Consider the paraboloid f (x comma y )equals14 minus startfraction x squared over 4 endfraction minus startfraction y squared over 16 endfraction and the point upper p (2 startroot 2 endroot comma 8 )on the level curve f (x comma y )equals8. compute the slope of the line tangent to the level curve at p and verify that the tangent line is orthogonal to the gradient at that point.

See steps below

Step-by-step explanation:

We have the paraboloid

The point

on the level curve f(x,y) = 8

The slope of the line tangent to the level curve at P is given by

Let's then compute the partial derivatives

so, the slope of the tangent at P would be

Hence, the line

is tangent to the level curve. As we know the line passes through , then

and the line  

is the tangent to the level curve at point

In order to find a vector that lies on this line, we take any two points on the line and subtract them.

For example, (0,16) and

The vector is parallel to line.

We have now to see that it is orthogonal to the gradient at

.

To verify this, we have to show that their inner product is 0

the gradient at is

so the inner product is

a) y = -3x +2

b) y = 1/2x -6

these 2 can be combined into one equation.

-3x +2 = .5x -6

8 = 3.5x

x = 8 / 3.5 = 2.2857142857

y = -3 * (8 / 3.5) +2

y = (-24 / 3.5) + 2

y = -4.8571428571

step-by-step explanation:

i need points

step-by-step explanation:

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