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Calculus Archive: Questions from November 03, 2022

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  1) Solve the differential equations with the given starting values i. \( y^{\prime}-2 y=4 ; \quad y(0)=0 \) ii. \( y^{\prime}=\frac{y+1}{x} ; \quad y(1)=0 \) iii. \( y^{\prime \prime}-5 y^{\prime}+4 y
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  Select the positive-definite quadratic forms: a. \( q(x, y)=x^{2}-3 x y+y^{2} \) b. \( q(x, y, z)=-2 x^{2}-5 y^{2}+12 y z+7 z^{2} \) c. \( q(x, y, z)=4 x y+3 y^{2}+z^{2} \) d. \( q(x, y, z)=25 x^{2}-7
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  Find \( y^{\prime}, y^{\prime \prime} \), and \( y^{\prime \prime \prime} \) for the functions in Exercises 1-12. 1. \( y=(3-2 x)^{7} \) 2. \( y=x^{2}-\frac{1}{x} \) 3. \( y=\frac{6}{(x-1)^{2}} \) 4.
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  \( g(x)=\int_{1}^{x^{2}} y+2 y^{3} d y \)
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  1. Find the derivatives of the following functions. (a) \( f(x)=e^{7 x^{3}+2 x} \) (f) \( f(x)=\sqrt{\frac{7 e^{5 x}}{x^{2}-5}} \) (b) \( y=\left(3^{x}\right)^{2} \) (c) \( y=3^{x^{2}} \) (g) \( y=\fr
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  Find \( y^{\prime \prime} \) for \( y=x^{5}\left(x^{7}-2\right)^{6} \) \[ y^{\prime \prime}= \]
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  Find \( y^{\prime} \) where \( y=\sin ^{14} x+e^{\cos (23 x)} \) Answer: \[ \begin{array}{l} 0 y^{\prime}=14 \sin ^{13} x \cos x+23 e^{\cos (23 x)} \sin (23 x) \\ y^{\prime}=14 \sin ^{13} x \cos x-23
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  Show the limit \( \lim _{\substack{x \rightarrow 2 \\ y \rightarrow 0}} \frac{\sin y}{2-\sqrt{4-x y}}= \) \( z=\ln \frac{x}{y} \), show \( \left.d z\right|_{(1,1)}= \)
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  \( 7: \) If \( \iiint_{E} f(x, y, z) d V=\int_{0}^{\pi / 2} \int_{0}^{\pi / 4} \int_{0}^{\cos (\theta)} 18 \rho^{2} \sin (\phi) \cos (\phi) d \rho d \theta d \phi \), find \( f(x, y, z) \) \[ f(x, y,
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  P4) Resuelva la siguiente ecuación diferencial usando el método de separación de variables. \( \operatorname{sen} x \operatorname{sen} y d x+\cos x \cos y d y=0 \)
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  P6) Resuelva las siguientes ecuaciones lineales. 1. \( \left(x^{4}+2 y\right) d x-x d y=0 \)
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  P8). La siguiente E.D. Autónoma es tanto separable como lineal. Resuelva la ED por ambos métodos: \[ y^{\prime}=0.3-0.2 y \]
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  Evaluate \( \iiint_{\mathcal{B}} f(x, y, z) d V \) for the specified function \( f \) and \( \mathcal{B} \) : \[ f(x, y, z)=\frac{z}{x} \quad 3 \leq x \leq 6,0 \leq y \leq 6,0 \leq z \leq 10 \] \[ \ii
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• differential equations
  1. Ley de enfriamiento de Newton (33 puntos) Un cuerpo a una temperatura desconocida se pone en un refrigerador a una temperatura de 1 \( { }^{\circ} \mathrm{F} \). Si después de 20 minutos la temper
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  Suppose that \( f^{\prime}(x)=2 x \) for all \( x \). a) Find \( f(3) \) if \( f(0)=0 \). b) Find \( f(3) \) if \( f(1)=0 \) c) Find \( f(3) \) if \( f(-4)=18 \).
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• differential equations
  Resuelve las siguientes ED's y de ser posible expresa la solución en forma explícita: b) ED de 2do orden: \( 5 \frac{d^{2} x}{d t^{2}}+26 \frac{d x}{d t}+5 x=0, \quad x(0)=-0,1, \quad x^{\prime}(0)=
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• The general solution of the differential equation is independently linear which is:
  La solución general de la ecuación diferencial \( x^{2} y^{\prime \prime}+x y^{\prime}-y=0, x \neq 0 \), si \( y_{1}=x \) es una solución linealmente independiente, es: а. \( y=c_{1} x+c_{2} x^{-3
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  Differentiate the function. \[ y=\ln \left(e^{x}+x e^{x}\right) \] \[ y^{\prime}= \]
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• The general solution of the differential equation is:
  ión general de la ED \( y^{\prime \prime}+2 y^{\prime}+y=\frac{e^{-x}}{x} \) \[ \begin{array}{l} y(x)=e^{-x}\left(c_{1}+c_{2} x-x \ln |x|\right) \\ y(x)=e^{-x}\left(c_{1}+c_{2} x+2 x \ln |x|\right) \
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• Find the general solution of the differential equation:
  Halle la solución general de la ED \( y^{\prime \prime}-4 y^{\prime}+4 y=x e^{2 x} \), es: a. \( y(x)=e^{2 x}\left(c_{1}+c_{2} x+\frac{x^{3}}{6}+3 x\right) \) b. \( y(x)=e^{2 x}\left(c_{1}+c_{2} x+\f
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  Evaluate \( \iiint_{E} 2 x z d V \) where \( E=\{(x, y, z) \mid 0 \leq x \leq 2, x \leq y \leq 2 x, 0 \leq z \leq x+2 y\} \)
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  Factor out all factors common to all terms. \[ 8 b d(6 k+u)-(6 k+u) \] \[ 8 b d(6 k+u)-(6 k+u)= \]
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  Evaluate \( \iiint_{E} 2 x z d V \) where \( E=\{(x, y, z) \mid 2 \leq x \leq 4, x \leq y \leq 2 x, 0 \leq z \leq x+2 y\} \)
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  Evaluate \( \iiint_{B}\left(4 z^{3}+3 y^{2}+2 x\right) d V \) \[ B=\{(x, y, z) \mid 0 \leq x \leq 9,0 \leq y \leq 10,0 \leq z \leq 1\} \]
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• help
  If \( f(x)=\int_{-1}^{x} t^{7} d t \) then \[ f^{\prime}(x)= \] \( f^{\prime}(3)= \)
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  Find \( \frac{d^{2} y}{d x^{2}} \) if \( y=\cos (4 x-1) \)
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  Find \( \frac{d^{2} y}{d x^{2}} \) if \( y=\cos (3 x+4) \)
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• differentiate its cot(cos^2
  1) \( y=\sqrt[3]{\sqrt[4]{\sqrt{\cot \left(\cos ^{2} \sqrt{5 x^{3}-7^{x}}\right.}}} \) 2) \( y=\frac{\tan x \sin x}{\csc x}-\frac{3 x^{21}-x \sqrt{3 x}-8}{x^{2}}+x^{-3 e} \)
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• pls show all steps
  \( y=\cot \left(11 x^{2}+24\right) \sqrt{13 x^{3}+11} \)
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  Find all the second partial derivatives. \[ f(x, y)=\sin ^{2}(m x+n y) \] \( f_{x x}(x, y)= \) \[ f_{x y}(x, y)= \] \( f_{y x}(x, y)= \) \[ f_{y y}(x, y)= \]
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• The interval centered at x=0 so that the initial value problem ( x − two ) Y " + 3 Y = x have unique solution is ( − 1 , 1 ) true or false
  El intervalo centrado en \( \mathrm{x}=0 \) para que el problema de valor inicial \( (x-2) y^{\prime \prime}+3 y=x \) tenga solución única es \( (-1,1) \) Seleccione una: Verdadero Falso
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• Determine the general solution of the differential equation given
  Halle la solución general de la ED \( y^{\prime \prime}+4 y=\sin x, y(0)=y^{\prime}(0)=1 \), es: a. \( y(x)=\cos 2 x+\frac{1}{3}(\sin x+2 \sin 2 x) \) b. \( y(x)=\cos 2 x+\frac{1}{3}(2 \sin x+\sin 2
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• Find dy/dx for it
  (a) \( y=\frac{x^{2 x}(x-1)^{3}}{(3+5 x)^{4}} \)
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  If \( y=\tan x-\cot x \), then \( \frac{d y}{d x}= \) (A) \( \sec x \csc x \) (B) \( \sec x-\csc x \) (C) \( \sec x+\csc x \) (D) \( \sec ^{2} x-\csc ^{2} x \) (E) \( \sec ^{2} x+\csc ^{2} x \) D B C
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  Suppose \( y=\sin ^{2} x+2^{\sin x} \) Answer:
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  Let \[ f(x)=\ln \left[x^{2}(x+5)^{3}\left(x^{2}+7\right)^{9}\right] \] \[ f^{\prime}(x)= \]
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  EXAMPLE Complete all possible steps for the curve \( \quad y=f(x)=\frac{\sin (x)}{2+\cos (x)} \)
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  9. Differentiate a) \( y=\frac{1}{2} \sin (3 x+2) \) b) \( y=\sin ^{4} x+\cos x \) c) \( v(t)=\frac{1}{\cos ^{2} x} \) d) \( y=6 \sin (x)-2 x^{4} \) e) \( y=3 x^{4} \sin \left(x^{2}\right) \) f) \( \m
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  Find \( y^{\prime} \) where \( y=\cot \left(12 x^{2}+18\right) \sqrt{13 x^{3}+16} \) Answer \[ y^{\prime}=-24 x \csc ^{2}\left(12 x^{2}+18\right) \sqrt{13 x^{3}+16}+\cot \left(12 x^{2}+18\right) \frac
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• Given the differential equation, find the particular solution but do not solve it.
  b. (10 puntos) Dada la ecuacion diferencial \( y^{(4)}+4 y^{\prime \prime}=x^{3}+x+\cos 2 x+e^{2 x} \), escriba pero no halle la solución particular.
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• HELP ME SOLVE THIS USING HIGHER DE
  \( y^{(I V)}+625 y=0 \) \( \left(D^{4}-81\right) y=0 \)
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