Calculus Archive: Questions from October 27, 2023
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Determine the Vertical Asymptotes: \[ y=\frac{x^{3}}{x^{2}+x-4} \] 1. \[ f(x)=\frac{1-x}{x^{2}+x-2} \] 2. \[ y=\frac{x^{3}}{4-x^{2}} \] 3. \[ y=\frac{x^{2}+1}{x-2} \]1 answer -
Hallar el área de la superficie dada por z=f(x,y) sobre la región R.
\[ f(x, y)=2 x+2 y \] \( R \) : triángulo cuyos vértices son \( (0,0),(4,0),(0,4) \)1 answer -
Hallar el área de la superficie dada por z = f (x, y) sobre la región R.
\( \begin{array}{l}f(x, y)=7+2 x+2 y \\ R=\left\{(x, y): x^{2}+y^{2} \leq 4\right\}\end{array} \)1 answer -
Let \[ f(x, y)=\left(x^{3}-y\right)(x-3 y)^{3} \] Compute the value of \[ \begin{array}{l} f_{x}(x, y)+f_{y}(x, y) \\ \text { if } x=2 \text { and } y=0 \end{array} \]1 answer -
Solve the following initial value problem \[ y^{\prime \prime}(x)+12 y^{\prime}(x)+100 y(x)=0, \quad y(0)=1, \quad y^{\prime}(0)=-6 \] \[ y(x)= \]1 answer -
HARRY
Let \[ f(x, y)=\sqrt{\frac{x-y}{y}} . \] The domain of \( f \) is equal to \[ \begin{array}{c} \left\{(x, y) \in \mathbb{R}^{2} \mid y>0 \text { and } x \geq y\right\} \cup\left\{(x, y) \in \mathbb{R}1 answer -
DANIEL
Let \[ f(x, y)=\frac{y-x}{\sqrt{9-x^{2}-y^{2}}} . \] The domain of \( f \) is equal to \[ \begin{array}{l} \left\{(x, y) \in \mathbb{R}^{2} \mid x^{2}+y^{2}1 answer -
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1. Hallar la derivada de las siguientes funciones. (24 puntos) a) \( Y(u)=\left(\frac{1}{u^{2}}+\frac{1}{u^{3}}\right)\left(u^{5}-2 u^{2}\right) \) b) \( g(t)=\frac{t-\sqrt{t}}{t^{1 / 3}} \) 2. Hallar1 answer -
If \( f(x) g(y)=2 x+\ln (y)-5 \), where \( f \) is a differentiable function of \( x \) and \( g \) is a differentiable function of \( y \), what is \( \frac{d y}{d x} \) ? \( \frac{2+\frac{1}{y}-f^{\1 answer -
2. Hallar la derivada de las funciones. (24 puntos) a) \( y=x^{2} \operatorname{sen}^{2}\left(\sqrt{1+x^{2}}\right) \) b) \( y=\left(\frac{1-\cos 2 x}{1+\cos 2 x}\right)^{4} \)1 answer -
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3. Hallar \( \frac{d x}{d y} \) y \( \frac{d y}{d x} \) por diferenciación implicita. (24 puntos) \[ y \operatorname{sen}\left(x^{2}\right)=x \operatorname{sen}\left(y^{2}\right) \] 3. Hallar \( \fr1 answer -
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5. Hallar el límite de la función. (14 puntos) \[ \lim _{x \rightarrow 0} \frac{\cos (2 x)-\cos (3 x)}{x^{2}} \]1 answer -
3. Hallar \( \frac{d x}{d y} \) y \( \frac{d y}{d x} \) por diferenciación implícita. (24 puntos) \[ y \operatorname{sen}\left(x^{2}\right)=x \operatorname{sen}\left(y^{2}\right) \]1 answer -
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linearization of the function at the given point. Find the f(x, y) = 6x2 +82-10 at (10,-2) L(x, y) = 12x + 16y +622 L(x, y) = 120x - 32y +622 L(x, y) = 12x + 16y − 642 - L(x, y) = 120x - 32y - 642
Find the linearization of the function at the given point. \[ \begin{array}{c} f(x, y)=6 x^{2}+8 y^{2}-10 \text { at }(10,-2) \\ L(x, y)=12 x+16 y+622 \\ L(x, y)=120 x-32 y+622 \\ L(x, y)=12 x+16 y-641 answer -
Find \( y^{\prime} \) and \( y^{\prime \prime} \). \[ y=x^{2} \ln (8 x) \] \[ y^{\prime}= \] \[ y^{\prime \prime}= \]1 answer -
Find the Jacobian. s,t,u) (x,y,z) (z,y,z) 1 where z = s - 5t+u, y = 2t - 5s +5u, z = 4t - 5s - 4u.
Find the Jacobian. \( \frac{\partial(x, y, z)}{\partial(s, t, u)} \), where \( x=s-5 t+u, y=2 t-5 s+5 u, z=4 t-5 s-4 u \). \( \frac{\partial(x, y, z)}{\partial(s, t, t)}= \)1 answer -
Hallar la derivada de las siguientes funciones. a) \( Y(u)=\left(\frac{1}{u^{2}}+\frac{1}{u^{3}}\right)\left(u^{5}-2 u^{2}\right) \) b) \( g(t)=\frac{t-\sqrt{t}}{t^{1 / 3}} \)1 answer -
Hallar la derivada de las funciones. a) \( y=x^{2} \operatorname{sen}^{2}\left(\sqrt{1+x^{2}}\right) \) b) \( y=\left(\frac{1-\cos 2 x}{1+\cos 2 x}\right)^{4} \)1 answer -
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Solve the initial value problem. \[ \begin{array}{c} y^{(4)}-y^{\prime \prime \prime}=e^{2 x}, \\ y(0)=0, \quad y^{\prime}(0)=0, \quad y^{\prime \prime}(0)=0, y^{\prime \prime \prime}(0)=0 \end{array}1 answer -
Find the first partial derivatives of the function. \[ \begin{array}{l} h(x, y, z, t)=x^{4} y \cos \left(\frac{z}{t}\right) \\ h_{x}(x, y, z, t)=4 x^{3} y \cos \left(\frac{z}{t}\right) \\ h_{y}(x, y,1 answer -
Find \( d y / d x \) for (a) \( y=8 e^{x} \) (b) \( y=5 e^{2 x}+7 \) (c) \( y=e^{x \cdot \ln (10)} \) (d) \( u=e^{x \cdot \ln (2)} \)1 answer -
Given the following functions: \[ \begin{aligned} f(x, y) & =x^{2}+y^{3}-\sin (x+y)+\ln \left(x^{2} y\right) \\ h(x, y, z) & =e^{x+y-z}+\sin \left(x z^{2}\right)+x^{3}-\frac{y}{z} \end{aligned} \] com1 answer -
7. Calculate e \( y^{\prime} \) : A. \( y=\left(x^{2}+x^{3}\right)^{4} \) H. \( y=\sin (\cos x) \) B. \( x y^{4}+x^{2} y=x+3 y \) I. \( y=\frac{1}{\sin (x-\sin x)} \) C. \( y=\frac{x^{2}-x+2}{\sqrt{x}1 answer -
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ich is the gradient of \( f(x, y)=\sqrt{x^{2}+4 y^{2}} \) ? \[ \begin{array}{l} \frac{4 x+y}{\sqrt{x^{2}+4 y^{2}}} \\ \frac{x+4 y}{\sqrt{x^{2}+4 y^{2}}} \\ \frac{1}{\sqrt{x^{2}+4 y^{2}}}\langle 4 x, y1 answer -
Find the second derivative \( y^{\prime \prime} \) for \( y=3 x \sin x \). Choose the correct answer. A. \( y^{\prime \prime}=6 \cos x-3 x \sin x \). B. \( y^{\prime \prime}=3 \cos x-6 x \sin x \). C.1 answer -
3. Hallar \( \frac{d x}{d y} \) y \( \frac{d y}{d x} \) por diferenciación implícita. (24 puntos) \[ y \operatorname{sen}\left(x^{2}\right)=x \operatorname{sen}\left(y^{2}\right) \] 4. Hallar la ter1 answer -
\[ y=\frac{x^{2}+2 x-2}{x^{2}-2 x+2} \] A. \( y^{\prime}=\frac{-4 x^{2}+8 x}{\left(x^{2}-2 x+2\right)^{2}} \) B. \( y^{\prime}=\frac{-4 x^{2}-8 x}{\left(x^{2}-2 x+2\right)^{2}} \) C. \( y^{\prime}=\fr1 answer -
d the derivative of \( y=\ln (\cos x) \) \[ \begin{array}{l} y^{\prime}=\tan x \\ y^{\prime}=-\tan x \\ y^{\prime}=\cot x \\ y=-\cot x \end{array} \]1 answer -
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Find the derivative of y = ln(cos x) A. y' = tan x B. y = - tan x S C. y' = cot r D. y cotx
\( \mathrm{d} \) the derivative of \( y=\ln (\cos x) \) \[ \begin{array}{l} y^{\prime}=\tan x \\ y^{\prime}=-\tan x \\ y^{\prime}=\cot x \\ y=-\cot x \end{array} \]1 answer -
2. Calculate \( y^{\prime} \) \begin{tabular}{|cc|c|c|c|c|} \hline a) & \( y=\left(x^{2}+x^{3}\right)^{4} \) & b) & \( y=e^{\sec (x)}+\csc \left(e^{x}\right) \) & C) & \( y+\sin (x y)=x^{2}-y \) \\ \h1 answer -
2. Hallar la derivada de las funciones. (24 puntos) a) \( y=x^{2} \operatorname{sen}^{2}\left(\sqrt{1+x^{2}}\right) \) b) \( y=\left(\frac{1-\cos 2 x}{1+\cos 2 x}\right)^{4} \)1 answer -
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Find y'. y = 5 sin(2x + 8) A. -10 sin(2x + 8) B. -20 sin(2x+8) c. 10 cos(2x + 8) D. -20 cos(2x + 8)
Find \( y^{\prime \prime} \). \[ y=5 \sin (2 x+8) \] A. \( -10 \sin (2 x+8) \) B. \( -20 \sin (2 x+8) \) C. \( 10 \cos (2 x+8) \) D. \( -20 \cos (2 x+8) \)1 answer -
Find the derivative. y = - + 6 sec x A. y' B. y = C. y' = D.y = OB O C O A - OD I² 8 + 6 tan² - 6 csc x -6 sec x tan r -6 sec r tan r x
Find the derivative. \[ y=\frac{s}{x}+6 \sec x \] A. \( y^{\prime}=-\frac{8}{x^{2}}+6 \tan ^{2} x \) B. \( y^{\prime}=-\frac{8}{x^{2}}-6 \csc x \) C. \( y^{\prime}=\frac{8}{x^{2}}-6 \sec x \tan x \) D1 answer -
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11. Solve the differential equation ہے dy dx = x sin x y y(0) = −1.
11. Solve the differential equation \( \frac{d y}{d x}=\frac{x \sin x}{y}, \quad y(0)=-1 \).1 answer -
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Find the derivative of y= = A. y' B. y' C. y' D. y' - = = e³r 1+e³ 3e³z 1+(er) 3e³x 1+9e91² 3e³r 1+9e6 arctan(e3)
Find the derivative of \( y=\arctan \left(e^{3 x}\right) \) A. \( y^{\prime}=\frac{e^{3 x}}{1+e^{3 x}} \) B. \( y^{\prime}=\frac{3 e^{3 x}}{1+\left(e^{6 x}\right)} \) C. \( y^{\prime}=\frac{3 e^{3 x}}1 answer -
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Given the following functions: compute (a) of af f(x, y) = x² + y³ - sin(x+y)+ln(x²y) h(x, y, z) = eª+v−² + sin(xz²) + x³ – 1/ - Z (b) a² f მყ2 (c) hx(x, y, z) (d) hzz(x, y, z)
Given the following functions: \[ \begin{aligned} f(x, y) & =x^{2}+y^{3}-\sin (x+y)+\ln \left(x^{2} y\right) \\ h(x, y, z) & =e^{x+y-z}+\sin \left(x z^{2}\right)+x^{3}-\frac{y}{z} \end{aligned} \] com1 answer -
AP#4: Find y' and y" when y = In(X² +5X + 8)
AP\#4: Find \( y^{\prime} \) and \( y^{\prime \prime} \) when \( y=\ln \left(X^{2}+5 X+8\right) \)1 answer -
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Find the gradient of \( f(x, y)=e^{1 \cdot x} \sin (3 \cdot y) \) at \( (x, y)=(-1,-3) \). \( \nabla f(x, y)=\langle A, B\rangle \) then \[ \begin{array}{l} A= \\ B= \end{array} \] Question Help: \( \1 answer -
If \( y^{\prime}=2 x^{2} y^{4} \) and \( y(1)=1 \), then \( y(0)=\ldots \) \( \ldots \approx 0.693 \) \( \ldots \approx 1.142 \) \( \ldots \approx 0.229 \) \( \ldots \approx 0.577 \)1 answer -
If \( y^{\prime}=5 x^{3} y^{3} \) and \( y(1)=1 \), then \( y(0)=\ldots \) \[ \ldots \approx 1.436 \] \[ \ldots \approx 0.535 \] \[ \ldots \approx 0.213 \] \[ \ldots \approx 0.722 \]1 answer -
Evaluate \( \iint_{R} y \cos x y \mathrm{~d} A \), where \( R=\{(x, y) \mid 0 \leq x \leq 1,0 \leq y \leq \pi / 3\} \).1 answer -
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compute f(x, y) = x²y³ +exy + ln(xy²) g(x, y, z) = cos(xyz²) + sin(yz) + xyz + ex+yz af (a) ay a² f (b) əxəy (c) gz(x, y, z) (d) 9zx (x, y, z)
\[ \begin{aligned} f(x, y) & =x^{2} y^{3}+e^{x y}+\ln \left(x y^{2}\right) \\ g(x, y, z) & =\cos \left(x y z^{2}\right)+\sin (y z)+x y z+e^{x+y z} \end{aligned} \] compute (a) \( \frac{\partial f}{\pa1 answer -
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