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March 1st, 2017, 02:35 PM   #1
b19
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Can someone please help me answer this? thanks

Use the gradient to find the directional derivative of the function at P in the direction of Q.
f(x, y) = 3x² - y² + 4, P(7, 5), Q(6, 8)

Last edited by skipjack; March 1st, 2017 at 02:56 PM.
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March 1st, 2017, 02:58 PM   #2
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So can you get the gradient vector $\displaystyle \begin{align*} \nabla f \end{align*}$? Do you know how to use this to get the direction vector $\displaystyle \begin{align*} \frac{\mathrm{d}f}{\mathrm{d}\mathbf{u}} = \nabla f \cdot \hat{ \mathbf{u} } \end{align*}$?
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March 1st, 2017, 03:01 PM   #3
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So can you get the gradient vector $\displaystyle \begin{align*} \nabla f \end{align*}$? Do you know how to use this to get the direction vector $\displaystyle \begin{align*} \frac{\mathrm{d}f}{\mathrm{d}\mathbf{u}} = \nabla f \cdot \hat{ \mathbf{u} } \end{align*}$?
No, I don't.

Last edited by skipjack; March 3rd, 2017 at 04:36 PM.
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March 1st, 2017, 04:10 PM   #4
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No, I don't.
What do you mean by this? That you do not know how to find $\displaystyle \nabla f$ or that you do not know how to use that to find $\displaystyle \nabla f\cdot \vec{v}$?

For any function of two variables, f(x, y), $\displaystyle \nabla f= \frac{\partial f}{\partial x}\vec{i}+ \frac{\partial f}{\partial y}\vec{j}$.
Here, $\displaystyle f(x, y)= 3x^2- y^2+ 4$ so $\displaystyle \nabla f= 6x\vec{i}- 2y\vec{j}$. Evaluate that at x = 7, y = 5 and take the dot product of that with $\displaystyle 6\vec{i}+ 8\vec{j}$.

Last edited by skipjack; March 3rd, 2017 at 05:16 PM.
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March 3rd, 2017, 03:16 PM   #5
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What do you mean by this? That you do not know how to find $\displaystyle \nabla f$ or that you do not know how to use that to find $\displaystyle \nabla f\cdot \vec{v}$?

For any function of two variables, f(x, y), $\displaystyle \nabla f= \frac{\partial f}{\partial x}\vec{i}+ \frac{\partial f}{\partial y}\vec{j}$.
Here, $\displaystyle f(x, y)= 3x^2- y^2+ 4$ so $\displaystyle \nabla f= 6x\vec{i}- 2y\vec{j}$. Evaluate that at x = 7, y = 5 and take the dot product of that with $\displaystyle 6\vec{i}+ 8\vec{j}$.
Ok, I know my answer is = -12/sqrt20(7) - -8/sqrt20 (5) so my answer in decimal form is -9.84 ... but I need my answer to be in a form for example -66 sqrt2/5, can someone convert that for me because I don't know how to... thanks.

Last edited by skipjack; March 3rd, 2017 at 05:16 PM.
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March 3rd, 2017, 05:24 PM   #6
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I haven't checked your calculation, but your expression is equivalent to -22√5/5.
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