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Calculus Archive: Questions from October 29, 2023

• \left(2xy-y^2\right)dx+\left(x^2-2xy\right)dy=0
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• Solve the ODE
  1. \( x \frac{d y}{d x}+y=y^{2} \) 2. \( (2 x+y) d y=(x+2 y) d x \) 3. \( y^{\prime}=\frac{x-y}{x+y} \)
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  Solve the given Initial-value problem [4) (1 Point) \[ \begin{array}{l} y^{\prime \prime}+-4 y=-\square \square 2, \quad y\left(\frac{\pi}{8}\right)=\frac{1}{2}, \quad y^{\prime}\left(\frac{\pi}{8}\ri
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• —e−2x) cos x sin (a) Determine the derivatives of y = (4x³-e-2x) cos 2 y and y = (cos x sin x) 4x³-e-2r and [6 marks]
  (a) Determine the derivatives of \( y=\left(4 x^{3}-e^{-2 x}\right)^{\cos x \sin x} \) and \( y=(\cos x \sin x)^{4 x^{3}-e^{-2 x}} \). [6 marks]
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• 1. Find dy/dx for the function y = ln (1 − 3x^2 )^3/3x + 5 (2x 3 + 4x)
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  Select all the second order, linear, differential equations in the variable \( y(t) \) A. \( y^{\prime \prime}+4 t^{2} y^{\prime}+\ln |t| y=0 \) B. \( y^{\prime}+5 y=t^{3} \) C. \( y^{\prime \prime}+3
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• describir como Howard Eves compara los métodos de Fermator-Barrow con los métodos modernos
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  Find \( \frac{d^{2} y}{d y^{2}} \) given \[ y=\sin x \cos x \] Find \( \frac{d y}{d x} \) given \[ y=\left(3 x^{2}+3 x-1\right)^{4} \]
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• ×^3 - 2×^2 -4x + 2 teorema del valor medio
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• Find if x²y - sinx - cos y = 10. dy dx dy dx dy dx dy dx Ody dx dy dx = = = = -2xy + cos x x² + sin y 2xy - cos x x² - sin y COS X 2x + sin y COS X 2x + sin y - 2xy + cos x x² + sin y
  \( \begin{array}{l}\frac{d y}{d x} \text { if } x^{2} y-\sin x-\cos y=10 \\ \frac{d y}{d x}=\frac{-2 x y+\cos x}{x^{2}+\sin y} \\ \frac{d y}{d x}=\frac{2 x y-\cos x}{x^{2}-\sin y} \\ \frac{d y}{d x}=\
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  \( \int \frac{\cos x}{3+2 \sin x+\sin ^{2} x} d x \)
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  4. Determine los valores de \( k \) tales que la integral \( \int_{1}^{\infty} \frac{(\ln x)^{k}}{x} d x \) converja.
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  Given \( f(x, y)=2 x^{3} \ln (y)+8 \), what is \( f_{x} \) ? \[ \begin{array}{l} f_{x}=6 x^{2} \ln (y) \\ f_{x}=\frac{6 x^{2}}{y}+8 \\ f_{x}=\frac{2 x^{3}}{y} \\ f_{x}=\frac{6 x^{2}}{y} \\ f_{x}=6 x^{
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  Let \( f(x, y)=\sqrt{125-2 x^{2}-3 y^{2}} \) Evaluate \( f(3,4) \)
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• find dy/dx y = 3 secx tan x Find
  Find \( \frac{d y}{d x} \) \[ y=3 \sec x \tan x \]
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• Evaluate the triple integral E f(x, y, z) dV over the solid E. f(x, y, z) = x2 + y2, E = {(x, y, z) | x2 + y2 ≤ 4, x ≥ 0, x ≤ y, 0 ≤ z ≤ 7}
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  Solve for \( x \) : \( \quad 3^{2 x}-90\left(3^{x}\right)+729=0\left[\right. \) HINT: let \( \left.u=3^{x}\right] \)
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  R2 Evaluate the double integrals: (a) \( \iint_{R}\left(x y+y^{2}\right) d x d y \), where \( R=\{(x, y): 0 \leq x \leq 1,0 \leq y \leq 1\} \). (b) \( \iint_{A}(x+2 y) d x d y \) where \( A=\{(x, y):
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• (1 point) Find y as a function of x if y(0) = -3, y (0) = -18, y" (0) = 324. y(x) = +81y = 0,
  (1 point) Find \( y \) as a function of \( x \) if \[ y^{\prime \prime \prime}+81 y^{\prime}=0 \] \[ \begin{array}{l} y(0)=-3, \quad y^{\prime}(0)=-18, \quad y^{\prime \prime}(0)=324 . \\ y(x)= \end{a
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• Find the total differential. w = 8x + y 7z − 4y
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• If f(x) = 3x³ – 4e², find: J f'(4) = SONDE
  If \( f(x)=3 x^{3}-4 e^{x} \), find: \[ \begin{array}{l} f^{\prime}(x)= \\ f^{\prime}(4) \end{array} \] \[ \begin{array}{l} f^{\prime \prime}(x) \\ f^{\prime \prime}(4) \end{array} \]
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• \lim_(x->-\infty )(3x^(7)-4x^(2)+1)/(5-10x^(2))\infty 5 0
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  \( f(x, y)=\sqrt{6-x^{2}-y^{2}}, x+y-2=0 \)
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  Find \( \iint_{D} \cos \left(x^{2}+y^{2}\right) d A \) where \( D=\left\{(x, y) \mid 25 \leq x^{2}+y^{2} \leq 49\right\} \)
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  If \( f(x, y, z)=x e^{4 y} \sin (9 z) \), then the gradient is \[ \nabla f(x, y, z)= \]
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  Establezca una integral que proporcione la longitud de la elipse \( x^{2} / a^{2}+y^{2} / b^{2}=1, a>b>0 \). Sea \( C \) la parte de la parábola \( y=a x^{2}-1 \) dentro del círculo \( x^{2}+y^{2}=1
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  La región acotada por abajo por la parábola \( y=x^{2} \) y por arriba por la recta \( y=4 \) se tiene que dividir en dos subregiones de la misma área, cortándolas con una recta horizontal \( y=c
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• tan(xy) = y use implicit differentiation to find dy/dx
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  Consider the function. \[ f(x, y)=y+x e^{y} \] (a) Find \( \int_{0}^{5} f(x, y) d x \). (b) Find \( \int_{0}^{1} f(x, y) d y \).
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• Find y^(') and y^('') by implicit differentiation. 2x^(3)-3y^(3)=2y^(')=y^('')=
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• Find 2 2 1² FO 3 1³ f(x, y) dx and [nd__ f(x, y) dx = 6³ f(x, y) = 11y √x+2 f(x, y) dy = f(x, y) dy.
  Find \( \int_{0}^{2} f(x, y) d x \) and \( \int_{0}^{3} f(x, y) d y \) \[ \begin{array}{l} f(x, y)=11 y \sqrt{x+2} \\ \int_{0}^{2} f(x, y) d x= \\ \int_{0}^{3} f(x, y) d y= \end{array} \]
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• La razón de cambio de nivel de agua en un contenedor está dada por: r(t)=4t^(3)-12t^(2)+8t centímetros/segundo Si inicialmente (t=0) el contenedor tenía un nivel de 5 centímetros, calcula el nive
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• La razón de cambio de nivel de agua en un contenedor está dada por: r(t)=4t^(3)-12t^(2)+8t centímetros/segundo Si inicialmente (t=0) el contenedor tenía un nivel de 5 centímetros, calcula el nive
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  Find \( \int_{0}^{2} f(x, y) d x \) and \( \int_{0}^{3} f(x, y) d y \) \[ f(x, y)=9 x+3 x^{2} y^{2} \] \[ \begin{array}{l} \int_{0}^{2} f(x, y) d x= \\ \int_{0}^{3} f(x, y) d y= \end{array} \]
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  Consider the function. \[ f(x, y)=y+x e^{y} \] (a) Find \( \int_{0}^{4} f(x, y) d x \). (b) Find \( \int_{0}^{1} f(x, y) d y \).
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• 1. Σ 9 11 46 WIWIW W WI WWI WWI Σ Σ n Inn In(Inn) all this DE TE INSOL z(41) Ε
  1. \( \sum_{n=1} \frac{1}{n \cdot \ln n \cdot \ln (\ln n)} \) 2. \( \sum_{n=2}^{\infty} \frac{1}{n(\ln n)^{n / 2}} \) 3. \( \sum_{n=1}^{\infty} \frac{3 n+2}{5 n+3} \) 4. \( \sum_{n=1}^{\infty} \frac{1
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• Evaluate the triple integral. y dv, where E = {(x,y,z)10SxS2,0SYSx,x-yszs - y ≤ z≤ x + y}
  Evaluate the triple integral. \[ \iiint_{E} y d V \text {, where } E=\{(x, y, z) \mid 0 \leq x \leq 2,0 \leq y \leq x, x-y \leq z \leq x+y\} \]
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• Si f(x)=sec(5x)tan(5x) , encuentra la derivada f^(')(3)
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  3. Let \( f(x)=3 x^{4}-4 x^{3}+6 \). Find all local extrema.
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  If \( f(x)=4 \tan ^{-1}(6 \sin (3 x)) \), find \( f^{\prime}(x) \).
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• Compute the limit/value of the following expressions f) \lim_(z->\infty )(z^(2)+i8)/(z-2) g) \lim_(z->\infty )(3z^(3)-i)/(2z^(2)+1+i)
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• 803. 3x3 +9xy2 = 5x³
  \( 3 x^{3}+9 x y^{2}=5 x^{3} \)
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