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Calculus Archive: Questions from September 16, 2022

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  \( y=\frac{1}{2}\left[x \sqrt{4-x^{2}}+4\right. \)
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• solve the linear differential equations
  (2) \( y^{\prime}-y \operatorname{cotan} x=2 x-x^{2} \operatorname{cotan} x, y\left(\frac{\pi}{2}\right)=\frac{\pi^{2}}{u}+1 \)
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  \( y=\tan \left(4 x^{2}\right) \) \( y=\tan ^{2}(\sin x) \) \( y=\cot (3 x-8) \) \( y=\sqrt{\cot x} \) \( y=\sec ^{2} x+\tan ^{2} x \)
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  Solve the given differential equation \[ (2 x+3 y-6) d x+(3 x+4 y-6) d y=0 \] \[ \begin{array}{l} 2(x+6)^{2}+4(x+6)(y-6)+(y-6)^{2}=C \\ 4(x-6)^{2}+6(x-6)(y+6)+2(y+6)^{2}=C \\ 2(x-6)^{2}+4(x-6)(y+6)+(y
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• Bernoulli's Differential Equations
  Solve the given differential equation \[ (4 x+y-1) d x+(x-y-4) d y=0 \] \[ \begin{array}{l} (x+1)^{2}-4(x+1)(y-5)-2(y-5)^{2} \\ 4(x+1)^{2}+2(x+1)(y-5)-(y-5)^{2} \\ 4(x+1)^{2}+(y-5)^{2}+(x+1)(y-5) \\ 2
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  5. \( \lim _{y \rightarrow 1} \sec \left(y \sec ^{2} y-\tan ^{2} y-1\right) \)
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  Find the potential function \( f \) for the field \( F \). \[ \begin{array}{l} \mathbf{F}=-\left(\frac{x}{\left(x^{2}+y^{2}+z^{2}\right)^{3 / 2}}\right) \mathbf{i}-\left(\frac{y}{\left(x^{2}+y^{2}+z^{
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  2. Solve the following equations and plot \( y(t) \) along \( t \) : (a) \( y^{\prime}+y=e^{2 t} \) (b) \( t y^{\prime}+2 y=4 t^{2}, y(1)=2 \). (c) \( y^{\prime \prime}+y^{\prime}+y=a \sin \omega t, y
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  Find all the second partial derivatives. \[ f(x, y)=\ln (a x+b y) \]
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  Consider the following. \[ f(x, y)=x^{4} y-3 x^{5} y^{2} \] Find the first partial derivatives. \[ f_{x}(x, y)= \] \[ f_{y}(x, y)= \] Find all the second partial derivatives. \[ f_{x x}(x, y)= \] \[ f
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  Find all the second partial derivatives. \[ \begin{array}{l} f(x, y)=x^{4} y-3 x^{5} y^{2} \\ f_{x x}(x, y)= \\ f_{x y}(x, y)= \\ f_{y x}(x, y)= \\ f_{y y}(x, y)= \end{array} \]
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  1. Derivative Practice (a) \( y=\sin 5 x \) (b) \( f(t)=\sec (3 t) \) (c) \( z(x)=3 / \sqrt{5} \) (d) \( g(x)=x^{3} \sin x \) (e) \( \mathrm{h}(\mathrm{x})=\frac{1+\sqrt[3]{x}}{\cos x} \) (f) \( \math
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  Find \( \frac{d^{2} y}{d x^{2}} \) \( y=-3 x^{6}+1 \) \( \frac{d^{2} y}{d x^{2}}= \) Find \( \frac{d^{2} y}{d x^{2}} \) \[ y=4 x+2 \] \[ \frac{d^{2} y}{d x^{2}}= \]
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• Evaluate the following integrals
  II. Evalúe los integrales. a) \( \int\left(9 x^{3}-x\right)^{2} d x \) b) \( \int \sec (x)\left(\frac{\operatorname{sen}(x)-1}{\cos (x)}\right) d x \) c) \( \int \frac{1}{(3 x)^{2}} d x \)
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  Find \( \frac{d^{2} y}{d x^{2}} \) \[ y=\frac{8 x^{5}}{5}-7 x \] \[ \frac{d^{2} y}{d x^{2}}= \] Find \( y^{\prime \prime} \) \[ y=\frac{3 x+4}{2 x-5} \]
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  Differentiate the function. \[ y=\frac{\sqrt{4 x-5}-\sqrt{x}}{7} \]
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  \( y^{\prime}(x) \) if \( y(x)=(x+3)^{x} \) \( y^{\prime}(x)= \) \( y(x)=(\ln (2 x))^{5 x}, \quad x>1 / 2 \) \( y^{\prime}(x)= \) \( y(x)=(\cos x)^{2 \sqrt{3}} \) \( y^{\prime}\left(\frac{\pi}{4}\r
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  9. Let \( z \) be a constant number. Let \( F(x, y)=x y z+e^{-x y^{2}} \). (a) Compute \( \frac{\partial}{\partial x} F(x, y) \). (b) Compute \( \frac{\partial}{\partial y} F(x, y) \). (c) Compute \(
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  Halle una solución particular para \( F^{\prime}(x)=2 x^{2}+4 x \) que satisfaga la condición inicial \( F(2)=3 \) Halle una solución particular para \( F^{\prime}(x)=\frac{48}{3} x^{3}+9 x^{2}+1 \
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  3. Un objeto es lanzado verticalmente hacia arriba con una velocidad inicial de \( 10 \mathrm{~m} / \mathrm{s} \) sobre desde una distancia inicial de \( 2 \mathrm{~m} \). Determine la altura máxima
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  Valor promedio de una función: 4. \( f(x)=\sqrt{x} \) intervalo: \( [0,3] \) 5. \( f(x)=2 \sec ^{2} x \) intervalo \( \left[-\frac{\pi}{4}, \frac{\pi}{4}\right] \) 6. \( f(x)=\frac{9}{x^{3}} \quad \)
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  1. \( y=\left(x^{3}+x^{2}+x+4\right)^{5} \Rightarrow d y / d x \) 2. \( y=\tan (\ln x) \Rightarrow d y / d x \) 3. \( y=\cos \left(\log _{2} x\right) \Rightarrow d y / d x \) 4. \( y=\left(x^{2}+1\rig
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  (1 point) Solve the initial-value problem \( x y^{\prime}=y+x^{2} \sin x, y\left(\frac{\pi}{4}\right)=0 \). Answer: \( y(x)= \)
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  Locate and classify all the citical pts of the function \[ f(x, y)=3 x^{2}+y^{2}-12 x-2 y \]
  2 answers