a) Sketch a few level surfaces of the function f(x, y, z) = 2x + y + 3z and use them to indicate the general direction in which the values of f increase.  b) The plane y = 3 intersects the...

a) Sketch a few level surfaces of the function f(x, y, z) = 2x + y + 3z and use them to indicate the general direction in which the values of f increase. 

b) The plane y = 3 intersects the surface z = 2x2 + y2 in a curve. Find a parametric equation for the tangent line to this curve at the point corresponding to x = 2. 

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aruv | High School Teacher | (Level 2) Valedictorian

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Let f(x,y,z)=c , where c is an arbitrary constant.

Thus

2x+y+3z=c

3z=c-(2x+y)

z=(c-2x-y)/3.0

Since c is an arbitrary constant.

1. Let c>0

z increases as x and y are decreases i.e. x,y <0 .

z decreases as x and y are increases i.e. x,y >0 .

2. Let c <0

z decreases as x and y are decreases i.e. x,y <0 .

z increases as x and y are increases i.e. x,y >0 .

Thus for different values of c, we have different level surfaces.

So f(x,y,z) decrease as x,y, and z decrease.

f(x,y,z) increase as x,y, and z increase.

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aruv | High School Teacher | (Level 2) Valedictorian

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(b).The plane y = 3 intersects the surface `z=2x^2+y^2`  in a curve.

Thus the intersecting curv will be

`z=2x^2+9`                       (i)

`gradz=4x+0y+0z`

`gradz=(4x,0,0)`

`gradz_{(2,0,0)}=4xx2`

`=8`

The corresponding tangent line in parametric form is given by

`(x(t),y(t),z(t))=(2,0,0)+t(8,0,0)`

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aruv | High School Teacher | (Level 2) Valedictorian

Posted on

(a).We have given

`f(x,y,z)=2x+y+z`                  (i)

Let us assume

`f(x,y,z)=c `    (ii)

Where c is an arbitrary constant.

So from (i) and (ii), we have

`2x+y+3z=c`

`3z=c-y-2x`

`z=(c-y-2x)/3`

Since c is an arbitrary constant. So we have following

1. If  c is positive i.e. c > 0 then z will increase if  x and y will decrease  i.e. x,y <0 .

If x and y will  increase then z will decrease  i.e. x,y >0 .

2. If c is negative i.e.  c <0 then z will dicrease if  x and y will decrease  i.e. x,y <0 .

If x and y will  increase then z will increase  i.e. x,y >0 .

Since for different values of c, we have different level surfaces.

So f(x,y,z) will decrease if  x,y, and z decrease and f(x,y,z) will  increase if  x,y, and z increase.

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