Calculus Archive: Questions from October 25, 2023
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Solve the following initial value problem \[ y^{(4)}-3 y^{\prime \prime \prime}+2 y^{\prime \prime}=4 x, y(0)=0, y^{\prime}(0)=0, y^{\prime \prime}(0)=0, y^{\prime \prime \prime}(0)=0 \]1 answer -
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Solve the following initial value problem \[ y^{(4)}-3 y^{\prime \prime \prime}+2 y^{\prime \prime}=4 x, y(0)=0, y^{\prime}(0)=0, y^{\prime \prime}(0)=0, y^{\prime \prime \prime}(0)=0 \]1 answer -
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\[ \sum_{k=1}^{\infty}\left(\frac{\sin ^{2}\left(\frac{k \pi}{2}\right)}{k^{2}}-\frac{\cos ^{2}\left(\frac{k \pi}{2}\right)}{4^{k}}\right) \] converges or diverges.1 answer -
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A. Given the multivariable functions f (x, y) = x³ + 4x³ y³ + 5yª, g (x, y) = 4x² + 3x6 y² + 2y and h (x, y) = x² + 2x8 y7 + 3y8 find: i. fx(x, y) ii. iii. iv. fy(x, y) fxy(x, y) fxx(x, y) V. v
A. Given the multivariable functions \( f(x, y)=x^{3}+4 x^{5} y^{3}+5 y^{4}, \mathrm{~g}(x, y)=4 x^{4}+3 x^{6} y^{4}+ \) \( 2 y^{6} \) and \( \mathrm{h}(x, y)=x^{7}+2 x^{8} y^{7}+3 y^{8} \) find: i. \1 answer -
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5. Encuentre la ecuación \( \rho=9 \csc \phi \csc \theta \) en coordenadas rectangulares a. \( x=9 \) b. \( y=9 \) c. \( x y=9 \) d. \( y=1 / 9 \)1 answer -
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2. Hallar los extremos relativos de a. \( f(x, y)=3(x-2)^{2}-2(y-3)^{2} \) b. \( g(x, y)=\sin (x) \cos (y) \quad \) donde \( \quad 01 answer -
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Differentiate. dy 증 y = 3x3 - 13x2 wwwwww 13x +16x+1
Differentiate. \[ y=3 x^{3}-13 x^{2}+16 x+1 \] \[ \frac{d y}{d x}= \]1 answer -
Si f(x,y) = x(y^2 -x) g. Halle el dominio de f h. grafique la region del dominio f
\( \operatorname{Sif}(x, y)=x \square\left(y^{2}-x\right) \), g. Halle Dominio de \( f \)1 answer -
Evaluate the integral. \[ \int e^{2 x} x^{2} d x \] A. \( \frac{1}{2} x^{2} e^{2 x}-\frac{1}{4} x e^{2 x}+\frac{1}{4} e^{2 x}+C \) B. \( \frac{1}{2} x^{2} e^{2 x}-\frac{1}{2} x e^{2 x}+C \) C. \( \fra1 answer -
Evaluate the integral. \[ \int 6 \cos ^{3} 4 x d x \] A. \( \frac{3}{2} \sin 4 x-\frac{1}{2} \sin ^{3} 4 x+C \) B. \( \frac{3}{2} \sin 4 x+\frac{1}{2} \sin ^{3} 4 x+C \) C. \( \frac{3}{2} \sin 4 x-\fr1 answer -
Evaluate the integral. \[ \int \tan ^{4} 5 t d t \] A. \( -\frac{1}{15} \tan ^{3} 5 t+\frac{1}{5} \tan 5 t+C \) B. \( \frac{1}{15} \tan ^{3} 5 t-\frac{1}{5} \tan 5 t+t+C \) C. \( \frac{1}{15} \tan ^{31 answer -
Consider the function. f(x,y) = y + xey
Consider the function. \[ f(x, y)=y+x e^{y} \] a) Find. \[ \int_{0}^{2} f(x, y) d x= \] b) Find. \[ \int_{0}^{1} f(x, y) d y= \]1 answer -
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15. ff arctan (y/x) dA, R where R = {(x, y) | 1 ≤ x² + y² ≤ 4, 0 ≤ y ≤x}
\( R=\left\{(x, y) \mid 1 \leqslant x^{2}+y^{2} \leqslant 4,0 \leqslant y \leqslant x\right\} \)1 answer -
HELP PORFAVOR
Determine the global extreme values of the function \( f(x, y)=4 x^{3}+4 x^{2} y+4 y^{2}, \quad x, y \geq 0, x+y \leq 1 \) \( f_{\min }= \) \( f_{\max }= \)1 answer -
En los siguientes problemas evalúe la integral definida dada empleando el Teorema Fundamental del Cálculo. (a) \[ \int_{-1}^{1}\left(\frac{1}{e^{x}}-\frac{1}{e^{-x}}\right) d x \] (b) \[ \int_{-1}^{1 answer -
Determine el área de la región \( R \) que yace bajo la curva dada \( y=f(x) \) en el intervalo \( a \leq x \leq b \). (a) Bajo \( y=\sqrt{x}(x+1) \), sobre \( 0 \leq x \leq 4 \). (b) Bajo \( y=x e^1 answer -
2. Dos barcos salen de un mismo puerto a la misma hora. El barco \( A \) viaja a \( 44 \mathrm{mi} / \mathrm{h} \) en dirección \( \mathrm{S} 75^{\circ} \mathrm{E} \). El barco \( B \) viaja a \( 361 answer -
3. Desde el suelo de un cañón se necesitan 59 pies de soga para alcanzar la cima de una pared del cañón y 94 pies para alcanzar la cima de la pared opuesta. Si las dos sogas forman un ángulo de \1 answer -
Surface integrals 1) evaluates
Intograles do superficio 1) Evalue \( \int_{s} f(x-2 y+z) d S \) para \( S: z=4-x, 0 \leq x \leq 4,0 \leq y \leq 3 \)1 answer -
Surface integrals 2) Find the flow of F through S.____________ where N is the unit vector normal to S directed upwards given F(x,y,z) = 3zi - 4j + yK and S:z=1-x-y in the first octant.
2) Halle el flujo de \( \mathrm{F} \) a través de \( \mathrm{S}, \int_{S} \int F \cdot N d S \) donde \( \mathrm{N} \) es el vector unitario normal a \( \mathrm{S} \) dinigido hacia arriba dado \( F(1 answer -
\[ \mathbf{u}=7 \mathbf{i}+\mathbf{j}, \mathbf{v}=-6 \mathbf{i}+8 \mathbf{j} \] (a) radians \[ \theta= \] (b) degrees0 answers -
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Differentiate the function. y' = y = In(e¯ˇ + xe¯X)
Differentiate the function. \[ y=\ln \left(e^{-x}+x e^{-x}\right) \] \[ y^{\prime}= \]1 answer -
answer this step by step to find the critical points and to find the maximum and minimum in the given interval
En los problemas del 5 al 26 identifique los puntos críticos y encuentre los valores máximo y mínimo en el intervalo dado. 5. \( f(x)=x^{2}+4 x+4 ; I=[-4,0] \) 6. \( h(x)=x^{2}+x ; I=[-2,2] \) 7. \1 answer -
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1. Find the absolute max and min for \( f(x, y)=(x-3)^{2}+y^{2} \) on \( D=\left\{(x, y): 0 \leq x \leq 4, x^{2} \leq y \leq 4 x\right\} \).1 answer -
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8. Find (a) \( \int 2 \cos 2 x \sin x d x \) (b) \( \int \sin ^{2} 2 x d x \) (c) \( \int \cos ^{3} x \sin ^{2} x d x \)1 answer -
Para la figura que se muestra determine la coordenada en " \( y \) " ( \( \bar{y}) \) del centroide del área compuesta sombreada. \( (30 \%) \)1 answer -
\( \begin{array}{l}\text { ferentiate } f(x)=10 \cdot 7^{x} \\ f^{\prime}(x)=10 \cdot 7^{x} \cdot \ln (10 \\ f^{\prime}(x)=10 x \cdot 7^{(x-1)} \\ f^{\prime}(x)=10 \cdot 7^{x} \\ f^{\prime}(x)=70^{x}1 answer -
ifferentiate \( f(x)=3^{\csc (x)} \). \[ \begin{array}{l} f^{\prime}(x)=\csc (x) \cdot 3^{(\csc (x)-1)} \\ f^{\prime}(x)=3^{\csc (x)} \end{array} \] \[ f^{\prime}(x)=-3^{\csc (x)} \cdot \csc (x) \cot1 answer -
\( \begin{array}{l}\text { ifferentiate } f(x)=\sin \left(5 \mathrm{e}^{3 x}\right) \\ f^{\prime}(x)=15 \mathrm{e}^{3 x} \cos \left(5 \mathrm{e}^{3 x}\right) \sin \left(5 \mathrm{e}^{3 x}\right) \\ f^1 answer -
\( \begin{array}{l}\text { ifferentiate } f(x)=\tan \left(3^{3 x^{2}}\right) \\ f^{\prime}(x)=3^{3 x^{2}}(6 x) \sec ^{2}\left(3^{3 x^{2}}\right) \\ f^{\prime}(x)=3^{3 x^{2}} \ln (3)(6 x) \tan \left(3^1 answer -
Find y' by implicit differentiation. Match the equations defining y implicitly with the letters labeling the expressions for y'. 1.5x cos y + 3 sin 2y = 2 sin y 2. 5x sin y + 3 sin 2y = 2 cos y 3. 5x
Find \( y^{\prime} \) by implicit differentiation. Match the equations defining \( y \) implicitly with the letters labeling the expressions for \( y^{\prime} \) 1. \( 5 x \cos y+3 \sin 2 y=2 \sin y \1 answer -
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Please do number 8 only
8. \( y^{\prime}+\frac{1}{x \ln x} y=9 x^{2} \). 9. \( y^{\prime}-y \tan x=8 \sin ^{3} x \).1 answer -
Prove the identity. sinh(x + y) = sinh(x) cosh(y) + cosh(x) sinh(y) sinh(x) cosh(y) + cosh(x) sinh(y) = [ze* - e-x][;(| = ( = = ²/1 [ex + y = e = (x + y)] D+C + ex [X] + ex-Y - e-x + y −e¯x - y) +
Prove the identity. \[ \begin{array}{l} \sinh (x+y)=\sinh (x) \cosh (y)+\cosh (x) \sinh (y) \\ \sinh (x) \cosh (y)+\cosh (x) \sinh (y)=\left[\frac{1}{2}\left(e^{x}-e^{-x}\right)\right]\left[\frac{1}{21 answer -
Evaluate the double integral. 7. integral of y² dA, D = {(x, y) | −1≤ y ≤ 1, -y - 2 ≤ x ≤ y} Y
\( 7-10 \) - Evaluate the double integral. 7. \( \iint_{D} y^{2} d A, \quad D=\{(x, y) \mid-1 \leqslant y \leqslant 1,-y-2 \leqslant x \leqslant y\} \)1 answer -
Select each correct particular solution of the differential equation. y/= y 2 sin a y = sin x + cos x Oy=2 Oy = cos x sin x + cos x + 2 y Oy=0 y = sin x
Select each correct particular solution of the differential equation. \[ \begin{array}{l} y=y-2 \sin x \\ y=\sin x+\cos x \\ y=2 \\ y=\cos x \\ y=\sin x+\cos x+2 \\ y=0 \\ y=\sin x \end{array} \]1 answer -
Express the integral \( \iiint_{E} f(x, y, z) d V \) as an iterated integral in six different ways, where \( \mathrm{E} \) is the solid bounded by \( z=0, z=7 y \) and \( x^{2}=4-y \). 1. \( \int_{a}^1 answer -
Calculate the double integral
\( \iint_{R} \frac{x y^{2}}{x^{2}+1} d A, \quad R=\{(x, y) \mid 0 \leqslant x \leqslant 1,-3 \leqslant y \leqslant 3\} \)1 answer -
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Hallar el máximo valor de la derivada direccional en el punto dado. \( g(x, y)=\ln \left(\sqrt[3]{x^{2}+y^{2}}\right. \). En el punto \( P(1,2) \)1 answer -
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Hallar una ecuación del plano tangente a la superficie en el punto dado. \( x y^{2}+3 x-z^{2}=4 \). En el punto \( P(2,1,-2) \)1 answer -
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Hallar una ecuación del plano tangente a la superficie en el punto dado. xy2 + 3x - z²=4. En el punto P (2,1, - 2)
Hallar una ecuación del plano tangente a la superficie en el punto dado. \( x y^{2}+3 x-z^{2}=4 \). En el punto \( P(2,1,-2) \)1 answer -
Hallar el máximo valor de la derivada direccional en el punto dado. \( g(x, y)=\ln \left(\sqrt[3]{x^{2}+y^{2}}\right. \). En el punto \( P(1,2) \)1 answer -
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Evaluate the triple integral. F 4y2 cos(z) dV where F = (x, y, z) | 0 ≤ y ≤ 𝜋 2 , 0 ≤ x ≤ y, 0 ≤ z ≤ xy
Evaluate the triple integral. \[ \iiint_{F} 4 y^{2} \cos (z) d V \text { where } F=\left\{(x, y, z) \mid 0 \leq y \leq \sqrt{\frac{\pi}{2}}, 0 \leq x \leq y, 0 \leq z \leq x y\right\} \]1 answer -
Differentiate the function. y' y = tan(In(ax + b)) sec²(In(ax + b)) b + ax
Differentiate the function. \[ \begin{array}{r} y=\tan (\ln (a x+b)) \\ y^{\prime}=\frac{\sec ^{2}(\ln (a x+b))}{b+a x} \end{array} \]1 answer -
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