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Calculus Archive: Questions from October 31, 2022

• If 𝑔(𝜃) = 𝜃 sin 𝜃, find 𝑔′′(𝜋/6).
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• (a) 𝑦 = √𝑒2𝑥 + 2𝑥3 (b)𝑦= tan𝑥 1+cos 𝑥 (c) 𝑦 = 𝑥3𝑒5𝑥 (d) 𝑦 = (𝑥2 + sin 𝑥)10 (e) 𝑦 = 𝑒sec 𝑥 (f) 𝑦 = tan2(sin 𝑥)
  Problem Set 8. Differentiate the following: (a) \( y=\sqrt{e^{2 x}+2 x^{3}} \) (b) \( y=\frac{\tan x}{1+\cos x} \) (c) \( y=x^{3} e^{5 x} \) (d) \( y=\left(x^{2}+\sin x\right)^{10} \) (e) \( y=e^{\sec
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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
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  Find partial derivatives \( f_{x}(x, y, z), f_{y}(x, y, z) \), and \( f_{z}(x, y, z) \). \[ f(x, y, z)=x e^{14 y}+y e^{17 z}+z e^{9 x} \] (Use symbolic notation and fractions where needed.) \[ f_{x}(x
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  Let \( f(x, y, z)=4 x y \sin (5 z)-3 y z \sin (3 x) \). Find \( f_{x}(x, y, z), f_{y}(x, y, z) \), and \( f_{z}(x, y, z) \). (Use symbolic notation and fractions where needed.) \[ f_{x}(x, y, z)= \] \
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  Let \( f(x, y, z)=5 x y+6 y z+7 x z \). Find \( f_{x}(x, y, z), f_{y}(x, y, z) \), and \( f_{z}(x, y, z) \). (Use symbolic notation and fractions where needed.) \[ f_{x}(x, y, z)= \] \[ f_{y}(x, y, z)
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  Given: \[ z=x^{3}+x y^{2}, \quad x=u v^{3}+w^{4}, \quad y=u+v e^{w} \] Find \( \frac{\partial z}{\partial u} \) when \( u=1, v=-1, w=0 \)
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  \( y=\frac{3}{5} x^{10}-\frac{1}{6} x^{3} \), find \( \frac{d^{2} y}{d x^{2}} \)
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  1. Find \( f_{x}(x, y) \) for \( f(x, y)=e^{x y}(\cos x \sin y) \) : (a) \( -y e^{x y}(\sin x \sin y) \) (b) \( e^{\mp}(\sin y)(y \cos x-\sin x) \) (c) \( e^{y}(\sin x \sin y) \) (d) \( e^{x y}(\sin y
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  Problem Set 8. Differentiate the following: (a) \( y=\sqrt{e^{2 x}+2 x^{3}} \) (b) \( y=\frac{\tan x}{1+\cos x} \) (c) \( y=x^{3} e^{5 x} \) (d) \( y=\left(x^{2}+\sin x\right)^{10} \) (e) \( y=e^{\sec
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  3. Find \( f_{4,}(x, y) \) for \( f(x, y)=\frac{4 x^{2}}{y}+\frac{y^{2}}{2 x^{2}} \). (a) \( -\frac{8 x}{y^{2}}-\frac{y}{x^{2}} \) (b) \( 8 x-y \) (c) \( 8 x+y \) (c) None of these
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  Find the derivative. \[ y=\left(\tan ^{-1} x\right)^{9} \]
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• If x = tsins and y =tcoss, find
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• #6,8,10 please
  In Exercises 1-10, find the partial derivative with respect to each variable. 1. \( f(x, y)=x^{2}-y^{2} \) 2. \( f(x, y)=x^{2} y^{2} \) 3. \( f(x, y)=\frac{x^{2}}{y^{2}} \) 4. \( f(x, y)=\cos (x y) \)
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• Determine whether lim (x,y)→(0,0)( x sin y )/ (x^2 + y^5/3 )exis
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  \( y=\left(1+\sin ^{3}\left(x^{2}\right)\right)^{9} \)
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  3. \( \int e^{3 x} \sin 4 x d x \) (6 pts) 4. \( \int \cos ^{3} 2 x \sin ^{5} 2 x d x \)
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  2. Solve the initial-value problem \( y^{\prime}=e^{-y} \sin x \) where \( y\left(\frac{\pi}{2}\right)=\frac{1}{2} \).
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  7-18 Evaluate the double integral. 7. \( \iint_{D} y^{2} d A, \quad D=\{(x, y) \mid-1 \leqslant y \leqslant 1,-y-2 \leqslant x \leqslant y\} \)
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• Show all work please. need asap
  Solve the initial value problem. \[ y^{\prime}+y=2 e^{x} ; y(0)=22 \] (A) \( y=22 e^{x} \) (B) \( y=4 e^{2}+20 e^{-x} \) (C) \( y=e^{x}+21 e^{-x} \) (D) \( y=2 e^{x}+19 e^{-x} \)
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  (1 point) Solve \( \begin{array}{ccc} & x^{\prime}=y & x(0)=0 \\ x(t)=[ & y^{\prime}=5 x+4 y & y(0)=5 \\ y(t)= & \text { help (formulas) } & \\ & \text { help (formulas) }\end{array} \)
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  Sea \( E=\left\{(z, y, z)\right. \) i \( \left.4 \leq z^{2}+y^{2} \leq 9,0 \leq z \leq 5-x-y\right\} \) y calculemos la integral \( \iint_{E} x d V \). Se puede calcular la integral triple en coordena
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  Use the Stokes' theorem to evaluate the line integral \( \int_{C} F \cdot d r \). Sketch \( S \) and \( C \). a. \( F(x, y, z)=(-y+z) i+(x-z) j+(x-y) k, S: z=9-x^{2}-y^{2}, z \geq 0 \) b. \( F(x, y, z
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• x = y + (x2 - 1)2
  Solve the differential equation. \[ x \frac{d y}{d x}=y+\left(x^{2}-1\right)^{2} \] (A) \( y=\frac{1}{3} x^{4}-2 x^{2}-1+C x \) B \( y=x^{4}-x^{2}-1+C x \) (C) \( y=\frac{1}{3} x^{3}-2 x-\frac{1}{x}+C
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• y ' + y = 2ex; y(0) = 22
  Solve the initial value problem. \( y^{\prime}+y=2 e^{x} ; y(0)=22 \) (A) \( y=22 e^{x} \) (B) \( y=4 e^{2}+20 e^{-x} \) (C) \( y=e^{x}+21 e^{-x} \) (D) \( y=2 e^{x}+19 e^{-x} \)
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• 2 - 4xy = 8x; y(0) = 23
  Solve the initial value problem. \[ 2 \frac{d y}{d x}-4 x y=8 x ; y(0)=23 \] (A) \( y=2+23 e^{x 2} \) (B) \( y=-2+25 e^{-x 2} \) (C) \( \mathrm{y}=-1+24 \mathrm{e}^{\mathrm{x} 2} \) (D) \( y=-2+25 \ma
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  Compute the second-order partial derivatives of \[ f(x, y)=e^{5 x^{2}+6 y^{2}} \] \[ \begin{array}{l} \frac{\partial^{2} f}{\partial x^{2}}(x, y)= \\ \frac{\partial^{2} f}{\partial x \partial y}(x, y)
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  Given \( f(x, y)=4 x^{5} \cos \left(y^{9}\right) \) \[ f_{x y}(x, y)= \] \( f_{y y}(x, y)= \)
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  Given \( f(x, y)=x^{6}+6 x^{2} y^{3}+4 y^{4} \) \[ f_{x}(x, y)= \] \[ f_{y}(x, y)= \] \[ f_{x x}(x, y)= \] \[ f_{x y}(x, y)= \]
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• Just 16. thank you!
  Compute the curl, \( \nabla \times \mathbf{F} \), of the vector fields in Exercises 13-16, 3.. \( \mathbf{F}(x, y, z)=x \mathbf{1}+y \mathbf{j}+z \mathbf{k} \) 14. \( \mathbf{F}(x, y, z)=y z \mathbf{i
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  Find each limit. \[ f(x, y)=9 x^{2}+4 y^{2} \] (a) \( \lim _{\Delta x \rightarrow 0} \frac{f(x+\Delta x, y)-f(x, y)}{\Delta x} \) (b) \( \lim _{\Delta y \rightarrow 0} \frac{f(x, y+\Delta y)-f(x, y)}{
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  \( f(x, y)=\left(3-12 x^{2}\right)\left(y^{2}+6 y\right) \)
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  (1 point) Find \( d y / d x \) if \( y=\sin ^{-1}(8 x) \) Answer:
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• Find y' and y" for x2+8xy-5y2=7
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  8. Find all possible functions for the derivative \( y^{\prime}=3 x^{3}+4 x^{2}-1 \). 9. Find all possible functions for the derivative \( y^{\prime}=2^{3 x}+\sec x \tan x \)
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• EVALUATE, BOUD BY Y=X^2, Y=2X
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  (1 point) Solve \[ \begin{array}{ll} x^{\prime}=y & x(0)=0 \\ y^{\prime}=5 x+4 y & y(0)=5 \end{array} \] \( x(t \quad \) help (formulas) \( y(t) \quad \) help (formulas)
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  (1 point) Differentiate \( y=\csc x(x+\cot x) \) \[ y^{\prime}= \] (1 point) If \( f(x)=\sqrt{x} \sin x \), \[ f^{\prime}(x)= \]
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• Solve the following de: y′′ = 24x^2 + 4x −8
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• Solve the following de: y′′ −2y′ + 10y = −104 sin^2(x)
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• Solve the following de: y′′ + 4y′ + 4y = x^(−2)e^(−2x), x > 0.
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• Calculate the integral (∭E x dV) E={(x,y,z): 4≤x2+y2≤9, 0≤z≤5−x−y}E={(x,y,z): 4≤x2+y2≤9, 0≤z≤5−x−y} The triple integral can be calculated in Cartesian coordinates or in polar
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  A) If \( x y^{2}+3 x+9 y=132 \) and \( y(2)=6 \) then \( y^{\prime}(2) \) B) If \( y^{3}+y^{2}=9 x^{2}+216 \) and \( y(2)=6 \) then \( y^{\prime}(2) \)
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• match the solutions ​​​​​​​
  1. \( \frac{d y}{d x}=-2 y \) 2. \( \frac{d^{2} y}{d x^{2}}=-4 y \) 3. \( \frac{d^{2} y}{d x^{2}}=4 y \) 4. \( \frac{d y}{d x}=2 y \) A. \( y=\sin (2 x) \) or \( y=2 \sin (x) \) B. \( y=2 \sin (x) \)
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  \( 3.4 \) wERK 8 HN HELP (4) \( C(x)=0.0002 x^{3}-0.06 x^{2}+120 x+5000 \) (a) ACTUAL COST OF MANUFACTURING \( n^{\text {th }} \) OVEN \( =C(n)-C(n-1) \) (i) \( C(101)-C(100)= \) (ii) \( C(201)-C(200)
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• 5 a,b,c,d,e
  (5) Evaluate the integral (a) \( \int_{-1}^{1} \frac{1}{16+t^{2}} d t \) (b) \( \int_{0}^{1 / 2} \tan ^{-1}(2 x) d x \) (c) \( \int(\ln (x))^{2} d x \) (d) \( \int_{0}^{t} e^{x} \cos 2 x d x \) (e) \(
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• Translation: E... calculate the integral The triple integral can be calculated in Cartesian coordinates or in polar coordinates. The answer is numerical, write it with at least two correct decimals.
  Sea \( E=\left\{(x, y, z): 4 \leq x^{2}+y^{2} \leq 9,0 \leq z \leq 5-x-y\right\} \) y calculemos la integral \( \iiint_{E} x d V \). Se puede calcular la integral triple en coordenadas cartesianas o e
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  Use logarithmic differentiation to find \( \mathrm{y}^{\prime} \). \[ y=\frac{\sqrt{4-9 x}\left(x^{2}+1\right)^{2}}{x^{2}+7 x+3} \] \[ y^{\prime}= \]
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  \( y=4 x^{4}-2 x^{5}-4 x+2 \)
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  \( y^{\prime \prime}+y=u_{1}(t)-u_{3}(t) \)
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  Let \( f(x, y, z)=4 x y+7 y z+7 x z \). Find \( f_{x}(x, y, z), f_{y}(x, y, z) \), and \( f_{z}(x, y, z) \). (Use symbolic notation and fractions where needed.) \[ f_{x}(x, y, z)= \] \[ f_{y}(x, y, z)
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  Given \( f(x, y)=2 x^{3} \ln \left(y^{6}\right) \) \[ f_{x x}(x, y)= \] \[ f_{y y}(x, y)= \]
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  Determine whether the sequence converges or diverges. If it converges, find the limit. (If an answer does not exist, enter DNE.) \[ \begin{array}{r} \left\{\frac{e^{n}+e^{-n}}{e^{2 n}-4}\right\} \\ \l
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  Evaluate the triple integral \( \iiint_{E} f\left(x_{r}, y, z\right) d V \) over the solid \( E \). \[ f(x, y, z)=z, E=\left\{(x, y, z) \mid x^{2}+y^{2} \leq 9, x \geq 0, y \geq 0,0 \leq z \leq 1\righ
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  Given \( f(x, y)=5 x^{3}-3 x^{2} y^{5}+6 y^{6} \), \( f_{x}(x, y)= \) \[ f_{y}(x, y)= \] \[ f_{x x}(x, y)= \] \[ f_{x y}(x, y)= \]
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  Given \( f(x, y)=-2 x^{4}-6 x y^{5}-4 y^{6} \) \( f_{x x}(x, y)= \) \[ f_{x y}(x, y)= \]
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• find the derivative of the function a) y' = b)F '(𝜃) = c)y' =
  \( y=\left(\tan ^{-1}(3 x)\right)^{2} \) \( F(\theta)=\arcsin (\sqrt{\sin (17 \theta)}) \) \( y=\arctan \sqrt{\frac{1-x}{1+x}} \)
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  Evaluate \( \iiint_{E}(x+y-3 z) d V \) where \( E=\left\{(x, y, z) \mid-3 \leq y \leq 0,0 \leq x \leq y, 0
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