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Calculus Archive: Questions from November 12, 2022

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  Let \( y=\exp \left(\sqrt{8+5 x^{7}}\right) \) \[ d y / d x= \]
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  Evaluate \( \int_{0}^{1}\left(x^{2} \sin y-x \cos (y)\right) d x \) \[ \begin{array}{l}\frac{2 \sin y-3 \cos y}{6}+C \\ \frac{-3 \sin y+2 \cos y}{6}+C \\ x^{2}(1-\cos (1)-x \sin (1)+C \\ x^{2}(1+\sin
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  Given \( f(x, y)=e^{x} \cos y \), Evaluate \( f_{x}(2,0) \) \( e^{2} \) 0 \( \cos 2 \) \( \sin 2 \)
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  Solve the differential equation. \[ \frac{d y}{d x}=9 \sqrt{x y} \] \[ y=\left(3 x^{3 / 2}+C\right)^{2} \] \[ y=9 x^{3}+C \] \[ y=3 x^{3 / 2}+C \] \[ y=3 x^{3}+x^{3 / 2}+C \]
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  Solve the DE \[ \left(D^{4}-24 D^{3}+192 D^{2}-512 D\right) y=0 ; y(0)=3, y^{\prime}(0)=15, y^{\prime \prime(0)}=108, y^{\prime \prime \prime}(0)=736 \] Evaluate \( y\left(\frac{1}{12}\right) \) to th
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  Solve the DE \[ \left(D^{3}+3 D^{2}+4 D+12\right) y=0 ; y(0)=4, y^{\prime}(0)=-9, y^{\prime \prime}(0)=23 \] Evaluate \( y\left(\frac{\pi}{12}\right) \) to the nearest integer.
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  Solve the DE \[ \left(D^{3}-13 D^{2}\right) y=0 ; y(0)=1, y^{\prime}(0)=12, y^{\prime \prime}(0)=169 \] Evaluate \( y\left(\frac{1}{7}\right) \) to the nearest integer.
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  \( \int_{0}^{\sqrt{\pi}} \int_{0}^{x} \int_{0}^{x z} x^{2} \sin y d y d z d x \)
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  1. Solve the initial value problem \[ \mathbf{y}^{\prime}=\left[\begin{array}{ccc} -1 & 4 & 2 \\ -2 & 5 & 2 \\ 1 & -2 & 0 \end{array}\right] \mathbf{y}, \quad \mathbf{y}(0)=\left[\begin{array}{c} 7 \\
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  3. [-/5 Points \( ] \) SCALCET9 10.1.001.MI. For the given parametric equations, find the points \( (x, y) \) corresponding to the parameter values \( t=-2,-1 \), 0 , 1,2 . \[ \begin{array}{ll} & x=9
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  \( 2 \quad \) Compute \( \frac{d y}{d x} \) if (a) \( y=\sin \left(e^{2 x}\right) \) (b) \( y=\ln \left(\frac{e^{x}}{1+e^{x}}\right) \) (c) \( y=x^{2} e^{2 x} \) (d) \( y=\left[\tan \left(e^{3 x}\righ
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  If \( \log _{3}\left(x^{7} \sqrt[3]{y^{17}}\right)=A \log _{3} x+B \log _{3} y \) then \( A= \) \( B= \)
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  Express the integral \( \iiint_{E} f(x, y, z) d V \) as an iterated integral in six different ways, where \( \mathrm{E} \) is the solid bounded by \( z=0, z=7 y \) and \( x^{2}=1-y \). 1. \( \int_{a}^
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  \( \iiint_{B} f(x, y, z) d V \) \( f(x, y, z)=x y \) \( B=\left\{(x, y, z) \mid x^{2}+y^{2} \leq 1, x \geq 0, x \geq y,-1 \leq z \leq 1\right\} \)
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• Compute dydxdydx for y=(3x2−x)23−x2y=(3x2−x)23−x2 Select one: a. 2(3x2−x)(6x−1)(3−x2)22(3x2−x)(6x−1)(3−x2)2 b. 2(3x2−x)(6x−1)(3−x2)−2x(3x2−x)2(3−x2)22(3x2−x)(6x−1)(
  Compute \( \frac{d y}{d x} \) for \( y=\frac{\left(3 x^{2}-x\right)^{2}}{3-x^{2}} \) Select one: a. \( \frac{2\left(3 x^{2}-x\right)(6 x-1)}{\left(3-x^{2}\right)^{2}} \) b. \( \frac{2\left(3 x^{2}-x\r
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• Compute dydxdydx for y=(3x2−x)23−x2y=(3x2−x)23−x2 Select one: a. 2(3x2−x)(6x−1)(3−x2)22(3x2−x)(6x−1)(3−x2)2 b. 2(3x2−x)(6x−1)(3−x2)−2x(3x2−x)2(3−x2)22(3x2−x)(6x−1)(
  Compute \( \frac{d y}{d x} \) for \( y=\frac{\left(3 x^{2}-x\right)^{2}}{3-x^{2}} \) Select one: a. \( \frac{2\left(3 x^{2}-x\right)(6 x-1)}{\left(3-x^{2}\right)^{2}} \) b. \( \frac{2\left(3 x^{2}-x\r
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  Q.1: Find \( \mathrm{dy} / \mathrm{dx} \), given that \[ y=\tan x \] ii) \( y=6^{x} \) Differentiate \( y=e^{\sin x} \). iii)
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  In Problems 1-18 solve each differential equation by variation of parameters. 2. \( y^{\prime \prime}+y=\tan x \)
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• find the derivative
  (c) \( y=\sqrt{\frac{e^{2 x}\left(3 x^{2}+4\right)^{3}(8 x-1)^{4}}{x^{2}+5 x+7}} \)
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  Problem 3 (4 pts). Compute \[ \int_{-1}^{1} \int_{|x|}^{1} e^{x^{2}+y^{2}} \sin (x y) d y d x \]
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• Please explain in detailed steps because I am very confused on this.
  Consider the integral \[ \int_{0}^{6} \int_{0}^{3-x / 2} \int_{0}^{2-x / 3-2 y / 3} f(x, y, z) d z d y d x \] Write this integral in the order \( d x d z d y \). \[ \int_{0}^{6} \int_{0}^{2-3 y / 2} \
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  14. Verify the following integral \[ \int_{0}^{\pi} \frac{\sin n \theta \sin \theta d \theta}{\cos \theta-\cos \eta}=-\pi \cos n \eta \] Hint: See von Mises
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  5. \( \int \frac{(2 x+1) d x}{(x+4)^{6}(x-3)^{6}} \) 6. \( \int \sin ^{4}\left(e^{3 x}\right) \cos \left(e^{3 x}\right) e^{3 x} d x \)
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  Solve the initial value problem \[ y^{\prime \prime}+1 x y^{\prime}-4 y=0, y(0)=2, y^{\prime}(0)=0 \] \[ y= \]
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  Solve the initial - value problem (IVP) 1) \( \frac{d y}{d x}=2 x \cos ^{2} y, \quad y(0)=\frac{\pi}{4} \) 2) \( \frac{d y}{d x}=\sec \frac{y}{x}+\frac{y}{x}, \quad y(1)=\frac{\pi}{2} \) 3) \( (\tan y
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  Dado que \( f(x, y)=6 x^{2}+5 x^{3} y^{4}-10 y^{5} \), luego \( f_{x}(1,1)+f_{y}(1,1)= \)
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• Find f(1,2,3)
  Halle \( f(1,3,2) \) dado que \( f(x, y, z)=\frac{\sqrt{x^{2}+y}}{z^{2}} \)
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  Dado que \( f(x, y)=x\left(e^{x y}+y\right) \), entonces \( f_{x}(x, y)= \)
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  Solve the initial value problem \[ y^{\prime \prime}+4 x y^{\prime}-16 y=0, y(0)=4, y^{\prime}(0)=0 \]
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  \( x=8 v-9 u, y=2 v-5 u \) implies \( \frac{\partial(u, v)}{\partial(x, y)}= \)
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  6. (21 pts) Find the derivative of each function. a. \( y=\ln (2 x)+5 e^{3 x} \) b. \( y=x^{6}-\pi x^{4}+2 x+e^{x} \) c. \( y=\sqrt{x}-4 x^{2} \) d. \( y=\cos \left(x^{4}\right) \) e. \( y=\ln (\sin x
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  Find the partial derivatives of the function \[ f(x, y)=x y e^{2 y} \] \[ \begin{array}{l} f_{x}(x, y)=y e^{2 y} \\ f_{y}(x, y)=2 x y e^{2 y} \\ f_{x y}(x, y)= \\ f_{y x}(x, y)= \end{array} \]
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  \[ \begin{array}{l} ), \quad R=\left\{(x, y) \mid \begin{array}{l} -a \leq x \leq a \\ 0 \leq y \leq \sqrt{a^{2}-x^{2}} \end{array}\right\} \\ A=\int_{R} y d x d y \end{array} \] HINT: Use the polar t
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