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Calculus Archive: Questions from August 08, 2022

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  7. Find the derivative \( d y / d x \) for each of the following functions: (a) \( y=\sqrt{x+\sqrt{1+x}} \) (b) \( y=\left(\frac{2 x+3}{4 x+5}\right)^{6} \) (c) \( y=\frac{x^{2}+\sqrt[3]{x}}{x^{3}+\sq
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  1. \( y=4 x^{2}-3 x-2 \) 11. \( y=\frac{1}{\sqrt{x}} \) 2. \( y=3+4 x-x^{2} \) 12. \( y=3 x^{2}-2 \sqrt{x} \) 3. \( y=\frac{1}{2}\left(3 x^{2}+1\right)^{2} \) 13. \( y=\frac{1}{\sqrt{x-2}} \) 4. \( y=
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  Given \( f(x, y)=y \ln (6 x-5 y) \) \[ f_{x}(x, y)= \] \[ f_{y}(x, y)= \]
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  Solve the differential equation \( \left(4+\tan ^{2} x\right) y^{\prime}=\sec ^{2} x \) \( y=\frac{1}{2} \arctan \left(\frac{\sec x}{2}\right)+C \) \( y=\arctan \left(\frac{\sec x}{2}\right)+C \) \( y
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  \( \int_{42}^{56} \int_{0}^{35} \int_{y}^{35} \frac{x \cos z}{z} d z d y d x . \)
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• y= ln(x+1)/(1+ln(x+1))
  \( y=\frac{\ln (x+1)}{1+\ln (x+1)} \quad \) find \( (-1 \)
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• Evaluate f(x, y, z) = x2 + y2 + z2 S: x2 + y2 = 9, 0 ≤ x ≤ 3, 0 ≤ z ≤ x
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• solve the following integrals , using the most appropiate technique , integration by parts or tabulation.
  Instrucciones al estudiante: En esta actividad usted contestará los ejercicios de práctica que se presentan a continuación. Es necesario que muestre su procedimiento claro y completo para que obten
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• evaluate the following integrals.
  Evaluar las siguientes integrales: 1. \( \int \ln x^{3} d x \) 2. \( \int\left(t^{3}-2 t^{2}+4 t-3\right) e^{2 t} d t \) 3. \( \int \sin ^{3} x \cos ^{2} x d x \) 4. \( \int \sec ^{3} \frac{x}{2} \tan
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• 1.find the area of the region. 2. Find the volume of the solid of revolution generated by rotating the given region around the indicated axis.
  1. Encuentre el área de la región \( y=e^{-x} \sin \pi x, y=0, x=0 \quad y x=1 \) 2. Halle el volumen del sólido de revolución generado al girar la región dada, alrededor del eje indicado \[ y=\o
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• evaluate the following integrals.
  Evalúe las siguientes integrales: 1. \( \int \frac{d x}{x^{2} \sqrt{9-x^{2}}} \) 2. \( \int \frac{\sqrt{x^{2}-3}}{x} d x \) 3. \( \int_{0}^{3} \frac{x^{3}}{\sqrt{x^{2}+9}} d x \) 4. \( \int \frac{d x
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  Let \( f(x, y)=\sqrt{175-3 x^{2}-4 y^{2}} \) Evaluate \( f(3,0) \)
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  Let \( f(x, y)=\sqrt{125-2 x^{2}-3 y^{2}} \) Evaluate \( f(4,5) \)
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  \[ B=\left\{(x, y, z) \mid 0 \leq x \leq \pi, \frac{3 \pi}{2} \leq y \leq 2 \pi, 1 \leq z \leq 3\right\} \] evaluate the triple integral \[ \iiint_{B} z \sin x \cos y d V= \]
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  Evaluate \( \iiint_{E} 2 x z d V \) where \( E=\{(x, y, z) \mid 2 \leq x \leq 4, x \leq y \leq 2 x, 0
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  1) [20 Points] Solve the following IVPs: a) \( y^{\prime}=\frac{2 t}{\sin y} \), \( y(1)=\pi \) b) \( x y^{\prime}+y=2 x \cos x \), \( y(\pi)=0 \)
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  Differentiate. 3) \( y=\frac{e^{x}}{7 x^{2}+8} \)
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• Can you show step by step? Thank you!
  Find \( y^{\prime} \) and \( y^{\prime \prime} \). \[ y=e^{5 e^{x}} \]
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  Evaluate \( \iiint_{E}(x+y-3 z) d V \) where \( E=\left\{(x, y, z) \mid-7 \leq y \leq 0,0 \leq x \leq y, 0
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  Evaluate the iterated integral. \[ \int_{0}^{\pi / 2} \int_{0}^{\cos \theta} 6 e^{\sin \theta} d r d \theta, \] Part 1 of 4 When evaluating \( \int_{0}^{\pi / 2} \int_{0}^{\cos \theta} 6 e^{\sin \thet
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  Given \( f(x, y)=-2 x^{2}-x y^{5}+y^{4} \) \[ f_{x}(x, y)= \] \[ f_{y}(x, y)= \]
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  Given \( f(x, y)=4 x^{4}+6 x y^{2}+y^{5}, f \) \[ f_{x x}(x, y)= \] \[ f_{x y}(x, y)= \]
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  Solve \( y^{\prime}+2 y^{\prime}-3 y=0, y(0)=2, y^{\prime}(0)=6 \) (12 marks)
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  Evaluate \( \iiint_{E} 2 x z d V \) where \( E=\{(x, y, z) \mid 2 \leq x \leq 4, x \leq y \leq 2 x, 0
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  Find \( \iint_{D}(3 x+5 y) d A \) where \( D=\left\{(x, y) \mid x^{2}+y^{2} \leq 4, x \geq 0\right\} \)
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  Find \( \iint_{D}(3 x+5 y) d A \) where \( D=\left\{(x, y) \mid x^{2}+y^{2} \leq 4, x \geq 0\right\} \)
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