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Calculus Archive: Questions from February 13, 2023

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  Exercises \( \mathbf{1 - 1 0} \). Find \( d y / d x \). 1. \( y=3 x^{4}-x^{2}+1 \). 2. \( y=x^{2}+2 x^{-4} \). 3. \( y=x-\frac{1}{x} \). 4. \( y=\frac{2 x}{1-x} \). 5. \( y=\frac{x}{1+x^{2}} \). 6. \(
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  Exercise 1. \( y=3 \cos x-4 \sec x \). 2. \( y=x^{2} \sec x \). 3. \( y=x^{3} \csc x \). 4. \( y=\sin ^{2} x \).
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  Find the second derivative, \( y^{\prime \prime} \). \[ y=\sqrt{2 x+1} \] A. \( y^{\prime \prime}=-\frac{1}{(2 x+1)^{\frac{3}{2}}} \) B. \( y^{\prime \prime}=-\frac{1}{4(2 x+1)^{\frac{3}{2}}} \) C. \(
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  Find the second derivative, \( y^{\prime \prime} \). Then simplify by factoring. \[ y=\left(x^{3}+7\right)^{5} \] A. \( y^{\prime \prime}=20\left(x^{3}+7\right)^{3} \) B. \( y^{\prime \prime}=30 x\lef
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  0. Find the derivative of the function: (a) \( y=\left(x^{5}-4 x+2\right) e^{x}= \) (b) \( y=\frac{3 x+1}{\left(x^{3}+1\right) \sin x} \) (c) \( y=\left(x^{2}-3 x+4\right)^{5} \) (d) \( y=\cos \left(x
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  \( \begin{array}{l}\text { Find } \frac{d^{2} y}{d x^{2}} \\ y=2 x+3 \\ \frac{d^{2} y}{d x^{2}}=\end{array} \)
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• Determine the horizontal asymptote. y = (2 marks) y= (2 marks) f (x)= (2 marks)
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• Please answer 2,4,10 without using the chain rule
  1-22 Differentiate. 1. \( f(x)=3 \sin x-2 \cos x \) 2. \( f(x)=\tan x-4 \sin x \) 3. \( y=x^{2}+\cot x \) 4. \( y=2 \sec x-\csc x \) 5. \( h(\theta)=\theta^{2} \sin \theta \) 6. \( g(x)=3 x+x^{2} \cos
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• Determine the horizontal asymptote. (2 marks) (2 marks) (2 marks)
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  Example 1: Solve the ODE: \[ y^{\prime}-\frac{1}{x} y=x y^{2} \] Example 2: Solve the ODE: \[ y^{\prime}-\frac{1}{x} y=\frac{-5}{2} x^{2} y^{3} \] Example 3: Solve the ODE: \[ y^{\prime}+\frac{1}{3} y
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• La suma de reas de la region triangular mayor y la region cuadrangular es 89 cm^(2). calcular el valor del perimetro del cuadrado en cm
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• que el corredor no ente para calcular el rsionista en cada com acciones de macdons nimero de accioner [[-1,-(3)/(2),-350],[(3)/(2),-(1)/(2),600]]
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  Ventas de Procesadores de Señales-Digitales: Las ventas de procesadores de señales digitales, en miles de millones de dólares, se proyecta como \[ S(t)=0.14 t 2+0.68 t+3.1 \quad(0 \leq t \leq 6) \]
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• Derivative of trigometric functions Derive the following.
  Example 1. \( v=3 \cos 2 u \) 2. \( w=2 \csc (1-3 x) \) 3. \( y=\sin \left(3 x^{2}-5\right) \) 4. \( y=\cos \sqrt{1-2 x+x^{2}} \) 5. \( y=\sin \beta \cos ^{2} \beta \) 6. \( y=\cot 2 x-\tan x^{2} \) 7
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• Let h(x)=f(g(x)). If: f(2)=−1, f′(34)=0, f′(2)=4, g(2)=34, & g′(2)=−2, then h′(2)=
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  For the following functions \( z=f[x, y] \) compute: 1.1) the general symbolic partial derivatives, \( \frac{\partial f}{\partial x}[x, y] \) and \( \frac{\partial f}{\partial y}[x, y] \), 1.2) the ge
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  Find \( \frac{d y}{d x} \) \[ y=4 \sin x \tan x \] \[ \frac{d y}{d x}= \]
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• Dividir. (18x^(3)-3x^(2)+11x+1)-:(6x+1) La respuesta debe incluir el cociente y el residuo. Cociente: Residuo: x
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• Dividir. (4x^(2)-25x+20)-:(x-5) La respuesta debe incluir el cociente yel residuo. Cociente:
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• Solving for a variabie unide perentheses in te Solye for y. C=(y-k)h
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• Factor the expression, if possible. Factor out any 32-a^(2)+4a
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• Solve cos^2 theta -2= 2sin theta on [0,2pi]
  Solve \( \cos ^{2} \theta-2=2 \sin \theta \) on \( [0,2 \pi) \)
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• s(x)=4x^(2) t(x)=x+6 Escribir las expresiones para (t+s)(x) y (t*s)(x) y evaluar (t-s)(-2)
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  If \( y=(8 x+5)^{2} \), find \( \left.\frac{d y}{d x}\right]_{x=1} \) A. 26 B. 208 C. 169 D. 104 E. 16
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• Subtract. Simplify the result if possible. (3x^(2)+x)/(x-9)-(23x+45)/(x-9)
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  Classify the critical points of a) \[ f(x, y)=(x-1) e^{x y} \] b) \[ f(x, y)=(2-x)(4-y)(x+y-3) . \]
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  Find the limit. \[ \lim _{x \rightarrow 0} \frac{\sin (7 x)}{4 x^{3}-3 x} \]
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  Differentiate the following function. \[ y=5 \csc (x)+5 \cos (x) \] \[ y^{\prime}= \]
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  Function Point \[ y=\frac{5+\csc (x)}{9-\csc (x)} \quad\left(\frac{\pi}{6}, 1\right) \]
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  Find all the second partial derivatives. \[ \begin{array}{l} f(x, y)=x^{6} y-3 x^{5} y^{2} \\ f_{x x}(x, y)= \\ f_{x y}(x, y)= \\ f_{y x}(x, y)=x^{6}-6 x^{5} y \\ f_{y y}(x, y)= \end{array} \]
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• Find dy/dx​​​​​​​
  \( y=\tan ^{-1}\left(\sqrt{7 x^{2}-1}\right) \)
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  Find all the second partial derivatives. \[ \begin{array}{l} f(x, y)=x^{8} y^{4}+3 x^{9} y \\ f_{x x}(x, y)= \\ f_{x y}(x, y)= \\ f_{y x}(x, y)= \\ f_{y y}(x, y)= \end{array} \]
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  1) Find the derivative \( \frac{d y}{d x} \). a) \( y=\sin \left(3 x^{2}+2 x+7\right) \) b) \( y=\tan ^{-1}\left(\frac{1}{x}\right) \) c) \( y=\frac{x+1}{x^{2}+1} \) d) \( x y+x^{2} y^{2}=1 \) e) \( y
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• Find limit
  46. \( \lim _{x \rightarrow 0} \frac{\sin x}{\sin \pi x} \)
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  Let \( A \) be a positive constant. Let \[ f(x)=(A \cos (x))^{x} \] Then \( f^{\prime}(2) \) equals \[ \begin{array}{l} \left(\cos ^{-1}(2)\right)^{2}\left(2 \operatorname { l n } \left(\cos ^{-1}(2)+
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  \( \begin{array}{l}\operatorname{Min} z=2 * x+2 * y \\ \text { subject to } \\ \qquad \begin{array}{l} x+2 * y \geq 3 \\ 2 * x+y \geq 5 \\ x, y \geq 0, \quad x, y \text { integer }\end{array}\end{arra
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• #2.) show all work
  2. Solve the differential equation using separation of variables: a. \( y^{\prime}=\frac{3 x^{2}+2 x+2}{y-1} \) b. \( x y^{\prime}+y^{2}+y=0 \) c. \( y^{\prime} \ln |y|+x^{2} y=0 \)
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