Calculus Archive: Questions from April 14, 2023
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Exercise: Differentiate (a) \[ y=\ln \left(x^{2} \sqrt{3 x-2)}\right) \] (b) \[ y=\log _{3}\left(x^{2}+2 x+3\right) \] (c) \[ y=\log _{e}\left[(5 x+2)^{4}(8 x-3)^{6}\right] \] (d) \[ y=x^{2} e^{2 x} \2 answers -
(8 points) Determine if \[ f(x, y)=\left\{\begin{array}{ll} \frac{x y^{2}}{\sqrt{x^{4}+y^{8}}} & ,(x, y) \neq(0,0) \\ 0 & ,(x, y)=(0,0) \end{array}\right. \] everywhere.0 answers -
differentiate
\( \begin{array}{c}y=e^{\sqrt{\ln x}} \\ y=e^{\sin \left(t^{2} \sin ^{2} t\right)}\end{array} \)2 answers -
differentiate
\( \begin{array}{c}y=e^{\sqrt{\ln x}} \\ y=e^{\sin \left(t^{2} \sin ^{2} t\right)}\end{array} \)2 answers -
\( \begin{array}{l}f(x)=\sec (3 x-1) \\ f(x)=\tan \left(x^{2}+1\right) \\ f(x)=\cot ^{3}(2 x) \\ f(x)=\cos (\tan x) \\ f(x)=\frac{\cos 2 x}{2+\sin x} \\ r(x)=\sqrt{\sin u \cos \frac{u}{2}}\end{array}2 answers -
Let \( F(x, y, z)=\left(7 x z^{2}, 6 x y z,-4 x y^{3} z\right) \) be a vector field and \( f(x, y, z)=x^{3} y^{2} z \). \[ \begin{array}{l} \nabla f=( \\ \nabla \times F=( \\ F \times \nabla f=( \\ F3 answers -
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Parameter a = 6 , b= 4
Hallar la solución de este problema de valor inicial usando el método de variación de parámetros. \[ y^{\prime \prime}+a^{2} y=b \cos ^{2}(a x), \quad y(0)=0, \quad y^{\prime}(0)=0 \] Usar tus val2 answers -
\( \nabla \phi=\left(2 x y+z^{2}\right) \hat{\mathbf{i}}+\left(x^{2}+z\right) \hat{\mathbf{j}}+(y+2 x z) \hat{\mathbf{k}} \)2 answers -
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Simplify the algebraic expression. \[ \frac{x^{2}+2 x y+y^{2}}{\frac{x}{y}-\frac{y}{x}} \] \[ \frac{x^{2}+2 x y+y^{2}}{\frac{x}{y}-\frac{y}{x}}= \]2 answers -
alủe \( \int(x+7) d x \) \[ \begin{array}{l} x^{2}+7 x+C \\ x^{2}-7 x+C \\ \frac{1}{2} x^{2}+7 x+C \end{array} \] Ninguna de las anteriores2 answers -
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Evalúe \( \int\left(\sec ^{2} \theta-\sin \theta\right) d \theta \) \[ \begin{array}{l} \csc \theta+\tan \theta+C \\ \tan \theta+\cos \theta+C \\ \sec ^{2} \theta+C \end{array} \] Ninguna de las ante2 answers -
Find the first partial derivatives. See Example 1. \[ \begin{array}{l} \quad h(x, y)=e^{-\left(x^{7}+y^{7}\right)} \\ h_{x}(x, y)= \\ h_{y}(x, y)= \end{array} \]2 answers -
Find the first partial derivatives. See Example 1. \[ \begin{array}{l} g(x, y)=9 e^{x / y} \\ g_{x}(x, y)= \\ g_{y}(x, y)= \end{array} \]2 answers -
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Find the first partial derivatives of the function. \[ \begin{array}{c} f(x, y)=\frac{4(x-y)}{x+y} \\ f_{x}(x, y)= \\ f_{y}(x, y)= \end{array} \]2 answers -
Find the first partial derivatives of the function. \[ \begin{array}{l} f(x, y, z)=\frac{8 x}{y+z} \\ f_{x}(x, y, z)= \\ f_{y}(x, y, z)= \\ f_{z}(x, y, z)= \end{array} \]2 answers -
Find the first partial derivatives of the function. \[ \begin{array}{l} f(x, y, z, t)=x y^{2} z^{9} t^{3} \\ f_{x}(x, y, z, t)= \\ f_{y}(x, y, z, t)= \\ f_{z}(x, y, z, t)= \\ f_{t}(x, y, z, t)= \end{a2 answers -
Find all the second partial derivatives. \[ \begin{array}{l} f(x, y)=x^{6} y^{7}+2 x^{7} y \\ f_{x x}(x, y)= \\ f_{x y}(x, y)= \\ f_{y x}(x, y)= \\ f_{y y}(x, y)= \end{array} \]2 answers -
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