Calculus Archive: Questions from October 08, 2023
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5 (Mixed examples). Solve the following differential equations or initial value problems. (a) \( \frac{d y}{d x}=\frac{y(y+1)}{x(x-1)} \) (f) \( \frac{d y}{d x}=(x+y+3)^{2} \) (b) \( \left(y+x^{2} y\r1 answer -
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Please show step by step
30. Find the domain of each function. (5 points) \begin{tabular}{|c|l|} \hline\( y=x^{2} \) & \\ \hline\( y=4 x \) & \\ \hline \( \mathrm{y}=\sqrt{x+2} \) & \\ \hline\( y=\frac{x+2}{x-3} \) & \\ \hlin1 answer -
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3. Solve the IVP using laplace transform; \[ y^{i v}-y=0, \quad y(0)=0, \quad y^{\prime}(0)=0, y^{\prime \prime}(0)=0, y^{\prime \prime \prime}(0)=2 \]1 answer -
Find the solution of the initial value problem: -e-cos(3x) + ¹5 sin (3x) 1/ y = ² e²+ = cos(3x) + +cos(3x)+sin (3x) - e-cos(3x) - 15 sin(3x) 1 - - 룸 + 를 cos(3x) -sin (3x) 15 y-e-cos(3x) - sin(3
Find the solution of the initial value problem: \[ \begin{array}{c} y^{\prime \prime \prime}-y^{\prime \prime}+9 y^{\prime}-9 y=0 \\ {\left[y(0)=0, y^{\prime}(0)=0, y^{\prime \prime}(0)=4\right]} \end1 answer -
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Find \( y^{\prime} \) and \( y^{\prime \prime} \) \[ \begin{array}{r} y=\sqrt{\sin (x)} \\ y^{\prime}=\frac{1}{2}(\sin (x)) \|^{-\frac{1}{2}}(\cos (x)) \end{array} \]1 answer -
In each of the following problems, determine whether or not the separable. Do not solve the equations! 1. \( y^{\prime}=2 y(5-y) \) 2. \( y y^{\prime}=1-y \) 3. \( t^{2} y^{\prime}=1-2 t y \) 4. \( \f1 answer -
1. Find the implicit derivative: a. \( \tan ^{3} x y=y^{4}-x^{2} y+7 \) b. \( \sqrt{\csc y}+y^{2}=\sin x-\sqrt{x} \)1 answer -
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Find dy/dx by implicit differentiation. tan(x- y) = y' = y 7 + x²
Find \( d y / d x \) by implicit differentiation. \[ \tan (x-y)=\frac{y}{7+x^{2}} \]1 answer -
Translation: sketch some solution curves of this differential equation Bosqueja algunas curvas solución de la ED. \[ \frac{\mathrm{dP}}{\mathrm{d} t}=k P\left(1-\frac{P}{N}\right) \]1 answer -
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Question 1. Use implicit differentiation to find \( d y / d x \). 1. \( 1+x=x \sin \left(y^{2}\right) \) 2. \( \sin x+\cos y=\sin x \cos y \)1 answer -
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Number 15
5. \( x=\sqrt{y}-y, \quad 1 \leqslant y \leqslant 4 \) 15. \( y=\frac{1}{4} x^{2}-\frac{1}{2} \ln x, \quad 1 \leqslant x \leqslant 2 \) 6. \( x=y^{2}-2 y, \quad 0 \leqslant y \leqslant 2 \) 16. \( y=\1 answer -
Find all the second partial derivatives. \[ f(x, y)=x^{8} y^{4}+5 x^{6} y \] \[ f_{x x}(x, y)= \] \[ f_{x y}(x, y)= \] \[ f_{y x}(x, y)= \] \[ f_{y y}(x, y)= \]1 answer -
3. Solve the initial value problem \[ \begin{array}{r} y^{(4)}+y^{\prime \prime \prime}-7 y^{\prime \prime}-y^{\prime}+6 y=0, \quad y(0)=5 \\ y^{\prime}(0)=-6, \quad y^{\prime \prime}(0)=10, \quad y^{1 answer -
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K Find y' y' = y = 3x + 4 2x - 5 46 (2x-5)²
Find \( y^{\prime \prime} \) \[ y=\frac{3 x+4}{2 x-5} \] \[ y^{\prime \prime}=-\frac{46}{(2 x-5)^{2}} \]0 answers -
Let \( f(x, y, z)=\frac{x^{2}-5 y^{2}}{y^{2}+6 z^{2}} \). Then \[ f_{x}(x, y, z)= \] \[ f_{y}(x, y, z)= \] \[ f_{z}(x, y, z)= \]1 answer -
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(1 point) Find \( d y / d x \) in terms of \( x \) and \( y \) if \( \cos ^{2}(4 y)+\sin ^{2}(4 y)=y+14 \) \[ \frac{d y}{d x}=-13 \]1 answer -
10. Find \( y " \) \[ y=\left(8+\frac{5}{x}\right)^{4} \] A) \( \frac{300}{x^{4}}\left(8+\frac{5}{x}\right)^{2}+\frac{40}{x^{3}}\left(8+\frac{5}{x}\right)^{3} \) B) \( -\frac{60}{x^{2}}\left(8+\frac{51 answer -
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Find the derivative of the function. \[ y=\frac{x^{2}+8 x+3}{\sqrt{x}} \] A. \( y^{\prime}=\frac{2 x+8}{x} \) B. \( y^{\prime}=\frac{3 x^{2}+8 x-3}{2 x^{3 / 2}} \) C. \( y^{\prime}=\frac{2 x+8}{2 x^{31 answer -
\[ x^{4 y}=y^{5 x} \] 1. \( y^{\prime}=\frac{y\left(4 x^{2} \ln y-5 y\right)}{x\left(5 y^{2} \ln x-4 x\right)} \) 2. \( y^{\prime}=\frac{y^{2}(\ln x-1)}{4 y \ln x-5 x} \) 3. \( y^{\prime}=\frac{y(5 x1 answer -
Determine \( y^{\prime} \) when \[ y=x^{1 / x} . \] 1. \( y^{\prime}=\frac{y}{x^{2}}(\ln x-1) \) 2. \( y^{\prime}=-x y(2 \ln x+1) \) 3. \( y^{\prime}=x y(2 \ln x+1) \) 4. \( y^{\prime}=y(\ln x+1) \) 51 answer -
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Find \( y^{\prime \prime} \) by implicit differentiation. \[ y^{\prime \prime}=\frac{14 x^{2}}{y^{2}} \]1 answer -
Find \( y^{\prime} \) and \( y^{\prime \prime} \) by implicit differentiation. \[ y^{\prime}=\frac{5 x^{3}-4 y^{3}=2}{\frac{5 x^{2}}{4 y^{2}}} \]1 answer -
5. Find the following derivatives. (a) \[ y=6 x^{12}+5 x \] (b) \[ y=32 x^{7} \] (c) \[ y=\left(x^{2}+3\right)\left(x^{4}-5\right) \] (d) \[ y=\frac{2-3 x+x^{2}}{\sqrt{x}} \] (e) \[ y=\frac{x-7}{x+5}1 answer -
Find dy/dx of the following
\( y=\left(e^{\sqrt{x}}+\cos \left(e^{x}\right)\right)^{3} \) \( y=\frac{\left(e^{\sqrt{x}}+\cos \left(e^{x}\right)^{3}\right.}{\sqrt{x-4}} \)1 answer -
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Compute the derivative. \[ \begin{array}{l} R(y)=\frac{6 \cos y-7}{\sin y} \\ R^{\prime}(y)= \end{array} \]1 answer -
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\( \begin{array}{l}y=\left(x^{3}-10 x+2\right)\left(x^{2}+7 x+1\right) . \\ y=\frac{2 x+3}{7-5 x} . \\ y=\frac{x^{2}+1}{2 x^{3}-1} .\end{array} \)1 answer -
6 (10 points) Evaluate \( \iint_{D} 4 x \sqrt{x^{2}+y^{2}} d A \), where \( D:=\left\{(x, y) \mid x^{2}+y^{2} \leq 1, x \leq 0, y \geq 0\right\} \)1 answer -
(b) Let \( w=f(x, y), f_{x}(x, y)=x+y \) and \( f_{y}(x, y)=x y \). If \( x=u^{2} \) and \( y=u v \), then find \( \frac{\partial w}{\partial u} \) in terms of \( u \) and \( v \).1 answer -
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