Get questions and answers for Calculus

Calculus Archive: Questions from October 23, 2022

• 
  

  
  Solve the following initial value problems: (i) \( y^{\prime}=8 x^{2} e^{-2 y} \) (ii)
  2 answers

• 

  
  Solve the following second order linear differential equations. \[ \begin{array}{l} y^{\prime \prime}+4 y^{\prime}+4 y=0 \\ y^{\prime \prime}+2 y^{\prime}+4 y=0 \\ y^{\prime \prime}+5 y^{\prime}+6 y=x
  2 answers
• 
  

  
  4. Determine the derivative. a) \( y=\frac{x^{3}-2 x+1}{x+3} \) (2 marks) b) \( y=\ln \left(4 x^{2}-5 x\right) \) (2 marks) c) \( y=\left(5 x^{3}-6 x^{2}+8 x+11\right)^{6} \)
  2 answers
• ​​​​​​​
  \( y(1-\ln y) \frac{d^{2} y}{d x^{2}}+(1+\ln y)\left(\frac{d y}{d x}\right)^{2}=0 \)
  2 answers

• 

  
  Find \( y \) as a function of \( x \) if \[ \begin{array}{l} y(0)=0, \quad y^{\prime}(0)=-8, \quad y^{\prime \prime \prime}-5 y^{\prime \prime}-y^{\prime}+5 y=0 \\ y(x)= \end{array} \]
  2 answers
• 4 and 5
  Question1: Find \( \frac{d y}{d x} \) 1. \( y=\frac{x+1}{\left(x^{3}-2\right)^{2}} \) 2. \( y=\left(x^{3}-1\right)^{100} \) 3. \( y=\frac{1}{\sqrt[3]{x^{2}+x+1}} \) 4. \( y=\frac{x+1}{\left(x^{3}+x-2\
  2 answers
• 
  

  

  
  \( \frac{d y}{d x}=\sin ^{2}\left(4^{2} x-4^{2} y\right) \quad y(0)=\frac{\pi}{4(4)^{2}} \) Resolver este PVI usando el método de reducción a separable explicado en clase. Sustituir los valores de
  2 answers
• 
  

  
  Evaluate the limit, using L'Hôpital's Rule if necessary. \[ \lim _{x \rightarrow 0} \frac{\sin 3 x}{\tan 8 x} \]
  2 answers
• 
  

  
  II. Trabaje los integrales 1) \( \int \frac{1}{x \sqrt{1-(\ln x)^{2}}} d x \) 2) \( \int \frac{\operatorname{sen}(x)}{4+\cos ^{2}(x)} d x \) 3) \( \int_{0}^{\sqrt{2}} \frac{1}{\sqrt{4-x^{2}}} d x \)
  2 answers

• 

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

• 

  
  Evaluate the double integral. \[ \iint_{D} 9 x d A, D=\{(x, y) \mid 0 \leq x \leq \pi, 0 \leq y \leq \sin x\} \]
  2 answers
• Calculate the double integral. R x sec2(y) dA, R = (x, y) | 0 ≤ x ≤ 2, 0 ≤ y ≤ 𝜋 4
  2 answers
• Calculate the double integral. 5 cos(x + 2y) dA R , R = (x, y) | 0 ≤ x ≤ 7𝜋, 0 ≤ y ≤ 𝜋 2
  2 answers
• Calculate the double integral. 3xy2 x2 + 1 dA, R = {(x, y) | 0 ≤ x ≤ 3, −2 ≤ y ≤ 2} R
  3 answers
• 
  

  
  Find the limit. \[ \lim _{(x, y) \rightarrow(3 \pi / 2, \pi)} y \sin (x-y) \]
  2 answers
• 
  

  
  FOR THE REMAINDER, FIND \( Y^{\prime} \) : a) \[ \begin{array}{l} y=3 x^{2}-4 x+3 \\ y=\left(3 x^{2}-4 x+9\right)\left(2 x^{2}-4 x\right) \\ y=\left(\frac{\left.3 x^{2}-4 x+9\right)}{\left(2 x^{2}-4 x
  2 answers
• 
  

  
  (1 point) Rewrite the triple integral \( \int_{0}^{1} \int_{0}^{x} \int_{0}^{y} f(x, y, z) d z d y d x \) as \( \int_{a}^{b} \int_{a \cdot l z)}^{g_{2}(z)} \int_{h_{1}(y, z)}^{h_{2}(y, z)} f(x, y, z)
  2 answers

• 

  
  \( \int \sin ^{2} x \cos ^{2} x d x \) \( \int_{0}^{\pi / 2} \sin (2 x) \sin (3 x) d x \) \( \int_{0}^{\pi} \sin (3 x) \cos (4 x) d x \)
  2 answers
• prove the identity
  50. \( \cos (x+y) \cos (x-y)=\cos ^{2} x-\sin ^{2} y \)
  2 answers
• please fine Y
  \( y=\sin ^{3}\left(7 x^{2}\right) \)
  2 answers
• ​​​​​​​
  Find \( y \) as a function of \( x \) if \[ y^{\prime \prime \prime}-4 y^{\prime \prime}-y^{\prime}+4 y=0 \] \[ \begin{array}{l} y(0)=8, \quad y^{\prime}(0)=-1, \quad y^{\prime \prime}(0)=68 \\ y(x)=
  2 answers

• 

  
  Evaluate the double integral. \[ \iint_{D} \frac{y}{x^{2}+1} d A_{,} \quad D=\{(x, y) \mid 0 \leq x \leq 8,0 \leq y \leq \sqrt{x}\} \]
  2 answers
• Calculate y'
  40. \( y=\sin ^{2}(\cos \sqrt{\sin \pi x}) \)
  2 answers
• 
  

  
  3. Solve each of the following IVP by the Laplace transform method: a. \( y^{\prime \prime}-6 y^{\prime}+9 y=t^{2} e^{3 t} \quad y(0)=2 \quad y^{\prime}(0)=17 \) b. \( y^{\prime \prime}+0.04 y=0.02 t^
  2 answers
• the enclosure y =x ,y=x'3 x>0
  \( \iint_{D}\left(x^{2}+2 y\right) d A \)
  2 answers
• 24,32,35
  23. \( \int_{0} \int_{0}^{1 / t^{2} \sin ^{3} \phi d \phi d t} \) 24. \( \int_{0}^{1} \int_{0}^{1} x y \sqrt{x^{2}+y^{2}} d y d x \) 25. \( \int_{0}^{1} \int_{0}^{1} v\left(u+v^{2}\right)^{4} d u d v
  2 answers
• ​​​​​​​
  Find \( y \) as a function of \( x \) if \[ y^{\prime \prime \prime}-4 y^{\prime \prime}-y^{\prime}+4 y=0 \] \[ \begin{array}{l} y(0)=2, y^{\prime}(0)=-4, y^{\prime \prime}(0)=62 \\ y(x)= \end{array}
  2 answers

• 

  
  Evaluate the double integral. \[ \iint_{D} \frac{y}{x^{2}+1} d A, \quad D=\{(x, y) \mid 0 \leq x \leq 2,0 \leq y \leq \sqrt{x}\} \]
  2 answers

• 

  
  Evaluate the double integral. \[ \iint_{D} 4 y^{2} e^{x y} d A, D=\{(x, y) \mid 0 \leq y \leq 4,0 \leq x \leq y\} \]
  2 answers
• 
  

  
  Find each limit. \[ f(x, y)=\sqrt{y}(y+7) \] (a) \( \lim _{\Delta x \rightarrow 0} \frac{f(x+\Delta x, y)-f(x, y)}{\Delta x} \) (b) \( \lim _{\Delta y \rightarrow 0} \frac{f(x, y+\Delta y)-f(x, y)}{\D
  2 answers
• 
  

  
  Find each limit. \[ f(x, y)=\sqrt{y}(y+7) \] (a) \( \lim _{\Delta x \rightarrow 0} \frac{f(x+\Delta x, y)-f(x, y)}{\Delta x} \) (b) \( \lim _{\Delta y \rightarrow 0} \frac{f(x, y+\Delta y)-f(x, y)}{\D
  2 answers
• 
  

  
  Find and simplify the function values. \[ \begin{array}{l} f(x, y)=5 x^{2}-7 y \\ \text { (a) } \frac{f(x+h, y)-f(x, y)}{h} \end{array} \] (b) \( \frac{f(x, y+h)-f(x, y)}{h} \)
  2 answers

• 

  
  Preguntas: 1. \( F(n)=F(n-1)+F(n-2) \) donde: \[ \begin{array}{l} F(0)=0 \\ F(1)=1 \end{array} \] 2. \( t(n)=6+(n-1)+4+(n-2) \) donde: \[ \begin{array}{l} +(0)=0 \\ +(1)=4 \sqrt{5} \end{array} \] 3. \
  2 answers
• ​​​​​​​
  LARSONET5 13.1.018. Find and simplify the function values. \[ \begin{array}{l} f(x, y)=4 x^{2}-10 y \\ \text { (a) } \frac{f(x+h, y)-f(x, y)}{h} \end{array} \] (b) \( \frac{f(x, y+h)-f(x, y)}{h} \)
  2 answers
• ​​​​​​​
  Find each limit. \[ f(x, y)=\sqrt{y}(y+5) \] (a) \( \lim _{\Delta x \rightarrow 0} \frac{f(x+\Delta x, y)-f(x, y)}{\Delta x} \) (b) \( \quad \lim _{\Delta y \rightarrow 0} \frac{f(x, y+\Delta y)-f(x,
  2 answers
• 
  
  

  
  

  
  B. ¿Antes o después? ¿Cuándo hace usted estas cosas? Haga oraciones lógicas usando antes de o después de, según el modelo. No debe cambiar el orden de las frases. Mcorio: estudiar las lecciones
  2 answers

• 

  
  Find the limit, if it exists. \( \lim _{(x, y) \rightarrow(0,0)} \frac{x^{3}-y^{3}}{x-y} \)
  2 answers
• let 2 sin x + 7 cos y = 7 dy/dx= d^2 y/dx^2=
  2 answers
• Calcular la transformada de Laplace y su inversa usando el primer teorema de traslación, calcular la transformada de Laplace y su inversa usando el segundo teorema de traslación, determinar la trans
  Paree/Match: 1. \( \mathrm{L}\left\{e^{-2 t} \cos ^{2} 4 t\right\} \) a) \( \frac{10 !}{(s+7)^{11}} \) 2. \( L\left\{t^{10} e^{-7 t}\right\} \) b) \( \left(\frac{4 s+3}{s^{2}}\right) e^{-s} \) 3. \( L
  2 answers

• 

  
  Find \( y^{\prime} \). \[ y=\ln \left(\left(x^{4}+7\right)^{8}\right) \] \[ y^{\prime}= \]
  2 answers
• 
  

  
  11. \( \frac{d}{d x}\left(\cos ^{-1}(-3 x)\right)= \) (A) \( \frac{3}{\sqrt{1-(-3 x)^{2}}} \) (B) \( \frac{-3}{\sqrt{1-(-3 x)^{2}}} \) (C) \( -\sin ^{-1}(-3 x) \cdot(-3) \) (D) \( -\cos ^{-2}(-3 x) \c
  2 answers

• 

  
  2. \( t(n)=6+(n-1)+4+(n-2) \) donde: \[ \begin{array}{l} t(0)=0 \\ +(1)=4 \sqrt{5} \end{array} \]
  2 answers
• 
  

  
  1. Find the first derivative \( \left(y^{\prime}\right) \) for the following functions: a) \( y=6 x^{4}-3 e^{x}+4 x^{-3}+17 \) b) \( y=\left(3 x^{3}+5 x^{2}\right)\left(x^{-3}+8 x\right) \) c) \( y=\f
  2 answers
• Q8
  Differentiate the function. \[ y=\left(6 x^{4}-x+5\right)\left(-x^{5}+3\right) \]
  2 answers

• 

  
  Find the partial derivatives of the function \[ \begin{array}{l} f(x, y)=x y e^{4 y} \\ f_{x}(x, y)= \\ f_{y}(x, y)= \\ f_{x y}(x, y)= \\ f_{y x}(x, y)= \end{array} \]
  2 answers

• 

  
  Given \( f(x, y)=-\left(4 x^{6} y+8 x y^{3}\right) \). Compute: \[ \begin{array}{l} \frac{\partial^{2} f}{\partial x^{2}}= \\ \frac{\partial^{2} f}{\partial y^{2}}= \end{array} \]
  2 answers