Calculus Archive: Questions from December 11, 2023
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triple integral
\( \int_{0}^{3 \sqrt{9-x^{2}}} \int_{0}^{2} \int_{0}^{2} \sqrt{x^{2}+y^{2}} d z d y d x \)1 answer -
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Find the derivative of the function. \( y=\ln \left(x^{3}+4 x-9\right) \) \[ \begin{array}{l} y^{\prime}=\frac{3 x^{2}+4}{x^{3}+4 x-9} \\ y^{\prime}=\frac{3 x^{2}-9}{x^{3}+4 x-9} \\ y^{\prime}=\frac{11 answer -
9. 2² +2² J Answer dV, where E = {(x, y, z) |1 ≤ y ≤ 4, y < z < 4,0 <=< z} A
\( \begin{array}{l}\iiint_{E} \frac{z}{x^{2}+z^{2}} d V \text {, where } \\ \qquad E=\{(x, y, z) \mid 1 \leqslant y \leqslant 4, y \leqslant z \leqslant 4,0 \leqslant z \leqslant z\}\end{array} \)0 answers -
Considere la ecuación diferencial \( \frac{d x}{d t}+x=t^{2} \) a. (6 puntos) ¿Cuáles deben ser los valores de las constantes \( a, b, c \) de modo que función cuadrática \( x_{p}(t)=a t^{2}+b t+1 answer -
Ejercicio 2 (1.5pto): Encuentra la distribución de la v.a. \( Y=F_{X}(x) \), donde \( F_{X}(x) \) es función de distribución. Ejercicio 3 (1.5pto): muestra que si una v.a. tiene función generadora1 answer -
El valor de \( x \) de \( \log _{3}(3 x-6)^{2}=6 \) es: (A) \( x=11 \) (B) \( x=6 \) (C) \( x=3 \) (D) \( x=24 \)1 answer -
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For the function, find the partials \( f_{x}(x, y) \) and \( f_{y}(x, y) \). \[ f(x, y)=3 \ln \left(x y^{3}\right) \] (a) \( f_{x}(x, y) \) (b) \( f_{y}(x, y) \)1 answer -
Problem 2: Evaluate the following integrals: (i) \( \int_{1}^{3} \int_{0}^{\frac{\pi}{2}} x \sin y d y d x \) Ans: 4 (ii) \( \int_{1}^{4} \int_{0}^{4} \sqrt{x y} d x d y \) Ans: \( \frac{224}{9} \) (i1 answer -
Find all the first order partial derivatives for the following function. \[ \begin{aligned} f(x, y, z) & =\frac{\cos y}{x z^{2}} \\ \frac{\partial}{\partial x} & =\frac{\cos y}{z^{2}} ; \frac{\partial1 answer -
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Evaluate the integral. \[ \begin{array}{l} \int 7 x \sin x d x \\ 7 \sin x-7 \cos x+C \\ 7 \sin x-x \cos x+C \\ 7 \sin x-7 x \cos x+C \end{array} \]1 answer -
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15. The integral \[ \int_{0}^{\ln 2} \frac{e^{3 x}}{1+e^{2 x}} d x \] equals (a) \( \frac{\pi+1}{4}-\tan ^{-1} 2 \). (b) \( \frac{\pi+1}{4}+\tan ^{-1} 2 \). (c) \( 1+\frac{\pi}{4}-\tan ^{-1} 2 \). (d)1 answer -
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A, B, C, and D please
Problem 1 (12 points) Find the derivatives (a) \( y=\frac{x^{3}}{6}-\frac{2}{\sqrt{x}} \) (b) \( y=3 e^{-5 x}-4 \ln x \) (c) \( y=\left(4 x^{3}-5 x\right)^{2} \) Problem 1 (12 points) Find the deriva1 answer -
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Suppose \( f(x)=h(g(x) k(x)) \). If \( g(1)=5, k(1)=0, h(1)=-7, g^{\prime}(1)=8, k^{\prime}(1)=-3, h^{\prime}(1)=4 \), and \( h^{\prime}(0)=1 \), find \( f^{\prime}(1) \). Answer: \( f^{\prime}(1)= \)1 answer -
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Let \( f(x, y, z)=6 x z e^{4 y z} \). 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}{\1 answer -
show all work
17. \( \mathbf{F}(x, y, z)=y z e^{x z} \mathbf{i}+e^{x z} \mathbf{j}+x y e^{x z} \mathbf{k} \), \( C: \mathbf{r}(t)=\left(t^{2}+1\right) \mathbf{i}+\left(t^{2}-1\right) \mathbf{j}+\left(t^{2}-2 t\righ1 answer -
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1. (10 pts) Evaluate the integer \[ \iint_{R} \frac{\sin x}{x} d A, \text { where } R=\{(x, y) \mid 0 \leq y \leq 1, y \leq x \leq 1\} \text {. } \]1 answer -
Use implicit differentiation to find \( \frac{d y}{d x} \) given \( \sin (x)+\cos (y)=2 x^{2}+3 y^{3} \). \( \frac{\cos (x)-4 x-9 y^{2}}{\sin (y)} \) \( \frac{\cos (x)-4 x}{9 y^{2}+\sin (y)} \) \( \fr1 answer -
Use laplace transforms to solve the initial value problem
Use Laplace transforms to solve the initial value problems: 2. \( y^{\prime \prime}+y=\sin 2 y ; \quad y(0)=0, y^{\prime}(0)=0 \) 3. \( y^{\prime \prime}+4 y^{\prime}+3 y=1 ; y(0)=0, y^{\prime}(0)=0 \1 answer -
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2. Find global extreme of \( f(x, y)=x^{2}+y^{2} \) over the domains (a) \( (x-3)^{2}+(y-3)^{2} \leq 1 \)1 answer -
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1. Let \( \mathbf{r}(x, y, z)=(x, y, z) \) and \( r(x, y, z)=\sqrt{x^{2}+y^{2}+z^{2}}=\|\mathbf{r}\| \). Verify the following identities. (a) \( \nabla\left(\frac{1}{r}\right)=-\frac{\mathbf{r}}{r^{3}1 answer -
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tan ²0-sin² 0 = tan² @ sin² 0
\( \tan ^{2} \theta-\sin ^{2} \theta=\tan ^{2} \theta \sin ^{2} \theta \)1 answer -
Given \( f(x, y)=-5 x^{2}+5 x y^{6}-4 y^{3} \), \[ \begin{array}{l} f_{x x}(x, y)= \\ f_{x y}(x, y)= \end{array} \]1 answer -
52. (Secreción hormonal) Un modelo de este proceso puede ser \[ y^{\prime}=a-b \cos \frac{2 \pi t}{24}-k y . \] Aquí, \( t \) es el tiempo [en horas, eligiéndose \( t=0 \) en una hora adecuada, por1 answer -
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11. Evaluate: a. \( \int\left(3 x^{4}+2 x-\sqrt[3]{x}+\frac{1}{x}\right) d x \) b. \( \int_{1}^{3} 4-3 e^{-5 x} d x \)1 answer -
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17. Differentiate: a) \( y=\frac{2 x}{\sin (x)+\cos (3 x)} \) b) \( Y=\ln \left|\frac{x^{2}-4}{2 x+5}\right| \) c) \( y=\frac{\tan (x)-1}{\sec (x)} \) d) \( Y=\cos \left(\sin \left(x^{3}\right)\right)1 answer