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Calculus Archive: Questions from August 09, 2022

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  Solve the following initial value problem. \[ y^{\prime}-(\tan x) y=\sin 2 x, \quad y(0)=2 \]
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  Calculate \( \iint_{S} f(x, y, z) d S \) For \[ y=9-z^{2}, \quad 0 \leq x, z \leq 5 ; \quad f(x, y, z)=z \] \( \iint_{S} f(x, y, z) d S= \)
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  Calculate \( \iint_{S} f(x, y, z) d S \) For \[ y=9-z^{2}, \quad 0 \leq x, z \leq 5 ; \quad f(x, y, z)=z \] \( \iint_{S} f(x, y, z) d S= \)
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  Compute the gradient vector fields of the following functions: A. \( f(x, y)=3 x^{2}+3 y^{2} \) \( \nabla f(x, y)=\quad \mathbf{i}+\quad \mathbf{j} \) B. \( f(x, y)=x^{8} y^{7} \), \( \nabla f(x, y)=\
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  Integrate \( \int e^{-\frac{x}{2}} d x \) \( -2 e^{-\frac{x}{2}}+C \) \( e^{-\frac{x}{2}}+C \) \( \frac{-e^{-\frac{x}{2}}}{2} \)
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• Please help!! Complete the table. y = f(g(x)) u = g(x) y = f(u) y = (7x − 5)7 u = y = u7
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• need help 8,9. no work needed, just answers
  \( d \operatorname{curl}(F \times G)=V \times(F \times G) \). \( F\left(x, y_{i} z\right)=i+4 x j+2 y k \) \( G(x, y, z)=x i-y j+2 k \) \( \operatorname{div}(F \times G)=\nabla \cdot(F \times G) \) \(
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• Solve please
  \( \mathbf{A r}(\mathbf{F} \times \mathbf{G})=\nabla \times(\mathbf{F} \times \mathbf{G}) \) \( \mathbf{F}(x, y, z)=\mathbf{i}+8 x \mathbf{j}+5 y \mathbf{k} \) \( \mathbf{G}(x, y, z)=x \mathbf{i}-y \m
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  a) \( y=e^{7 x^{3}-2 x^{2}+6} \) FinD \( y^{\prime} \) b) \( y=\ln \left(12 x^{4}-6 x+2\right) \) Find \( y^{\prime} \)
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• Solve the ODE and determine the interval of validity. (a) y = (1 2r)y 2 (b) y (c) y 21 1+2y 3r 2 302-4
  3. Solve the ODE and determine the interval of validity. (a) \( y^{\prime}=(1-2 x) y^{2} \) (b) \( y^{\prime}=\frac{2 x}{1+2 y} \) (c) \( y^{\prime}=\frac{3 x^{2}}{3 y^{2}-4} \)
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• Please show all work, will leave a thumbs up! :)
  Compute the gradient vector fields of the following functions: A. \( f(x, y)=2 x^{2}+7 y^{2} \) \( \nabla f(x, y)=\quad \mathbf{i}+\quad \mathbf{j} \) B. \( f(x, y)=x^{7} y^{7} \), \( \nabla f(x, y)=\
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  \( f(x, y, z)=x^{3} \sec \left(x-y^{2} z\right) \)
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  Find the partial derivatives of the function \[ f(x, y)=x y e^{5 y} \] \[ \begin{array}{l} f_{x}(x, y)= \\ f_{y}(x, y)= \\ f_{x y}(x, y)= \\ f_{y x}(x, y): \end{array} \]
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• One of the following vector fields is conservative. Find a function for it. F = (15x^4 y^2+ e^x cos y, 6x^5 y - e^x sin y), G = (10x^4 y^2+ e^x sin y, 2x^5 y + e^x cos y)
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