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Calculus Archive: Questions from November 15, 2022

• Prove (cos^(2)x+tan^(2)x-1)/(sin^(2)x)=tan^(2)x, if possible.
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• Prove (1+cosx)/(1-cosx)=1+(2cosx(1+cosx))/(sin^(2)x)
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• list all possible rational zeros f(x)=6x^(2)+13x-5
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• If m angle w =(5x+1) and m angle y =(13x-37), find m angle y
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  Find \( d y / d x \) and simplify your answers whenever possible. 1. \( y=\log \sqrt{2 x+5} \) 2. \( y=\log \left[(x-1)^{3}(x+2)^{4}\right] \) 3. \( y=\ln ^{4}(x+3) \) 4. \( y=\ln (\ln \sec x) \)
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  1. Examine the following function for extreme values: \[ f(x, y)=x^{4}+y^{4}-2 x^{2}+4 x y-2 y^{2} . \]
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  \( \left(3 x y^{2}+2 \sin y-4 x^{2}\right) d x+\left(2 x^{2} y+x \cos y\right) d y=0, \quad y(1)=0 \)
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• please solve 4 thru 12
  4. \( y=e^{x}, y=0, x=-1, x=1 \); about the \( x \)-axis 5. \( y=\sqrt{25-x^{2}}, y=0, x=2, x=4 \); about the \( x \)-axis 6. \( 2 x=y^{2}, x=0, y=4 \); about the \( y \)-axis 7. \( y=\ln x, y=1, y=2,
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  \( \tan (x y)=x \), then \( \frac{d y}{d x}= \) A) \( \frac{1-y \tan (x y) \sec (x y)}{x \tan (x y) \sec (x y)} \) B) \( \frac{\sec ^{2}(x y)-y}{x} \) C) \( \cos ^{2}(x y) \) D) \( \frac{\cos ^{2}(x y
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• EVALUATE THE INTEGRALS (1) (2) (3)
  \( \int \frac{e^{2 x} d x}{e^{x}+1} \) \( \int e^{2 x}\left(1+e^{2 x}\right)^{2} d x \) \( \int \sqrt{\cos ^{2}(2 x)+2 \sin ^{2}(2 x)-1} d x \)
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• plzz solve all plzz plzz
  27-28 Describe the domain of \( f \) in words. 27. (a) \( f(x, y)=x e^{-\sqrt{y+2}} \) (b) \( f(x, y, z)=\sqrt{25-x^{2}-y^{2}-z^{2}} \) (c) \( f(x, y, z)=e^{x y z} \) 28. (a) \( f(x, y)=\frac{\sqrt{4-
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• problem 17 (a,b) please
  In Problems 13-22, sketch each pair of graphs on the same axes. 13. a. \( y=2^{x} \) b. \( y=\left(\frac{1}{2}\right)^{x} \) 14. a. \( y=3^{x} \) b. \( y=3^{-x} \) 15. a. \( y=4^{x} \) b. \( y=-4^{x}
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• problem 19(a,b) please
  In Problems 13-22, sketch each pair of graphs on the same axes 13. a. \( y=2^{x} \) b. \( y=\left(\frac{1}{2}\right)^{x} \) 14. a. \( y=3^{x} \) b. \( y=3^{-x} \) 15. a. \( y=4^{x} \) b. \( y=-4^{x} \
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• #28 and #34
  Find derivatives of the functions defined as follows. 1. \( y=e^{4 x} \) 2. \( y=e^{-2 x} \) 3. \( y=-8 e^{3 x} \) 4. \( y=1.2 e^{5 x} \) 5. \( y=-16 e^{2 x+1} \) 6. \( y=-4 e^{-0.3 x} \) 7. \( y=e^{x
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• please solve all the questions
  \( 39-50 \) Use logarithmic differentiation to find the derivative of the function. 39. \( y=(2 x+1)^{5}\left(x^{4}-3\right)^{6} \) 40. \( y=\sqrt{x} e^{x^{2}}\left(x^{2}+1\right)^{10} \) 41. \( \sqrt
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  (1) Find \( \frac{d y}{d x} \) for the following functions: (i) \( y=\frac{(5 x+1)^{3}(6 x+1)^{2} \sin ^{3} x}{(2 x+4)^{4} e^{5 x}} \) (ii) \( y=2^{x} 5^{x^{2}} 7^{x^{3}} \) (iii) \( y=\ln \left[\frac
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  \( \int \frac{\sin x}{4 \cos x+9 \cos ^{3} x} d x \) \( \int \frac{x^{2}-x+2}{3 x^{3}-2 x^{2}+6 x-4} d x \) \( \int \frac{x^{2}-x+1}{(x+1)^{3}} d x \)
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  Suppose that \( f(x, y)=x+3 y \) on the domain \( D=\left\{(x, y) \mid 1 \leq x \leq 2, x^{2} \leq y \leq 4\right\} \). Then the double integral of \( f(x, y) \) over \( D \) is \[ \iint_{D} f(x, y) d
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  VALE 10 PUNTOS: MUESTRE TODO SU TRABAJO: JUSTIFIQUE TODAS SUS RESPUESTAS I. Determine cual o cuales de las siguientes ecuaciones es (son) Ecuaciones diferenciales: homogéneas, de Bernoulli o ninguna
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  Find the first partial derivatives. See Example 1. \[ \begin{array}{l} g(x, y)=4 e^{x / y} \\ g_{x}(x, y)= \\ g_{y}(x, y)= \end{array} \]
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  Find the first partial derivatives. \[ g(x, y)=\ln \left(x^{8}+y^{8}\right) \] \[ g_{x}(x, y)= \] \[ g_{y}(x, y)= \]
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  Find the first partial derivatives. See Example 1 . \[ \begin{array}{c} f(x, y)=\frac{3 x y}{x^{2}+y^{2}} \\ f_{x}(x, y)= \\ f_{y}(x, y)= \end{array} \]
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  Snlve the fallowinn initial-valup nonhlems
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  Find \( y^{\prime \prime} \). 4) \( y=\left(8+\frac{5}{x}\right)^{4} \) A) \( \frac{300}{x^{4}}\left(8+\frac{5}{x}\right)^{2}+\frac{40}{x^{3}}\left(8+\frac{5}{x}\right)^{3} \) B) \( -\frac{60}{x^{2}}\
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• solve q48 and 50
  39-50 Use logarithmic differentiation to find the derivative of the function. 39. \( y=(2 x+1)^{5}\left(x^{4}-3\right)^{6} \) 40. \( y=\sqrt{x} e^{x^{2}}\left(x^{2}+1\right)^{10} \) 41. \( y=\sqrt{\fr
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  Find the Jacobian \( \partial(\mathrm{x}, \mathrm{y}, \mathrm{z}) / \partial(\mathrm{u}, \mathrm{v}, \mathrm{w}) \) of the transformations below. a. \( x=u \cos v, y=3 u \sin v, z=6 w \) b. \( x=4 u-3
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  Find the Jacobian \( \partial(\mathrm{x}, \mathrm{y}, \mathrm{z}) / \partial(\mathrm{u}, \mathrm{v}, \mathrm{w}) \) of the transformations below. a. \( x=u \cos v, y=3 u \sin v, z=6 w \) b. \( x=4 u-3
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  Find the divergence of F. \[ \mathbf{F}(x, y, z)=x^{2} z \mathbf{i}+y^{2} x \mathbf{j}+(y+4 z) \mathbf{k} \]
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• I. Evaluate the integral in the indicated order. II. Evaluate the integral by changing the order of the integration
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  Find \( y^{\prime} \) for \( 4 x y^{3}-3 x^{3}+23 x y=0 \) at \( (x, y)=(3,1) \).
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  Let \( \frac{d y}{d x}=6 x^{-2}+2 x^{-1}-5 \). Find \( y \) if \( y(1)=4 \) \[ y= \]
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  Given \( f(x, y)=6 x^{2} y-7 x y^{6} \) \[ \frac{\partial^{2} f}{\partial x^{2}}= \] \[ \frac{\partial^{2} f}{\partial y^{2}}= \]
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  Given \( z=f(x, y)=5 e^{5 x}-2 x y^{6}+2 y^{3}+6 \ln (y) \), find \[ z_{x}(x, y)= \] \[ z_{y}(x, y)= \]
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  Given \( y^{\prime \prime}(x)=6-4 x^{3}+60 x^{2}, y^{\prime}(0)=-4 / 5 \), and \( y(0)=3 \), find \( y=f(x) \). \[ y^{\prime \prime}=6-4 x^{3}+60 x^{2} \]
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• find dy/dx
  \( x^{2}+y^{6}=4 x^{2} y^{2}-3 y^{2} x^{4}-3 x^{2} y^{4} \) \( y=(x+1)^{x} \)
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  \( x=3 u+10 v, y=10 u+10 v \) implies \( \frac{\partial(x, y)}{\partial(u, v)}= \)
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  1. If \( \sin (x+y)=y^{2} \cos (x) \), then find \( y^{\prime} \). 2. If \( x^{4}+y^{4}=16 \), then find \( y^{\prime} \). 3. If \( 2 \sqrt{x}+\sqrt{y}=3 \), then find \( y^{\prime} \). 4. If \( x^{4}
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  2) For each of the following vector fields \( \vec{F}(x, y, z) \), compute \( \operatorname{curl}(\vec{F}) \). (a) \( \vec{F}(x, y, z)=\left\langle x^{2}, y^{2}, z^{2}\right\rangle \) (b) \( \vec{F}(x
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  Find the first partial derivatives. See Example 1 . \[ \begin{aligned} & g(x, y)=4 e^{x / y} \\ g_{x}(x, y)=& \frac{4}{y} e^{\left(\frac{x}{y}\right)} \\ g_{y}(x, y)=&-\frac{4}{2} e^{\left(\frac{x}{y}
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  3) For each of the following vector fields \( \vec{F}(x, y, z) \), compute \( \operatorname{div}(\vec{F}) \). (a) \( \vec{F}(x, y, z)=\left\langle x^{2}, y^{2}, z^{2}\right\rangle \) (b) \( \vec{F}(x,
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  Consider the function \[ f(x, y, z)=8 x^{5} y^{2} z-3 x y^{3} z^{6} \] Find the following partial derivatives. \[ \begin{array}{l} \frac{\partial f}{\partial x}(x, y, z)= \\ \frac{\partial f}{\partial
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  1) Calculate the gradient field of \( f(x, y, z)=z \sec (x y) \) at \( (x, y, z)=\left(\frac{\pi}{4}, 1, \frac{\pi}{4}\right) \).
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• Find dy
  Find \( d y \). \[ y=\sin (7 \sqrt{x}) \]
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  \( x=10 v-7 u, y=5 u+10 v \) implies \( \frac{\partial(u, v)}{\partial(x, y)}= \)
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  9. Find function F given gradient \( \langle z y, x z, x y-2 z\rangle \) b. Apply Ftoc
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• MATH 1400
  Differentiate the function \( y=\left(x^{2}+7\right) \ln \left(x^{2}+7\right) \) \[ y^{\prime}= \]
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  Compute \( \frac{\partial^{4} f}{\partial x^{2} \partial y^{2}}(x, y) \), where \( f(x, y)=x \cos (y)+y e^{x} \) Compute \( \frac{\partial^{3} g}{\partial x \partial y \partial z}(x, y, z) \), where \
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• 17 and 21 please
  In Problems 17-22 find the range of each function \( f(x, y) \), when defined on the specified domain \( D \). 17. \( f(x, y)=x^{2}+y^{2} ; D=\{(x, y): 0 \leq x \leq 1,0 \leq y \leq 1\} \) 21. \( f(x,
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