the derivative of the function. 45. \( y=\left(x^{2}+2\right)^{2}\left(x^{4}+4\right)^{4} \) 46. \( y=\frac{e^{-x} \cos ^{2} x}{x^{2}+x+1} \) 47. \( y=\sq
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• Please solve Question 50. Please show all work .
  45-56 Use logarithmic differentiation to find the derivative of the function. 45. \( y=\left(x^{2}+2\right)^{2}\left(x^{4}+4\right)^{4} \) 46. \( y=\frac{e^{-x} \cos ^{2} x}{x^{2}+x+1} \) 47. \( y=\sq
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• Please solve Question 51. please show all work.
  45-56 Use logarithmic differentiation to find the derivative of the function. 45. \( y=\left(x^{2}+2\right)^{2}\left(x^{4}+4\right)^{4} \) 46. \( y=\frac{e^{-x} \cos ^{2} x}{x^{2}+x+1} \) 47. \( y=\sq
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• Differentiate the function. y = 1.3x + ln x y' =
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• y(3) - y'' - y' +y = 0 general solution?
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• Differentiate the function. y = (ln x)9 y' =
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• El vector v tiene punto inicial (1, 2, 3) y punto terminal (−1, −2, 3). Hallar un vector unitario en la dirección de v. Hallar un vector de longitud 1 y punto inicial P (0, 0, 0).
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  5. Encuentre un conjunto de ecuaciones paramétricas de las siguientes líneas: a. La línea que pasa por el punto \( (1,1,4) \) y es perpendicular al plano dado por la ecuación \( x-y+z=2 \). b. La
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  4. Hallar el área del triángulo con los vértices dados a. \( P(0,0,0), Q(3,4,0) \) y \( R(3,4,2) \).
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  Sean \( \boldsymbol{u}= \) y \( \boldsymbol{v}= \) dos vectores. a. Encuentre la proyección de \( \boldsymbol{v} \) sobre \( \boldsymbol{u} \). b. Encuentre la componente vectorial de \( \boldsymbol{
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  2. Encuentre (a) \( \boldsymbol{u} \cdot \boldsymbol{v} \) y (b) \( \boldsymbol{u} \times \boldsymbol{v} \) para los siguientes vectores: \[ \begin{array}{l} \boldsymbol{u}=-i-2 j+k \\ \boldsymbol{v}=
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• Subtract. Simplify the result if possible. (7x^(2)+2x)/(x-8)-(57x+8)/(x-8)
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• methods to determine the actual roo p(x)=x^(3)+x^(2)-25x-25
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• tion. Simplify your answer if possible. (v+2)/(3)+(v+6)/(2)=(v-5)/(7)+3
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• Find the orthogonal trajectory. y = ke^(−3x)
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  6. Hallar una ecuación del plano que muestrea en la figura.
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  7 Hallar la distancia entre los planos que se muestran en la siguiente figura \[ x-y+z=5 \text { (Plano azul) y } 3 x-3 y+3 z=-2 \text { (plano rojo). } \]
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• differentiate and simplify when possible.
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• 5 por favor
  OBLEMAS (a) Partiendo directamente entonces \( f^{\prime}(a)=-1 / a^{2} \), (b) Demostrar que la tangen de \( f \) más que en el pur (a) Demostrar que si \( f(x)= \) (b) Demostrar que la tanger punto
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  Compute the gradient vector fields of the following functions: \[ \begin{array}{l} \text { A. } f(x, y)=5 x^{2}+10 y^{2} \\ \nabla f(x, y)=\quad \mathbf{i}+\quad \mathbf{j} \end{array} \] B. \( f(x, y
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• find the dy/dx,show steps please
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  Find the derivative. 1. \( y=\frac{1}{x}+\sqrt{x}-\frac{1}{\sqrt{x}} \) 2. \( y=x^{3}\left(x^{2}+1\right)^{1 / 3} \) 3. \( y=\ln \sqrt{5 x^{2}-4} \) 4. \( y=\left(e^{x^{2}+2}\right)^{2} \) 5. \( y=\ln
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  Differentiate the function. \[ y=\sqrt{5+6 e^{7 x}} \] \[ y^{\prime}= \]
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  1. Prove the following identities: (a) \( \sec x-\cos x=\tan x \sin x \) (b) \( \sin (x+y) \sin (x-y)=\sin ^{2} x-\sin ^{2} y \)
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  Help Entering Answers (1 point) Find the gradient vector field of the following functions: \[ \begin{array}{ll} f(x, y, z)=\sqrt{x+y+4 z}, & \nabla f(x, y, z)= \\ g(x, y, z)=6 \tan (-3 x-y z), & \nabl
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  Help Entering Answers (1 point) Find the gradient vector field of the following functions: \[ f(x, y)=2 x \ln \left(y^{2}-2\right), \quad \nabla f(x, y)= \] \[ g(x, y)=\sqrt{2 x^{2}-3 y^{2}}, \quad \n
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  (1 point) 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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• Find the Derivatives
  11. \( y=\sqrt{x} \cos \sqrt{x} \) 15. \( y+x \cos y=x^{2} y \)
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  if \( y=\frac{\cos x}{x} \), then \( y^{\prime}= \) and \( y^{\prime \prime}= \) Hence, \( x y^{\prime \prime}+2 y^{\prime}= \)
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  Compute the gradient vector fields of the following functions: A. \( f(x, y)=10 x^{2}+6 y^{2} \) \( \nabla f(x, y)=\quad \mathbf{i}+\quad \mathbf{j} \) B. \( f(x, y)=x^{9} y^{7} \), \( \nabla f(x, y)=
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• number 6
  3-14 Use Lagrange multipliers to find the maximum and mini mum values of the function subject to the given constraint. 3. \( f(x, y)=x^{2}+y^{2} ; \quad x y=1 \) 4. \( f(x, y)=3 x+y ; \quad x^{2}+y^{2
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• y = sec2(x), y = 8 cos(x), −𝜋/3 ≤ x ≤ 𝜋/3
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• y = 4 cos(𝜋x), y = 8x2 − 2 find the area
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  (1 point) Let \( f(x, y, z)=\frac{x^{2}-6 y^{2}}{y^{2}+6 z^{2}} \). 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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• y= 1/x+ 1, y= 0, x= 0, x= 2 around the y axis .
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  Compute the gradient vector fields of the following functions: A. \( f(x, y)=7 x^{2}+4 y^{2} \) \( \nabla f(x, y)= \) B. \( f(x, y)=x^{1} y^{7} \), \( \nabla f(x, y)= \) C. \( f(x, y)=7 x+4 y \) \( \n
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• No supe como resolverla alguien que me pueda explicar?
  2. Calcular la derivada direccional de la función \( z=f(x, y)=y^{2}(\tan x)^{2} \) en la dirección del vector \( v=-\frac{1}{2} 3 i+\frac{1}{2} j \) en el punto \( \left(\begin{array}{l}\pi \\ 3\en
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  \( y^{\prime \prime}+49 y=0, \quad y(0)=7, \quad y^{\prime}(0)=-6 \) \( y(x)=-6 \cos (7 x)+\left(\frac{6}{7}\right) \sin (7 x) \)
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  \[ f(x)=x^{3}-12 x+1 \] relative maximum \( (x, y)=(\quad) \) relative minimum \( \quad(x, y)=(\quad \)
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• Arc Length
  \( x=\frac{1}{3} \sqrt{y}(y-3), \quad 1 \leqslant y \leqslant 9 \)
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  (1 point) Find the gradient vector field of the following functions: \[ f(x, y, z)=x^{2}+2 x y z+\frac{3 z^{3}}{x}, \quad \nabla f(x, y, z)= \] \[ g(x, y, z)=6 \tan (-2 x-y z), \quad \nabla g(x, y, z)
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  Find the gradient vector field of the following functions: \[ f(x, y)=\sqrt{2 x^{2}-y^{2}}, \quad \nabla f(x, y)= \] \[ g(x, y)=7 x e^{3 x y}, \quad \nabla g(x, y)= \] \[ h(x, y)=3 x \ln \left(2+y^{2}
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  3. Demostrar que si \( f(x)=\sqrt{x} \), entonces \( f^{\prime}(a)=1 / 2 \sqrt{a} \), para \( a>0 \). (La expresión que se obtenga para \( [f(a+h)-f(a)] / h \) requerirá algún trabajo algebraico, p
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  6. Demostrar lo siguiente, partiendo de la definición (y trazando un dibujo explicativo): (a) Si \( g(x)=f(x)+c \), entonces \( g^{\prime}(x)=f^{\prime}(x) \); (b) Si \( g(x)=c f(x) \), entonces \( g
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• 13&15
  13. \( \frac{1+\tan ^{2} \theta}{\csc ^{2} \theta}+\sin ^{2} \theta+\frac{1}{\sec ^{2} \theta} \) 14. \( \left(\frac{\tan x}{\csc ^{2} x}+\frac{\tan x}{\sec x}\right)\left(\frac{1+\tan x}{1+\cot x}\ri
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  \( y^{\prime}+\frac{3}{x} y=\sin x, y(\pi)=1 \)
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  \[ y^{\prime}+(\tan x) y=\frac{\sin x}{y} \] (Find family of implicit solutions
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• PLEASE SOLVE ASAP
  Evaluate: \[ \frac{d}{d x} \ln \left(\frac{(3 x+1)^{8}}{\sqrt[3]{5 x^{2}+7 x+16}}\right)= \]
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• PLEASE SOLVE ASAP
  Find \( \frac{d y}{d x} \) for \( y=\frac{e^{x^{3}}}{\sqrt{8-x^{2}}} \) \[ \frac{d y}{d x}=\sqrt{\frac{\left(3 x^{2} e^{x^{3}}\left(8-x^{2}\right)-x e^{x^{3}}\right)}{\left(8-x^{2}\right)^{\frac{3}{2}
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• PLEASE SOLVE ASAP
  Find \( \frac{d y}{d x} \) where \[ y^{4} \sin (10 x)-4 x^{3}-y^{9}=2 \] \[ \frac{d y}{d x}=\frac{\left(12 x^{2}-10 y^{4} \cos (10 x)\right)}{4 y^{3} \sin \left(10 x 0-9 y^{8}\right)} \times \]
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• PLEASE SOLVE ASAP
  Find \( \frac{d y}{d x} \) for \( y=\frac{\sin (3 x)}{\sqrt{9-x^{5}}} \). \[ \frac{d y}{d x}=\frac{5 x^{4} \sin (3 x)}{2\left(9-x^{5}\right)^{\frac{3}{2}}}+\frac{3 \cos (3 x)}{\sqrt{9-x 5}} \times \]
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• PLEASE SOLVE ASAP
  Find \( \frac{d y}{d x} \) for \( y=\frac{\ln (3 x+6)}{8 e^{2 x}} \) \[ \frac{d y}{d x}= \]
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• PLEASE SOLVE ASAP
  Find \( \frac{d y}{d x} \) for \( y=\frac{\sin (4 x)}{9 e^{2 x}} \) \[ \frac{d y}{d x}= \]
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• PLEASE SOLVE ASAP
  \[ y=\left(-5 x^{2}+8 x-7\right) e^{-3 x} \] en \( \frac{d y}{d x}= \)
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  \( \begin{array}{c}y=\frac{t \sin (t)}{1+t} \\ y^{\prime}=\frac{t \cos (t)+t^{2} \cos (t)}{(1+t)^{2}}\end{array} \)
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  ( 1 point) Find \( y \) as a function of \( x \) if \[ y^{\prime \prime \prime}-12 y^{\prime \prime}+27 y^{\prime}=64 e^{x} \] \[ y(0)=11, y^{\prime}(0)=19, y^{\prime \prime}(0)=23 \text {. } \] \( y(
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  If \( y^{\prime \prime}=\frac{1}{(x+1)^{2}} \cdot y(0)=2 \cdot y^{\prime}(0)=1^{\prime} \)
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