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Eva8 [605]
3 years ago
13

Chef Amy does beginning inventory on Thursday night and finds that she has $4194 in food products in the restaurant. Throughout

the week she purchases: $2088 produce, $1678 meat, $870 dry goods, and $3914 dairy. The following Thursday she does ending inventory and finds that she has $3464 in food. She looks at her sales and finds that she made $30541 over the same 7 day period. What is her total food cost?
Mathematics
1 answer:
Darya [45]3 years ago
5 0

Answer:

Step-by-step explanation:

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Find the directional derivative of the function at the given point in the direction of the vector v. G(r, s) = tan−1(rs), (1, 3)
alexandr1967 [171]

The <em>directional</em> derivative of f at the given point in the direction indicated is \frac{5}{2}.

<h3>How to calculate the directional derivative of a multivariate function</h3>

The <em>directional</em> derivative is represented by the following formula:

\nabla_{\vec v} f = \nabla f (r_{o}, s_{o})\cdot \vec v   (1)

Where:

  • \nabla f (r_{o}, s_{o}) - Gradient evaluated at the point (r_{o}, s_{o}).
  • \vec v - Directional vector.

The gradient of f is calculated below:

\nabla f (r_{o}, s_{o}) = \left[\begin{array}{cc}\frac{\partial f}{\partial r}(r_{o},s_{o})  \\\frac{\partial f}{\partial s}(r_{o},s_{o}) \end{array}\right]   (2)

Where \frac{\partial f}{\partial r} and \frac{\partial f}{\partial s} are the <em>partial</em> derivatives with respect to r and s, respectively.

If we know that (r_{o}, s_{o}) = (1, 3), then the gradient is:

\nabla f(r_{o}, s_{o}) = \left[\begin{array}{cc}\frac{s}{1+r^{2}\cdot s^{2}} \\\frac{r}{1+r^{2}\cdot s^{2}}\end{array}\right]

\nabla f (r_{o}, s_{o}) = \left[\begin{array}{cc}\frac{3}{1+1^{2}\cdot 3^{2}} \\\frac{1}{1+1^{2}\cdot 3^{2}} \end{array}\right]

\nabla f (r_{o}, s_{o}) = \left[\begin{array}{cc}\frac{3}{10} \\\frac{1}{10} \end{array}\right]

If we know that \vec v = 5\,\hat{i} + 10\,\hat{j}, then the directional derivative is:

\nabla_{\vec v} f = \left[\begin{array}{cc}\frac{3}{10} \\\frac{1}{10} \end{array}\right] \cdot \left[\begin{array}{cc}5\\10\end{array}\right]

\nabla _{\vec v} f (r_{o}, s_{o}) = \frac{5}{2}

The <em>directional</em> derivative of f at the given point in the direction indicated is \frac{5}{2}. \blacksquare

To learn more on directional derivative, we kindly invite to check this verified question: brainly.com/question/9964491

3 0
3 years ago
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