Advanced Math Archive: Questions from September 06, 2022
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1 answer
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Differentiate the following to \( x \) and simplify where possible: 2.1 \( y=x^{4} \tan 3 x \) 2.2 \( y=\ln \left(2 x^{2}+1\right) \) \( 2.3 \quad y=\frac{e^{x}+1}{e^{x}-1} \) 2.4 \( y=\frac{2 \sin x}1 answer -
only A C D
4. Solve the initial value problem. (a) \( y^{\prime}=-x e^{x}, \quad y(0)=1 \) (b) \( y^{\prime}=x \sin x^{2}, \quad y\left(\sqrt{\frac{\pi}{2}}\right)=1 \) (c) \( y^{\prime}=\tan x, \quad y(\pi / 4)1 answer -
Let \( f(x, y)=x^{2}+x y+y \). Give an " \( \epsilon, \delta " \) proof that \( \lim _{(x, y) \rightarrow(1,1)} f(x, y)=3 \)1 answer -
\[ g(x, y, z)=\ln (x+y+z) \] a) \( g(0,0,0) \) \( g(1,0,0) \) \( g(0,1,0) \) \( g(z, x, y) \) \( g(x+h, y+k, z+l) \)1 answer -
2. For the vector field \( F(x, y, z)=\left(x^{2} y, x y z,-x^{2} y^{2}\right) \) \( F(x, y, z)=(x 2 y, x y z,-x 2 y \) 2find: (a) \( \operatorname{div}(F) d i v(F) \) (b) \( \operatorname{curl}(F) \o1 answer -
uelva \( 2 x^{2} y d x=\left(3 x^{3}+y^{3}\right) d y \) \[ y^{9}=c\left(x^{3}+y^{3}\right)^{2} \] \[ \frac{c^{3}}{y^{9}}=\left(x^{3}+y^{3}\right)^{2} \] \[ \frac{2}{3} \ln \left(x^{3}+y^{3}\right)^{21 answer -
A=1.8
\( y=\ln (a x) \) \( y=-a x^{2}+6 a \cos x \) \( y=3 a \tan x-2 a \cos x+x-1 \) \( y=\frac{\sin x}{a x} \) \( y=\arccos (a x) \)0 answers -
Problem 5. Let \( X \) be the plane \( \mathbb{R}^{2} \), and let \( \mu_{1}, \mu_{2}, \mu_{3}, \mu_{4}: X \times X \rightarrow \mathbb{R} \) be given by \( \mu_{1}\left((x, y),\left(x^{\prime}, y^{\p1 answer -
\( e c=2 x+y+5 z \) subje \( x+y+z \geq 90 \) \( 2 x+y \quad \geq 50 \) \( y+z \geq 50 \) \( x \geq 0, y \geq 0, z \geq 0 \)1 answer -
2. For the vector field \[ F(x, y, z)=\left(x^{2} y, x y z\right. \] \( F(x, y, z)=(x 2 y, x y z,-x 2 \) y 2 fir (a) \( \operatorname{div}(F) d \operatorname{iv}(\mathrm{F}) \) (b) \( \operatorname{cu1 answer -
Solve the separable initial value problem. 1. \( y^{\prime}=9 x^{2} \sqrt{1+x^{3}}\left(1+y^{2}\right), y(0)=3 \Rightarrow y^{\prime} \) 2. \( y^{\prime}=2 x \cos \left(x^{2}\right)\left(1+y^{2}\right1 answer