Get questions and answers for Calculus

Calculus Archive: Questions from October 25, 2022

• 

  
  For \( f(x, y, z)=x^{2}+2 x y+2 y^{3} \), compute \( \frac{\partial f}{\partial x}(x, y) \) \[ \frac{\partial f}{\partial x}(x, y)= \]
  2 answers

• 

  
  For \( f(x, y, z)=\sqrt{x y^{5} z+5 x^{2} y^{7}} \), defined for \( x, y, z \geq 0 \), compute \( f_{x} \)
  2 answers
• ​​​​​​​
  \( \left.d \frac{d^{2} y}{d \theta^{2}}\right|_{\theta=\frac{\pi}{4}} \), if \( y=\sec \theta \) \( 2 \sqrt{2} \) \( 4 \sqrt{2} \) \( 3 \sqrt{2} \) \( \sqrt{2} \)
  2 answers

• 

  
  Find \( f_{x} \) and \( f_{y} \). \[ f(x, y)=\int_{y}^{x} 6 e^{t^{8}} d t \] \[ f_{x}(x, y)= \] \[ f_{y}(x, y)= \]
  2 answers

• 

  
  Find \( f_{x}(x, y) \) and \( f_{y}(x, y) \) \[ f(x, y)=\left(y^{7} \tan (7 x)\right)^{-\frac{5}{4}} \] \[ f_{x}(x, y)= \] \[ f_{y}(x, y)= \]
  2 answers
• 
  

  
  1. Dibuje la gráfica utilizando un graficador en linea, Desmos o geogebray describa la superficie generada. Debe pegarla en el programa Microsoft Word. 2. Determine el área de la superficie generada
  2 answers
• ​​​​​​​
  2 answers
• Find all first-order partial derivatives
  8. \( f(x, y)=\int_{x}^{x+y} e^{y^{2}-t^{2}} d t \) 9. \( f(x, y, z)=3 x \ln \left(x^{2} y z\right)+x^{y / z} \) 10. \( f(x, y, z)=\frac{2}{\sqrt{x^{2}+y^{2}+z^{2}}}-x^{2} e^{x y / z} \)
  2 answers
• number 16 , 18 and 20
  Finding a Derivative In Exercises 7-26, use the rules of differentiation to find the derivative of the function. 41. \( g(t)=t^{2}-\frac{4}{t^{3}} \) 8. \( f(x)=-9 \) 7. \( y=12 \) 43. \( f(x)=\frac{x
  2 answers
• 
  

  
  Find the first and second derivatives of the function. \[ \begin{array}{l} f(x, y)=e^{x y} \sin (y) \\ f_{x}(x, y)= \\ f_{y}(x, y)= \\ f_{x x}(x, y)= \\ f_{x y}(x, y)= \\ f_{y y}(x, y)= \end{array} \]
  2 answers

• 

  
  If the function \( f(x, y) \) satisfies the following equations \[ \frac{\partial f}{\partial x}=-\sin y+\frac{1}{1-x y} \quad \text { and } \quad f(0, y)=2 \sin y+y^{3} \] find \( f(x, y) \).
  2 answers

• 

  
  (1 point) Match the functions and their derivatives: 1. \( y=\sin (x) \tan (x) \) 2. \( y=\cos (\tan (x)) \) 3. \( y=\cos ^{3}(x) \) 4. \( y=\tan (x) \) A. \( y^{\prime}=-\sin (\tan (x)) / \cos ^{2}(x
  2 answers
• 
  

  
  In exercises 1-10, find all first-order partial derivatives. 1. \( f(x, y)=x^{3}-4 x y^{2}+y^{4} \) 2. \( f(x, y)=x^{2} y^{3}-3 x \) 3. \( f(x, y)=x^{2} \sin x y-3 y^{3} \) 4. \( f(x, y)=3 e^{x^{2} y}
  2 answers
• Evaluate
  I. Evalúe \( \int F \cdot d r \) donde \( C \) está representada por \( r(t) \). 3a) \( F(x, y)=3 x i+4 y j ; C: r(t)=\cos (t) i+\operatorname{sen}(t) j \) donde \( 0 \leq t \leq \pi / 2 \) 3 b) \(
  2 answers
• 
  

  
  

  
  Calculate \( y^{\prime \prime} \) and \( y^{\prime \prime \prime} \) \( y=x^{2}(9 x+17) \) \( y^{\prime \prime}= \) \( y^{\prime \prime \prime}= \)
  2 answers
• 
  

  
  Find \( y^{\prime} \) and \( y^{\prime \prime} \). \[ \begin{array}{l} y=\ln (2+\ln (x)) \\ y^{\prime}=\square \\ y^{\prime \prime}=\square \end{array} \]
  2 answers

• 

  
  3. \( \int_{0}^{2} \int_{0}^{z^{2}} \int_{0}^{y-z}(2 x-y) d x d y d z \)
  2 answers

• 

  
  7. Find \( \frac{d y}{d x} \) using implicit differentiation. (a) \( \cos (x y)=1+\sin y \) Answer: \( \frac{d y}{d x}=-\frac{y \sin (x y)}{x \sin (x y)+\cos y} \) (b) \( y^{5}+x^{2} y^{3}=1+x^{4} y \
  2 answers
• 
  

  
  Compute the following. \[ \left.\frac{d^{2}}{d x^{2}}\left(3 x^{3}-x^{2}+6 x-8\right)\right|_{x=4} \]
  2 answers
• 
  

  
  Situación 2: Comparar los métodos de discos y arandelas con el método de capas cilíndricas y enumere al menos dos similitudes y/o diferencias.
  2 answers

• 

  
  Consider the function. \[ f(x, y)=3-\frac{x}{10}-\frac{y}{2} \] Find \( \nabla f(x, y) \). \[ \nabla f(x, y)= \]
  2 answers
• Find dy/dx for y2 =
  2 answers
• will you please do 64
  55-66. Combining rules Use the Chain Rule combined with other differentiation rules to find the derivative of the following functions. 55. \( y=\left(\frac{x}{x+1}\right)^{5} \) 56. \( y=\left(\frac{e
  2 answers

• 

  
  (4) Find the partial derivatives of the following functions. ( 20 points total). (a) \( f(x, y)=2 x e^{3 y^{2}} \) (b) \( g(x, y)=\frac{x \ln (y)+2}{x^{2}-y} \) (c) \( h(x, y)=\sqrt{x^{4} y^{2}+y^{3}}
  2 answers
• Need 58
  55-66. Combining rules Use the Chain Rule combined with other differentiation rules to find the derivative of the following functions. 55. \( y=\left(\frac{x}{x+1}\right)^{5} \) 56. \( y=\left(\frac{e
  2 answers
• need 62
  55-66. Combining rules Use the Chain Rule combined with other differentiation rules to find the derivative of the following functions. 55. \( y=\left(\frac{x}{x+1}\right)^{5} \) 56. \( y=\left(\frac{e
  2 answers
• need 64
  55-66. Combining rules Use the Chain Rule combined with other differentiation rules to find the derivative of the following functions. 55. \( y=\left(\frac{x}{x+1}\right)^{5} \) 56. \( y=\left(\frac{e
  2 answers
• need 66 please
  55-66. Combining rules Use the Chain Rule combined with other differentiation rules to find the derivative of the following functions. 55. \( y=\left(\frac{x}{x+1}\right)^{5} \) 56. \( y=\left(\frac{e
  2 answers

• 

  
  Find \( d y / d x \) by implicit differentiation. \[ \tan (x-y)=\frac{y}{1+x^{2}} \]
  2 answers
• differentiate each function
  a. \( y=\left(\sqrt{x}+\frac{1}{\sqrt[3]{x}}\right)^{2} \) b. \( F(y)=\left(\frac{1}{y^{2}}-\frac{3}{y^{4}}\right)\left(y+5 y^{3}\right) \) c. \( y=\cos \sqrt{\sin (\tan \pi x)} \) d. \( f(s)=\sqrt{\f
  2 answers
• 
  

  
  3. Hallar un vector unitario y ortogonal a los vectores \( \vec{A}=i-4 j+k, \vec{B}=2 i+3 j \) \[ \begin{array}{l} -\frac{\sqrt{3}}{\sqrt{134}} i+\frac{2}{\sqrt{134}} j+\frac{11}{\sqrt{134}} k \\ -\fr
  2 answers
• 
  

  
  3. Hallar un vector unitario y ortogonal a los vectores \( \vec{A}=i-4 j+k, \vec{B}=2 i+3 j \) \[ -\frac{\sqrt{3}}{\sqrt{134}} i+\frac{2}{\sqrt{134}} j+\frac{11}{\sqrt{134}} k \] \[ \begin{array}{l} -
  2 answers
• 
  

  
  2. Given \( \cos \left(\frac{\pi}{12}\right)=\frac{\sqrt{2}}{4}(1+\sqrt{3}) \) and \( \sin \left(\frac{\pi}{12}\right)=\frac{\sqrt{2}}{4}(\sqrt{3}-1) \), Find a.) \( \cos \left(\frac{13 \pi}{12}\right
  2 answers

• 

  
  3. Sea \( \mathbf{r}(\mathrm{x})=\left(\sqrt{2-x}, e^{x}-1, \ln (x+1)\right) \) a. Halle el dominio de \( \mathbf{r} \) b. Halle \( \lim _{x \rightarrow 0} \mathbf{r}(\mathrm{x}) \) c. Halle \( \lim _
  2 answers

• 

  
  5. \( \operatorname{Si~} \mathrm{f}(\mathrm{x}, \mathrm{y})=\frac{\sqrt{x+y+1}}{x-1} \), a. Halle Dominio de \( \mathrm{f} \) b. Grafique la región del dominio de f
  2 answers

• 

  
  Si \( f(x, y)=x \ln \left(y^{2}-x\right) \), c. Halle Dominio de \( \mathrm{f} \) d. Grafique la región del dominio de \( \mathrm{f} \)
  2 answers

• 

  
  7. Halle los siguientes limites, si existen: a. \( \lim _{(x, y) \rightarrow(3,2)}\left(x^{2} y^{3}-4 y^{2}\right) \) b. \( \lim _{(x, y) \rightarrow(2,-1)} \frac{x^{2} y+x y^{2}}{x^{2}-y^{2}} \)
  2 answers
• 
  

  
  Let \( y=\ln \left(x^{2}+y^{2}\right) \). Determine the derivative \( y^{\prime} \) at the point \( \left(\sqrt{e^{8}-64}, 8\right) \). \[ y^{\prime}\left(\sqrt{e^{8}-64}\right)= \]
  2 answers

• 

  
  (18 marks) Solve the following initial value problems. (i) \( y^{\prime \prime \prime}-3 y^{\prime \prime}+3 y^{\prime}-y=x e^{x}+e^{x} ; \quad y(0)=0, \quad y^{\prime}(0)=1, \quad y^{\prime \prime}(0
  1 answer
• Find the approximate value of y: y = (2x-1)^4 when x = 0.98
  2 answers
• 
  

  
  For \( f(x, y) \), find all values of \( x \) and \( y \) such that \( f_{x}(x, y)=0 \) and \( f_{y}(x, y)=0 \) simultaneously. \[ f(x, y)=\ln \left(5 x^{2}+2 y^{2}+1\right) \] \[ (x, y)=( \]
  2 answers

• 

  
  Find \( f_{x}(x, y) \) and \( f_{y}(x, y) \) \[ f(x, y)=\left(y^{5} \tan (5 x)\right)^{-\frac{4}{3}} \] \[ f_{x}(x, y)= \] \[ f_{y}(x, y)= \]
  2 answers
• 
  
  

  
  
  

  
  

  
  
  

  
  

  
  I. Evalúe \( \int F \cdot d r \) donde \( C \) está representada por \( r(t) \). a) \( F(x, y)=3 x i+4 y j ; C: r(t)=\cos (t) i+\operatorname{sen}(t) j \) donde \( 0 \leq t \leq \pi / 2 \) b) \( F(x
  2 answers
• 
  

  
  II: Evalúe el integral utilizando el Teorema fundamental del integral de línea a) \( \int_{c}(3 y i+3 x j) \cdot d r \) C. curva suave desde \( (0,0) \) hasta \( (3,8) \) b) \( \int_{c} \cos (x) \op
  2 answers
• 
  

  
  1. Determine si los siguientes campos vectoriales son conservativos, de no serlo, explique. a) \( F(x, y)=3 x^{2} y^{2} i+2 x^{3} y j \) b) \( F(x, y)=x e^{x^{2} y}(2 y i+x j) \) c) \( F(x, y, z)=x y^
  2 answers

• 

  
  Evaluate the triple integral \( \iiint_{B} g(x, y, z) d V \) over solid \( \mathrm{B} \). \( B=\left\{(x, y, z) \mid x^{2}+y^{2} \leq 2^{2}, x \geq 0, y \geq 0,0 \leq z \leq 1\right\} \) and \( g(x, y
  2 answers