Calculus Archive: Questions from May 01, 2023
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Q19: Solve \( \frac{d x}{d y}=\frac{\cos x \log (y+1)}{y+1} \) when \( x=\frac{\pi}{4}, y=0 \) (a) \( 2 \log \left|\frac{\sec x+\tan x}{\sqrt{2}+1}\right|=\log ^{2}|1+y| \) (b) \( 2 \log \left|\frac{\2 answers -
Find the following for the function \( f(x, y)=3 x^{3} y^{2}-5 x+26 y-9 \) a. \( f(1,2)= \) b. \( f_{y}(x, y)= \) c. \( f_{x}(x, y)= \) d. \( f_{x}(1,2)= \) e. \( f_{y y}(x, y)= \) f. \( f_{x x}(x, y)2 answers -
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3) If Show that:
3) If \( y=\sqrt{\left(5 x^{2}+3\right)} \) Show that: \[ y \frac{d^{2} y}{d x^{2}}+\left(\frac{d y}{d x}\right)^{2}=5 \]2 answers -
5) If y = ln(1+sin x), show that:
If \( y=\ln (1+\sin x) \) \[ \frac{d^{2} y}{d x^{2}}+e^{-y}=0 \]2 answers -
2. Find the Derivative of each function and simplify fully. a) \( y=e^{-3 x} \cos 2 x \) b) \( y=\frac{x+e^{x}}{e^{2 x}} \) 18 c) \( y=\ln \left[3 x^{5}+\frac{1}{x}\right] \) d) \( y=\left[\ln \left(x2 answers -
¿en que puntos de la particula pasa por el plano z=0? ¿cuales son su velocidad y aceleracion en los puntos del inciso a?
Supongo que \( \bar{r}(t)=t^{2} \vec{\imath}+\left(t^{2}-2 t\right) \vec{J}+\left(t^{2}-5 t\right) \bar{k} \) es el vectar de posición de una partícula. en movimientc a. ¿ En qué puntos la partíc2 answers -
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determina la proyeccion y el componente
2. Determine: proy \( \overline{\bar{b}} \) y \( \operatorname{com} p_{\vec{a}} b \) \[ \text { donde } \vec{a}=\langle 1,2,3\rangle, \vec{b}=\langle 0,1,0\rangle \text {. } \]2 answers -
\[ \int e^{9 x} \cos (2 x) d x= \] A. \( \frac{e^{9 x}}{77}(9 \cos 2 x+2 \sin 2 x)+C \) B. \( \frac{e^{9 x}}{85}(9 \cos 2 x+2 \sin 2 x)+C \) C. \( \frac{e^{9 x}}{121}(9 \cos 2 x+2 \sin 2 x)+C \) D. \(2 answers -
derivatives
\( y=\frac{1}{7 x^{2}}+\frac{1}{11 x} \) \( y=\frac{x^{2}+1}{\sqrt{x}} \) \( y=\frac{\csc x}{x}+\frac{x}{\csc x} \) \( y=\frac{1+\sec x}{1-\sec x} \) \( y=\frac{4 e^{\prime}}{2 e^{\prime}+1} \) \( y=42 answers -
derivatives
\( y=\cos (\sqrt{x}) \) \( y=\sqrt{\cos x} \) \( y=x\left(x^{2}+1\right)^{13} \) \( y=(\cos 2 x+\sin 2 x)^{-7} \) \( y=(\cos x)^{\sqrt{2}} \)2 answers -
of of Let f(x,y,z) = 6xze5yz. Find (x, y, z),(x, y, z), and дх of ax (x, y, z) = Əf -(x, y, z). əz
Let \( f(x, y, z)=6 x z e^{5 y z} \). Find \( \frac{\partial f}{\partial x}(x, y, z), \frac{\partial f}{\partial y}(x, y, z) \), and \( \frac{\partial f}{\partial z}(x, y, z) \). \[ \frac{\partial f}{2 answers -
Evaluate the integral. \[ \begin{array}{l} \int_{0}^{10 \pi} \int_{11 \pi}^{11 \pi}(\sin x+\cos y) d x d y \\ 10 \pi \\ 20 \pi \\ 21 \pi \end{array} \]2 answers -
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1. Resolver: (i) \[ y^{\prime}=2 y-y^{2} \] (ii) Con la condición inicial \( y(0)=1 \) (iii) Supongamos que la condición inicial es \( y(0)=3 \). ¿ Es la solución cretiente, decreciente o ninguna2 answers -
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Evaluate g(x, y, z) = z2; G: x2 + y2 + z2 = 3, z ≥ 0
Evaluate \( \iint_{G} g(x, y, z) d S \). \[ g(x, y, z)=z^{2} ; G: x^{2}+y^{2}+z^{2}=3, z \geq 0 \]2 answers -
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Calculate all four second-order partial derivatives of \( f(x, y)=(5 x+4 y) e^{y} \). \[ f_{z x}(x, y)= \] \[ f_{x y}(x, y)= \] \[ f_{y x}(x, y)= \] \[ f_{y y}(x, y)= \]2 answers -
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\( x=-(10 u+10 u v) . y=9 u v+7 u v w \), and \( z=-6 u v w \) implies \( \frac{\partial(x, y, z)}{\partial(u, v, w)}= \)2 answers -
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\[ y^{\prime \prime}-y^{\prime}-12 y=0 \] ous \( y^{\prime \prime}-y^{\prime}-12 y=2 e^{2 x} \) ous \( y^{\prime \prime}-y^{\prime}-12 y=5 e^{4 x} \)2 answers -
\( x=5 w-(3 u+10 v), y=3 u-7 v-2 w \), and \( z=9 v-u-8 w \) implies \( \frac{\partial(x, y, z)}{\partial(u, v, w)}= \)2 answers -
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find \( \frac{d y}{d x} \) if: A) \( y=e^{-x} \sec 4 x \) B) \( x 2^{y}+y 2^{x}=4 \) C) \( x=\sqrt{1+e^{2 y}} \) D) \( y=x \operatorname{In} x \) E) \( y=\frac{\ln 2 x}{\sin 3 x} \) F) \( y=\operatorn2 answers -
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Numbers 5,24,26
3-26 Differentiate. 3. \( f(x)=\left(3 x^{2}-5 x\right) e^{x} \) 4. \( g(x)=(x+2 \sqrt{x}) e^{x} \) 5. \( y=\frac{x}{e^{x}} \) 6. \( y=\frac{e^{x}}{1-e^{x}} \) 7. \( g(x)=\frac{1+2 x}{3-4 x} \) 8. \(2 answers -
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Find the domain of the function of two variables. 24) \( f(x, y)=\frac{x}{y}+\frac{1}{y-2} \) A) \( \{(x, y) \mid x \neq 0 \) and \( y \neq 0 \) and \( y \neq 2\} \) B) \( \{(x, y) \mid y \neq 0 \) an2 answers -
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Find the gradient vector field \( (\vec{F}(x, y, z)) \) of \( f(x, y, z)=z e^{-2 y x} \). \[ \vec{F}(x, y, z)= \]2 answers -
Find the gradient vector field \( (\vec{F}(x, y, z)) \) of \( f(x, y, z)=y \cos \left(\frac{5 z}{x}\right) \). \[ \vec{F}(x, y, z)= \]2 answers -
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Find the gradient vector field \( (\vec{F}(x, y, z)) \) of \( f(x, y, z)=z^{2} \sin (5 y x) \). \[ \vec{F}(x, y, z)= \]2 answers -
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Solve the initial-value problem \( y^{\prime}=\frac{2 \sin (x)}{\sin (y)}, \quad y(0)=\frac{\pi}{2} \) \[ -\cos (y)=-2 \cos (x)+1 \]2 answers -
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Usando la forma normalizada: \( \frac{d y}{d x}+P(x) y=Q(x) \), y el factor integrante: \( u(x)=e^{\int P(x) d x} \), usa la ecuación general: \( \frac{1}{u(x)} \int Q(x) u(x) d x \) y encuentra la s2 answers -
Para el campos vectorial, revisa si es conservativo. Si lo es, encuentra su función Potencial: \[ F=2 x y i+x^{2} j \] Si es conservativo. Función: \( f(x, y)=2 x+K \) Si es conservativo. Función:2 answers -
Find \( y^{\prime} \) and \( y^{\prime \prime} \) by implicit differentiation. Simplify where possible. \[ \begin{array}{l} x^{2}+3 y^{2}=3 \\ y^{\prime}=-\frac{x}{3 y} \\ y^{\prime \prime}=\frac{\sqr2 answers -
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