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Problem 11

Find the first partial derivatives of the function. \(f(s, t)=\left(s^{2}-s t+t^{2}\right)^{3}\)

Expert verified

The first partial derivatives of the function \(f(s, t) = (s^2 - st + t^2)^3\) are:
\(\frac{\partial f}{\partial s} = 3(s^2 - st + t^2)^2 \cdot (2s - t)\)
\(\frac{\partial f}{\partial t} = 3(s^2 - st + t^2)^2 \cdot (-s + 2t)\)

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Chapter 12

Boor An empirical formula by E. F. Dubois relates the surface area \(S\) of a human body (in square meters) to its weight \(W\) (in kilograms) and its height \(H\) (in centimeters). The formula, given by $$S=0.007184 W^{0.425} H^{0.725}$$ is used by physiologists in metabolism studies. a. Find the domain of the function \(S\). b. What is the surface area of a human body that weighs \(70 \mathrm{~kg}\) and has a height of \(178 \mathrm{~cm}\) ?

Chapter 12

Find the first partial derivatives of the function. \(f(x, y)=\frac{x}{1+y}\)

Chapter 12

Find the critical point(s) of the function. Then use the second derivative test to classify the nature of each point, if possible. Finally, determine the relative extrema of the function. \(f(x, y)=2 x^{2}+y^{2}-4 x+6 y+3\)

Chapter 12

Find the first partial derivatives of the function. \(h(r, s, t)=e^{r s t}\)

Chapter 12

The efficiency of an internal combustion engine is given by $$E=\left(1-\frac{v}{V}\right)^{04}$$ where \(V\) and \(v\) are the respective maximum and minimum volumes of air in each cylinder. a. Show that \(\partial E / \partial V>0\) and interpret your result. b. Show that \(\partial E / \partial v<0\) and interpret your result.

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