Question

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Answer 1

Answer:

Well the directional derivative of the given two variable function will be it's gradient taken at point (2,-1)

The function is

z = y^2 * e^2x

So,

first of all it's gradient will be

Grad z = d/dx y^2e^2x i^ + d/dy y^2e^2x j^

= 2y^2e^2x i^ + 2ye^2x j^

Grad evaluated at (2,-1) = 2e^4 i^ - 2e^4 j^

Now,

because the vector b is a unit vector and also the directional derivative along it's is zero.

Which means that the dot product of this vector b along with grad at (2-1) must be zero.

Therefore,

a 2e^4 - b 2e^4 = 0

So,

a = b

But also since b is a unit vector therefore,

a^2 + b^2 = 1

Or,

a = b = +-1/√2

So,

|a + b| = √2

I'm sorry but don't confuse the vector b with the a and b components that i used in place of alpha and beta.

Hope this helps you !

Thank you so much !

Answer 2

Answer:

hope it helps you see the attachment for further information .....