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Calculus Archive: Questions from September 27, 2022

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  Find \( d y / d x \) by implicit differentiation. \[ \sin x+\cos y=\sin x \cos y \]
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  Solve the IVP: \( (2 x \sin y) d x+\left(x^{2} \cos y-1\right) d y=0 ; \quad y(0)=\frac{1}{2} \) \( -x^{2} \sin (y)-y=-\frac{1}{2} \) \[ \begin{array}{l} x^{2} \sin (y)-y=\frac{1}{2} \\ x^{2} \sin (y)
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  using Laplace transform solve, \( x y^{\prime \prime \prime} y^{4}-2 y^{\prime}+y=(\infty+1) \) givien, \( \quad y(0)=4 \quad \) in \( y^{\prime}(0)=-2 \)
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  (1 point) If \[ Y(s)=\frac{1}{6 s^{2}}-\frac{6}{s} \] then \[ Y^{\prime}(s)= \]
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  2. a. \( f^{\prime}(0) \) b. \( f^{\prime}(1) \) c. \( f^{\prime}(2) \) d. \( f^{\prime}(3) \) e. \( f^{\prime}(4) \) f. \( f^{\prime}(5) \) g. \( f^{\prime}(6) \) h. \( f^{\prime}(7) \)
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  (1 point) Consider the function \( f(x, y)=-2 x^{2} y+(-8) x y^{3} \). Calculate the followin \( f_{x}(x, y)= \) \( f_{x}(2,-1)= \) \( f_{x x}(x, y)= \) \( f_{x x}(2,-1)= \) \( f_{y}(x, y)= \) \( f_{y
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  (1 point) Consider the function \( f(x, y)=\frac{4 x}{3 y} \). Calculate the following: \( f_{x}(x, y)= \) \( f_{x}(1,-1)= \) \( f_{x x}(x, y)= \) \( f_{x x}(1,-1)= \) \( f_{y}(x, y)= \) \( f_{y y}(x,
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  (1 point) Consider the function \( f(x, y)=x^{y} \). \( f_{x}(x, y)= \) \[ f_{x}(1,3)= \] \( f_{x x}(x, y)= \) \( f_{x x}(1,3)= \) \( f_{y}(x, y)= \) \( f_{y y}(x, y)= \) \( f_{y x}(x, y)= \) \[ f_{x
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  HW Sec 3.5 Derivatives Involving Trigonometric Functions: Problem 7 Find the Second Derivative of a given function. If \( y=\tan x \), then \( y^{\prime \prime}= \) If \( y=\cot x \) then \( y^{\prime
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• Resolve: Find the critical points of the functions below and determine whether it is a relative maximum, relative minimum, or a saddle point.
  Resuelva: Halle los puntos críticos de las funciones que se presentan a continuación y determine si es un máximo relativo, mínimo relativo o un punto de silla. 1. \( f(x, y)=80 x+80 y-x^{2}-y^{2}
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  the domain and the range of \( g(x, y, z)=\sqrt{z^{2}-y^{3}-x-11} \) \[ \begin{array}{l} D_{g}=\left\{(x, y, z) \in \mathbb{R}^{3} \mid z^{2}-y^{3}-x-11 \geq 0\right\}, R_{g}=(-\infty, 0] \\ D_{g}=\le
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  1. Let \( f(x, y)=\ln \left(y^{2}+x^{3}\right) \). Compute \( D \) \( \left\langle\frac{1}{\sqrt{2}}, \frac{1}{\sqrt{2}}\right\rangle^{f(-2,-3)} \)
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• Evaluate the limit ​​​​​​​
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• 7 and 9
  5-20 Find \( f^{\prime}(x) \) 5. \( f(x)=\left(3 x^{2}+6\right)\left(2 x-\frac{1}{4}\right) \) 6. \( f(x)=\left(2-x-3 x^{3}\right)\left(7+x^{5}\right) \) 7. \( f(x)=\left(x^{3}+7 x^{2}-8\right)\left(2
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  Find \( y^{\prime \prime} \) for the following function. \[ y=\sec x \csc x \] \[ \mathrm{y}^{\prime \prime}= \]
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  Solve the initial value problem \[ y^{\prime \prime}-12 y^{\prime}+36 y=0, y(0)=0, y^{\prime}(0)=5 \] \[ \begin{array}{l} y=5 e^{6 t} \\ y=e^{6 t}+e^{6 t} \\ y=5 t e^{6 t} \\ y=e^{6 t} \\ y=5 e^{6 t}+
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• find the limit
  \[ (x, y) \underset{y \neq-7}{\lim _{y \neq-7}\left(4,-\frac{y+7}{x^{2} y+2 y+7 x^{2}+14}\right.} \] \( (x, y) \underset{x \neq y}{\lim _{x \neq y}(0,0)} \frac{7 \sqrt{y}-7 \sqrt{x}+\sqrt{x y}-x}{\sqr
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  Find \( y^{\prime} \) and \( y^{\prime \prime} \). \[ \begin{array}{c} y=e^{6 e^{x}} \\ y^{\prime}=6 e^{6 e^{x}+x} \end{array} \]
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  11. Calculate the derivative a) \( \mathrm{f}(\mathrm{x})=\left(3-4 x+2 x^{2}\right)^{-2} \) b) \( y=\frac{-3 x^{4}}{5 x^{2}-2 x+3} \) c) \( f(x)=(5 x+2)(6 x+4) \) d) \( \mathrm{f}(\mathrm{x})=x^{5} \
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  Solve: \[ \begin{array}{l} y^{\prime \prime \prime}-y^{\prime \prime}-14 y^{\prime}+24 y=108 e^{5 t} \\ y(0)=5, \quad y^{\prime}(0)=2, y^{\prime \prime}(0)=76 \\ y(t)= \end{array} \]
  1 answer

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  Help Entering Answers (1 point) Consider the function \( f(x, y)=x^{y} \). Calculate the following: \[ f_{x}(x, y)= \] \[ f_{x}(4,-1)= \] \[ f_{x x}(x, y)= \] \[ \begin{array}{l} f_{x x}(4,-1)= \\ f_{
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  2.)(2pts)Solve the DE \( \overline{d x}=\frac{\cos ^{2} x+1}{} \) 3.) (2pts)Solve the IVP \( (x+1)^{2} \frac{d y}{d x}+y=\frac{1}{x+1}, \quad y(0)=0 \).
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  1. (6 pts) Solve the following ODEs. a) (1 pt) \( y^{\prime \prime}-7 y^{\prime}+10 y=e^{t} \); b) (1 pt) \( y^{\prime \prime}=y+e^{2 t} \cos t \); c) (2 pts) \( y^{\prime \prime}-2 y^{\prime}+y=4 \co
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  e \( \frac{d y}{d x}=\frac{-\sin x}{y} \), if \( y(0)=-2 \) A. \( y=-\sqrt{2 \cos x+2} \) B. \( y=-\sqrt{2 \cos x-2} \) C. \( y=-\sqrt{2 \cos x+6} \) D. \( y=-\sqrt{2 \cos x-6} \) E. \( y=-\sqrt{-2 \c
  1 answer
• question 3 f please
  3. Evaluate the double integral: (a) \( \int_{0}^{4} \int_{0}^{\sqrt{y}} x y^{2} d x d y \) (b) \( \int_{0}^{\pi / 2} \int_{0}^{\cos \theta} e^{\sin \theta} d r d \theta \) (c) \( \int_{0}^{1} \int_{2
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  (1 point) If \( f(x)=4 x^{2}+3 x+9 \) \[ f^{\prime}(x)= \]
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  If \( f(x, y)=4 x \tan ^{-1}(x y) \), evaluate \( f_{x}(-1,1) \) \[ \begin{array}{l} -\pi-1 \\ -\pi+2 \\ \pi+1 \\ -\pi-2 \\ \pi-2 \end{array} \]
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  1. \( P=\left\{(x, y) \in \mathbb{Z}^{*} \times \mathbb{Z}^{*}: x+y=0\right\} \) 2. \( G=\left\{(x, y) \in \mathbb{Z}^{*} \times \mathbb{Z}^{*}: x y>0\right\} \) 3. \( L=\left\{(x, y) \in \mathbb{Z}^{
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  In exercises \( 1-10 \), find \( \frac{d y}{d x} \) for the given functions. 1) \( y=x^{2}-\sec x+1 \) Answer 2) \( y=3 \csc x+\frac{5}{x} \) 3) \( y=x^{2} \cot x \) Answer 4) \( y=x-x^{3} \sin x \) 5
  1 answer
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  (1 point) Consider the function \( f(x, y)=-2 x^{2} y+4 x y^{3} \) \( f_{x}(x, y)= \) \( f_{x}(2,2)= \) \( f_{x x}(x, y)= \) \( f_{x x}(2,2)= \) \( f_{y}(x, y)= \) \( f_{y y}(x, y)= \) \( f_{y x}(x, y
  1 answer
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  (1 point) Consider the function \( f(x, y)=\frac{6 x}{5 y} \). Calculate the following: \( f_{x}(x, y)= \) \[ f_{x}(2,1)= \] \[ f_{x x}(x, y)= \] \( f_{x x}(2,1)= \) \[ f_{y}(x, y)= \] \[ f_{y y}(x, y
  1 answer
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  (1 point) Consider the function \( f(x, y)=x^{y} \). Calculate the following: \( f_{x}(x, y)= \) \( f_{x}(1,1)= \) \( f_{x x}(x, y)= \) \( f_{x x}(1,1) \) \( f_{y}(x, y)= \) \( f_{y y}(x, y)= \) \( f_
  2 answers

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  Complete the table. Complete the table.
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  Give a possible formula for the functions in Problems \( 6-9 \). 6. 7. 8. 9.
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  Differentiate. \[ y=\sec (\theta) \tan (\theta) \]
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  Differentiate. \[ y=\frac{t \sin (t)}{1+t} \]
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  Differentiate. \[ y=5 x^{2} \cos (x) \cot (x) \]
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  Given \( f(x, y)=-3 x^{4}-2 x^{2} y^{3}+2 y^{2} \) \[ f_{x}(x, y)= \] \( f_{y}(x, y)= \) \[ f_{x x}(x, y)= \] \( f_{x y}(x, y)= \) Question Help:
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  If \( f(x)=x^{3} \), compute \( f(-8) \) and \( f^{\prime}(-8) \). \[ f(-8)= \]
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• differentate
  ferentiate. \( \begin{array}{ll}=3 \sin x-2 \cos x & \text { 2. } f(x)=\tan x-4 \sin x \\ x^{2}+\cot x & \text { 4. } y=2 \sec x-\csc x\end{array} \) 13. \( f(\theta)=\frac{\sin \theta}{1+\cos \theta}
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  Problem 20. Sketch and identify the following surfaces: (a) \( y=x^{2}+1 \). (b) \( z=4 x^{2}+y^{2} \). (c) \( z=\sqrt{x^{2}+y^{2}} \). (d) \( z^{2}=x^{2}+4 y^{2} \). (e) \( z=4-x^{2}-y^{2} \). (f) \(
  1 answer
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  Given \( f(x, y)=6 x^{9} \cos \left(y^{5}\right) \) \[ f_{x y}(x, y)= \] \[ f_{y y}(x, y)= \]
  1 answer
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  Given \( f(x, y)=-5 x^{3}+6 x^{2} y^{4}+3 y^{6} \) \( f_{x}(x, y)= \) \( f_{y}(x, y)= \) \( f_{x z}(x, y)= \) \( f_{x y}(x, y)= \)
  2 answers