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Calculus Archive: Questions from February 26, 2023

• Find the gradient Please show your work, thanks
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  In Exercises 5-8, calculate the gradient. 5. \( f(x, y)=\cos \left(x^{2}+y\right) \) \( g(x, y)=\frac{x}{x^{2}+y^{2}} \) 7. \( h(x, y, z)=x y z^{-3} \) 8. \( r(x, y, z, w)=x z e^{y w} \)
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  Evaluate \( \int \sin ^{5}(7 x) d x \) \[ \int \sin ^{5}(7 x) d x= \]
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  Determine an expression for \( \frac{d y}{d x} \) if \( \quad y^{2}-\tan y=\pi^{2} \). A) \( \frac{d y}{d x}=\frac{2 \pi-y^{2}}{2 x y-\sec ^{2} y} \) B) \( \frac{d y}{d x}=\frac{\sec ^{2} y}{y^{2}+2 x
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  Differentiate. \[ y=\sec (\theta) \tan (\theta) \] \[ y^{\prime}= \]
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  4. Find all possible values of \( a \) that satisfy \[ \int_{0}^{a}\left[\left(t \sqrt{1+t^{2}}\right) \mathbf{i}+\left(\frac{1}{1+t^{2}}\right) \mathbf{j}\right] d t=\frac{1}{3}(2 \sqrt{2}-1) \mathbf
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  Find the first partial derivatives of the function. \[ \begin{array}{c} f(x, y, z)=x z-7 x^{9} y^{9} z^{2} \\ f_{x}(x, y, z)=-6 \cdot x^{8} \cdot y^{9} \cdot z^{2}+z \\ f_{y}(x, y, z)=2 \cdot x^{9} \c
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  Find the general solution to the following first order linear differential equation. \[ y^{\prime}=\frac{x^{6} y}{\left(x^{7}+2\right)^{2}} \] (A) \( y=C e^{-1 /\left[6\left(x^{7}+2\right)\right]} \)
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  Problem \#5: Find the general solution to the following differential equation. \[ \frac{d y}{d x}=\frac{6 y}{7 x} \] (A) \( y=C e^{(6 / 7) x} \) (B) \( y=e^{(6 / 7) x}+C \) (C) \( y=C x^{7 / 6} \) (D)
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  En los problemas \( 17-24 \), encuentre el área de la región que está acotada por la gráfica de la ecuación polar que se indica. 17. \( r=2 \operatorname{sen} \theta \) 18. \( r=10 \cos \theta \)
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• Solve the given differential equation by undetermined coefficients. a) y''' − y'' − 25y' + 25y = 4 − ex + e5x b) y'' − 4y = (x2 − 8) sin(2x) c) y'' + 4y' + 4y = (7 + x)e−2x, y(0) = 4, y'
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• #9
  7-18 Find the exact length of the curve. 7. \( y=1+6 x^{3 / 2}, \quad 0 \leqslant x \leqslant 1 \) 8. \( y^{2}=4(x+4)^{3}, \quad 0 \leqslant x \leqslant 2, \quad y>0 \) 9. \( y=\frac{x^{3}}{3}+\frac{1
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  Problem 6 (24pts). Compute the derivative \( y^{\prime}=\frac{d y}{d x} \). Do not simplify. Show all work! (a) \( y=\frac{x^{3}}{2}+9 x^{2 / 3}-2 x+6+10 x^{-1 / 2} \) (b) \( y=\frac{4}{\sqrt[3]{x}}-3
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  Show all work on a separate sheet of paper, numbered. Write the answer to every problem below or beside the problem. For questions 1 through 3 , find the mass and the center of mass of the lamina that
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  \( \begin{array}{l}\lim _{x \rightarrow 0} \frac{\tan x^{2}}{\sin ^{2} x} \\ \lim _{x \rightarrow 0}\left[\sin 2 x+\frac{\tan 4 x}{6 x}\right] \\ \lim _{x \rightarrow 0} \frac{1-\cos x}{x^{2}}\end{arr
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• help me with 19,21 and 22 pls
  5-22 Find \( d y / d x \) by implicit differentiation. 5. \( x^{2}-4 x y+y^{2}=4 \) 6. \( 2 x^{2}+x y-y^{2}=2 \) 7. \( x^{4}+x^{2} y^{2}+y^{3}=5 \) 8. \( x^{3}-x y^{2}+y^{3}=1 \) 9. \( \frac{x^{2}}{x
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  1.- Encuentre las siguientes derivadas, \( \frac{\partial^{2} f}{\partial x^{2}}, \frac{\partial^{2} f}{\partial y \partial x}, \frac{\partial^{2} f}{\partial y^{2}} \), and \( \frac{\partial^{2} f}{\
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• #11 and # 15
  1-18 Find the exact length of the curve. 7. \( y=1+6 x^{3 / 2}, \quad 0 \leqslant x \leqslant 1 \) 8. \( y^{2}=4(x+4)^{3}, \quad 0 \leqslant x \leqslant 2, \quad y>0 \) 9. \( y=\frac{x^{3}}{3}+\frac{1
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  Differentiate the following function: a. \( \mathrm{Y}=6 x^{4}-7 x^{3}+2 x+\sqrt{2} \) e. \( y=\left(x^{3}+2 x-7\right)\left(3+x-x^{2}\right) \) b. \( y=\frac{2-x^{2}}{3 x^{2}+1} \) f. \( f(x)=\frac{3
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• Question 13 options: Solve y = 2(1.5)x for y = 20. Solve y = 2(0.5)x for y = 4. Solve y = 10(1.5)x for x = 20. Solve y = 2(0.5)x for x = 20. Solve y = 2(1.5)x for y = 4. Solve y = 10(1.5)x for y = 20.
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• find dy/dx when y = 6(ln(x))^4 find dy/dx when y = ln(11+square root of x)
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  n the 1st and 2 nd partial derivatives of the function \( f(x, y)=e^{x \sin y} \) with correct expressions. \[ \begin{array}{ll} e^{x \sin y} \sin y & \text { 1. } f_{x} \\ e^{x \sin y} \cos y(x \sin
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  LARCALC11 2.4.008. Comploto the tahle
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  t \( 3 x=2 \sqrt{y}, x=0, y=9 \) about the \( y \)-axis section \( 6.2 \) \( \# 6 y=\frac{1}{4} x^{2} ; y=5-x^{2} \) abunt the \( x \)-axis \#9 \( y=x, y=\sqrt{x} \) about \( y=2 \)
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  Find \( d y / d x \) by implicit differentiation. \[ \begin{array}{r} \tan (x-y)=\frac{y}{8+x^{2}} \\ y^{\prime}=\frac{\sec ^{2}(x-y)-2 x}{\frac{1}{8}+\sec ^{2}(x-y)} \end{array} \]
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  Find the derivative of the function. \[ \begin{array}{c} y=\left[x+\left(x+\sin ^{2} x\right)^{7}\right]^{3} \\ y^{\prime}=3\left(x+\left(x+\sin (x)^{2}\right)^{7}\right)^{2} \cdot\left(1+7\left(x+\si
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• Find y'= dy/dx for the curve x2y3=(x2+y2-1)3
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• FIND DY/DX x2+ y3 = 2y PLEASE <33333
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  b) Evaluate the integral. \[ \iint_{D}\left(x^{2} \tan x+y^{3}+4\right) d A, \] where \( D=\left\{(x, y) \mid x^{2}+y^{2} \leq 2\right\} \)
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  2) Find the derivative of a) \( y=\left(x-2 x^{4}\right)^{5}(\sin 2 x) \) b) \( y=\frac{e^{\tan x}}{\sqrt{3 x^{6}-\sec \frac{1}{x}}} \) c) \( y=\frac{6 e^{x}}{\sqrt{8 x+\ln x}} \) d) \( f(x)=\cos \lef
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  Find all possible values of \( a \) that satisfy \[ \int_{0}^{a}\left[\left(t \sqrt{1+t^{2}}\right) \mathbf{i}+\left(\frac{1}{1+t^{2}}\right) \mathbf{j}\right] d t=\frac{1}{3}(2 \sqrt{2}-1) \mathbf{i}
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  5. If \( \cot x=\frac{4}{3} \) and \( \cos y=-\frac{1}{2} \), where \( x \) and \( y \) lie in the interval \( [\pi, 2 \pi] \), evaluate (a) \( \sin (2 x+y) \) (b) \( \sec (x+2 y) \) (c) \( \tan (x-y)
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  \( \frac{d y}{d x} \) if \( 2 e^{5 x} y^{4}+\ln \left(9 y^{7}\right)=6-(4 x+5 y)^{-3} \)
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  Find the linear approximation of \( w=f(x, y, z)=\sin \left(x e^{y z}\right) \) at the origin. \[ L(x, y, z)=x \] \[ L(x, y, z)=z \] \[ L(x, y, z)=y \] \[ L(x, y, z)=x+y+z \]
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  Let \( f(x, y)=2 x^{2}-y^{2} \). Find \( f(2,3) \) \( -1 \) \( -4 \) 2 5
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• complete the table
  1. [-/1 Points] LARCALC11 2.4.003. Complete the table. 2. [-/1 Points] LARCALC11 2.4.511.XP. Complete the table.
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  2. If \( \mathbf{F}(x, y)=\left\langle 4 x^{3} y^{5}-2 y, 5 x^{4} y^{4}+17 x\right\rangle \) and \( \mathcal{C} \) is the circle \( (x-100)^{2}+(y+179)^{2}=717 \) then: \[ \int_{C} \mathbf{F} \cdot d
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• P.59 Operaciones con matrices En los ejercicios 15 a 20, realice las operaciones indicadas siempre que c=-2 y A=[[1,2,3],[0,1,-1]],B=[[1,3],[-1,2]],C=[[0,1],[-1,0]] (cB)(C+C)
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  if \( f(n)=\sin \left(x^{5}\right) \)
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  2. (3 pts) Find the derivative, \( y^{\prime} \). \[ y=\left(5 x^{2}-7 x\right)^{-5}\left(x^{3}+1\right)^{3} \]
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  Differentiate. \[ y=8 \sec (x)-\csc (x) \] \[ y^{\prime}= \]
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  Find the intersection. 19) \( x+y+z=1, x+y=10 \)
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  Differentiate. \[ y=6 x^{2} \cos (x) \cot (x) \]
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