Calculus Archive: Questions from July 28, 2022
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\( [(\mathrm{a})] \) Verify whether \( f(x, y, z)=3 x^{2} y z-4 y z^{2}+2 x \) is a potential for \[ \mathbf{F}(x, y, z)=\left\langle 6 x y z+2,3 x^{2} z-4 z^{2}, 3 x^{2} y-4 y z\right\rangle \] \( [(1 answer -
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3. Evaluate \( \iiint_{E} x d V \), where \( E=\{(x, y, z) \mid-1 \leq x \leq 1,0 \leq y \leq 4,0 \leq z \leq y\} \) A. 8 B. 4 C. \( -1 \) D. 1 E. 01 answer -
Find the domain of the function. \[ \begin{array}{l} f(x, y)=\sqrt{y-6 x} \ln (y+6 x) \\ x>0, y>6 x \\ x6 \\ -y6 x \\ -y0 \\ x>0, y>0 \end{array} \] Sketch the domain of the function.1 answer -
Evaluate \( \iiint_{E} x d V \), where \( E=\{(x, y, z) \mid-1 \leq x \leq 1,0 \leq y \leq 4,0 \leq z \leq y\} \). A. 8 B. 4 C. \( -1 \) D. 1 E. 01 answer -
Calculate \( \iint_{\mathcal{S}} f(x, y, z) d S \) For \[ y=5-z^{2}, \quad 0 \leq x, z \leq 5 ; \quad f(x, y, z)=z \] \( \iint_{\mathcal{S}} f(x, y, z) d S= \)1 answer -
60 please
In Problems \( 51-60 \), find the indicated function or value if \( C(x, y)=3 x^{2}+10 x y-8 y^{2}+4 x-15 y-120 \). 51. \( C_{x}(x, y) \) 52. \( C_{y}(x, y) \) 53. \( C_{x}(3,-2) \) 54. \( C_{y}(3,-2)1 answer -
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(1 point) 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 \( z=0, z=4 y \) and \( x^{2}=25-y \). 1. \1 answer -
Evaluate \( \iiint_{E} x d V \), where \( E=\{(x, y, z) \mid-1 \leq x \leq 1,0 \leq y \leq 4,0 \leq z \leq y\} \)1 answer -
Given \( y=f(u) \) and \( u=g(x) \), find \( \frac{d y}{d x}=f^{\prime}(g(x)) g^{\prime}(x) \). \[ y=\sqrt[3]{u^{2}}, u=\sin x \] \[ \frac{d y}{d x}= \]1 answer -
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\( f(x, y, z)=5 y-x \) \( S: r(u, v)=\cos (u) \mathbf{i}+\sin (u) j+v k \) \( 0 \leq u \leq \frac{\pi}{3}, 0 \leq v \leq 1 \)1 answer -
PLEASE SOLVE EACH
2. Find \( \frac{d y}{d x} \) for each of the following. (a) \( y=\left(x^{2}+4\right) \cos ^{-1}(2 x)+\tan ^{-1}\left(x^{2}\right) \) (b) \( y=\log _{3}\left(x^{2}+9\right)+e^{4 x^{3}+7 x}+5^{\cos x}3 answers -
PLEASE HELP ME SOLVE EACH
(b) \( \int\left(3^{4 x-7}+\frac{1}{2+5 x}+\frac{7}{1+x^{2}}\right) d x \) (c) \( \int_{\pi / 3}^{\pi / 2} \sin x \sqrt{1-2 \cos x} d x \)1 answer -
1. \( y=(4 x-1)(3 x+1)^{4} \) 2. \( y=(2 x+1)^{5 / 2}(4 x-1)^{3 / 2} \) 3. \( y=\frac{x^{2}-6 x}{x-2} \) 4. Let \( f(x)=(3 x+1)^{4}(3-x)^{5} \). Find all \( x \)-values such that \( f^{\prime}(x)=0 \)1 answer -
13. Differentiate a) \( y=\cot (\sqrt{\cos x}) \) b) \( g(x)=e^{3 x} \cos \left(2^{x}\right) \) c) \( h(x)=\frac{\sin (x)}{3-2 \cos (x)} \) d) \( y=\ln (\sin x)+\log _{6} 5 x \) e) \( f(x)=\arctan \le1 answer -
1. Differentiate the functions with respect for \( \mathrm{x} \) : a) \( y=\frac{1}{\sqrt[3]{x^{2}}} \) [5 marks] b) \( y=\operatorname{Sin}\left(e^{2 x}\right)+\tan (-3 x) \) [5 marks] c) \( y=\sqrt{1 answer -
1. \( y=(4 x-1)(3 x+1)^{4} \) 2. \( y=(2 x+1)^{5 / 2}(4 x-1)^{3 / 2} \) 3. \( y=\frac{x^{2}-6 x}{x-2} \) 4. Let \( f(x)=(3 x+1)^{4}(3-x)^{5} \). Find all \( x \)-values such that \( f^{\prime}(x)=0 \)1 answer -
Q1-Q12. Sketch the vector field F by drawing a diagram. Q1. \( F(x, y)=i+\frac{1}{2} j \) Q3. \( F(x, y)=i+\frac{1}{2} y j \) Q5. \( F(x, y)=-\frac{1}{2} i+(y-x) j \) Q9. \( F(x, y, z)=i \) Q11. \( F(1 answer -
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The iterated integral \( \int_{0}^{1} \int_{z^{2}}^{1} \int_{0}^{1-y} g(x, y, z) d x d y d z \) is equivalent to which of the following : \[ \begin{array}{l} \int_{0}^{1} \int_{\sqrt{y}}^{1} \int_{z^{1 answer -
The iterated integral \( \int_{0}^{1} \int_{z^{2}}^{1} \int_{0}^{1-y} g(x, y, z) d x d y d z \) is equivalent to which of the following : \[ \begin{array}{l} \int_{0}^{1} \int_{\sqrt{y}}^{1} \int_{z^{1 answer -
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Given \( f(x, y)=3 x^{6}+6 x y^{5} \) \[ f_{x x}(x, y)= \] \( f_{x y}(x, y)= \) \[ f_{y y}(x, y)= \]1 answer -
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Determine how the following lines interact. a. \( (x, y, z)=(-2,1,3)+t(1,-1,5) ;(x, y, z)=(-3,0,2)+s(-1,2,-3) \) b. \( (x, y, z)=(1,2,0)+t(1,1,-1) ;(x, y, z)=(3,4,-1)+s(2,2,-2) \) c. \( x=2+t, y=-1+21 answer -
Determine how the following lines interact. a. \( (x, y, z)=(-2,1,3)+t(1,-1,5) ;(x, y, z)=(-3,0,2)+s(-1,2,-3) \) b. \( (x, y, z)=(1,2,0)+t(1,1,-1) ;(x, y, z)=(3,4,-1)+s(2,2,-2) \) c. \( x=2+t, y=-1+23 answers -
Determine how the following lines interact. a. \( (x, y, z)=(-2,1,3)+t(1,-1,5) ;(x, y, z)=(-3,0,2)+s(-1,2,-3) \) b. \( (x, y, z)=(1,2,0)+t(1,1,-1) ;(x, y, z)=(3,4,-1)+s(2,2,-2) \) c. \( x=2+t, y=-1+21 answer -
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Solve by separating variables: \[ 3 y^{2} \frac{d y}{d x}=2 x \] \[ y= \] Solve by separating variables. \[ \frac{d y}{d x}=\frac{14 x}{y} \] \( y= \) (Type an exact answer, using radicals as needed.1 answer -
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Find the divergence of the vector field. \[ \mathbf{F}(x, y, z)=7 x^{7} \mathbf{i}-7 x y^{7} \mathbf{j} \] \( \operatorname{div} F(x, y, z)=7 x^{7} \mathbf{i}-8 x y^{7} \mathbf{j} \) \( \operatorname{1 answer -
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Consider the Cobb-Douglas Production function: \[ P(L, K)=13 L^{0.2} K^{0.8} \] Find the total units of production when \( L=15 \) units of labor and \( K=17 \) units of capital are invested. (Give yo3 answers -
Q13-Q16. Fid the gradient vector field \( \nabla f \) of \( f \). Q13. \( f(x, y)=y \sin (x y) \) Q15. \( f(x, y, z)=\sqrt{x^{2}+y^{2}+z^{2}} \)1 answer -
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