1) Do the following for each proposed equation. (Preferably done with geogebra) Plots the part of the surface that would be seen in the proposed constraint. Denotes traces and slice curves, as well as slices on the coordinate axes. Briefly describe the surface that is formed.

Use the chain rule to find $\frac{dV}{dt}$ by differentiating the volume formula $V = \frac{1}{3} π r^2 h$ with respect to $t$.

Resuelto 1) Do the following for each proposed equation.

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Pregunta: 1) Do the following for each proposed equation. (Preferably done with geogebra) Plots the part of the surface that would be seen in the proposed constraint. Denotes traces and slice curves, as well as slices on the coordinate axes. Briefly describe the surface that is formed.
1) Do the following for each proposed equation. (Preferably done with geogebra)

Plots the part of the surface that would be seen in the proposed constraint.
Denotes traces and slice curves, as well as slices on the coordinate axes.
Briefly describe the surface that is formed.

2) The volume of a cone is changing and at a certain time the height of the cone is 15 cm and is decreasing at a rate of 0.2 cm/min and the radius of the base is 10 cm and is decreasing at a rate of 0.3 cm/min.

a) Find the rate at which the volume of the cone changes in this time.

b) Interpret the result of part a)

Remember that the volume of the cone is V = 1/3 π r^2 h

3)

Consider the equation x^2 + y^2 + z^2 = 9

a) Find the equation of the tangent plane to the graph at the point P(0,-3,0)

b) Draw the surface and the plane in the same system of coordinate axes (A sketch)

c) Comment on the graph of the surface and indicate the slopes of the tangent plane (specify its value).

Muestra el texto de la transcripción de la imagen
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Solución
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Use the chain rule to find by differentiating the volume formula with respect to .
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Texto de la transcripción de la imagen:

Realiza lo siguiente para cada ecuación propuesta. 1. Gráfica la parte de la superficie que se vería en la restricción propuesta. 2. Denota las trazas y las curvas de corte, así como también los cortes en los ejes coordenados. 3. Describe brevemente la superficie que se forma. a)

z = 4 - x - y

con la restricción

- 2 \leq x \leq 2; - 2 \leq y \leq 2

z = y x^{2}

con restricción

- 2 \leq x \leq 2; - 1 \leq y \leq 1

x^{2} + z^{2} = y + 16

en el primer octante El volumen de un cono está cambiando y en un cierto tiempo la altura del cono mide

15 cm

y disminuye a razón de

0.2 cm / min

y el radio de la base mide

10 cm

y disminuye a razón de

0.3

cm / min .

a) Hallar la razón con la que cambia el volumen del cono en ese tiempo b) Interpreta el resultado del inciso a) Recuerda que el volumen del cono es

V = \frac{1}{3} π r^{2} h

¡Tu solución está lista!

Pregunta: 1) Do the following for each proposed equation. (Preferably done with geogebra) Plots the part of the surface that would be seen in the proposed constraint. Denotes traces and slice curves, as well as slices on the coordinate axes. Briefly describe the surface that is formed.

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