Find Directional Derivative at Given Point in Direction of Given Vector

[betamu]

Homework Statement

[/B]
Find the directional derivative of the function at the given point in the direction of the vector v.

$$g(s,t)=s\sqrt t, (2,4), \vec{v}=2\hat{i} - \hat{j}$$

Homework Equations

$$\nabla g(s,t) = <g_s(s,t), g_t(s,t)>\\
\vec{u} = \vec{v}/|\vec{v}|\\
D_u g(s,t) = \nabla g(s, t) \cdot \vec{u}$$

The Attempt at a Solution

I found $$\nabla g(s, t) =<\sqrt{t}, s/(2\sqrt{t})>$$ which gives $$\nabla g(2,4) = <2, 1/2>$$ and the directional vector to be $$<2/\sqrt{5}, -1/\sqrt{5}>$$ Which gives a dot product of $$5/2\sqrt{5}$$ but my book says that it should be $$7/2\sqrt{5}$$.

 

[fresh_42]

Homework Statement

[/B]
Find the directional derivative of the function at the given point in the direction of the vector v.

$$g(s,t)=s\sqrt t, (2,4), \vec{v}=2\hat{i} - \hat{j}$$

Homework Equations

$$\nabla g(s,t) = <g_s(s,t), g_t(s,t)>\\
\vec{u} = \vec{v}/|\vec{v}|\\
D_u g(s,t) = \nabla g(s, t) \cdot \vec{u}$$

The Attempt at a Solution

I found $$\nabla g(s, t) =<\sqrt{t}, s/(2\sqrt{t})>$$ which gives $$\nabla g(2,4) = <2, 1/2>$$ and the directional vector to be $$<2/\sqrt{5}, -1/\sqrt{5}>$$ Which gives a dot product of $$5/2\sqrt{5}$$ but my book says that it should be $$7/2\sqrt{5}$$.

What is $\langle (2,1/2)\,,\,(2,-1)\rangle \,?$

 

[Mark44]

Homework Statement

[/B]
Find the directional derivative of the function at the given point in the direction of the vector v.

$$g(s,t)=s\sqrt t, (2,4), \vec{v}=2\hat{i} - \hat{j}$$

Homework Equations

$$\nabla g(s,t) = <g_s(s,t), g_t(s,t)>\\
\vec{u} = \vec{v}/|\vec{v}|\\
D_u g(s,t) = \nabla g(s, t) \cdot \vec{u}$$

The Attempt at a Solution

I found $$\nabla g(s, t) =<\sqrt{t}, s/(2\sqrt{t})>$$ which gives $$\nabla g(2,4) = <2, 1/2>$$ and the directional vector to be $$<2/\sqrt{5}, -1/\sqrt{5}>$$ Which gives a dot product of $$5/2\sqrt{5}$$

Check your work on the dot product. I think you missed that the second fraction has a 2 in the denominator.

but my book says that it should be $$7/2\sqrt{5}$$.

I get this, as well, but it should be written as $.7/(2\sqrt{5})$ or better, as $$\frac 7 {2\sqrt 5}$$

 

[betamu]

What is $\langle (2,1/2)\,,\,(2,-1)\rangle \,?$

I'm unsure what you mean. You're putting the gradient vector and v into one vector together?

 

[betamu]

Check your work on the dot product. I think you missed that the second fraction has a 2 in the denominator. I get this, as well, but it should be written as $.7/(2\sqrt{5})$ or better, as $$\frac 7 {2\sqrt 5}$$

Haha wow. Yeah you're right. Well at least I gained some experience in writing out latex code in posting this. Thanks!

 

[fresh_42]

I'm unsure what you mean. You're putting the gradient vector and v into one vector together?

Yes, it is $\nabla f \cdot \vec{v}$, I simply left out the norm $\sqrt{5}$ since your mistake was the dot product, not the factor.

 

[Mark44]

I'm unsure what you mean. You're putting the gradient vector and v into one vector together?

That should be the dot product of two vectors. fresh_42's notation might be for the inner product, a generalization of the dot product.

 

[betamu]

Oh alright, I'd never seen that notation before. Thank you both!