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Chapter 12: TF. 2 (page 946)

$A chain rule for functions of two variables : Let f (x, y) = x^{2} y^{3}, x = s \cos t, and y = s \sin t . Find \frac{\partial x}{\partial s} and \frac{\partial x}{\partial t} .$

Short Answer

Expert verified

$\frac{\partial x}{\partial s} = \cos t \frac{\partial x}{\partial t} = - s \sin t$

Step by step solution

Step 1. Given information

$f (x, y) = x^{2} y^{3}, x = s \cos t, and y = s \sin t .$

Step 2. Finding partial derivatives

$x = s \cos t \Rightarrow \frac{\partial x}{\partial s} = \cos t \Rightarrow \frac{\partial x}{\partial t} = - s \sin t$

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Most popular questions from this chapter

Explain the steps you would take to find the extrema of a function of two variables, $f (x, y), if (x, y)$ is a point in the rectangle $R$ defined by role="math" localid="1649881836115" $a \leq x \leq b and c \leq y \leq d .$

In Exercises $21 - 28$ , find the directional derivative of the given function at the specified point $P$ and in the direction of the given unit vector $u$ .

$f (x, y) = \sqrt{\frac{y}{x}}$ at $P = (4, 9), u = (- \frac{\sqrt{17}}{17}, - \frac{4 \sqrt{17}}{17})$

Find the directional derivative of the given function at the specified point P in the direction of the given vector. Note: The given vectors may not be unit vectors.

$f (x, y, z) = x y^{2} z^{3}, P = (0, 0, 0), v = ⟨ 1, - 2, - 1 ⟩$

$A chain rule for functions of two variables : Let f (x, y) = x^{2} y^{3}, x = s \cos t, and y = s \sin t . Substitute the expressions for x and y into f (x, y) .$

Explain how you could use the method of Lagrange multipliers to find the extrema of a function of two variables, $f (x, y),$ subject to the constraint that $(x, y)$ is a point on the boundary of a triangle $T$ in the xy-plane.

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Achainruleforfunctionsoftwovariables:Letf(x,y)=x2y3,x=scost,andy=ssint.Find∂x∂sand∂x∂t.

Short Answer

Step by step solution

Step 1. Given information

Step 2. Finding partial derivatives

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$A chain rule for functions of two variables : Let f (x, y) = x^{2} y^{3}, x = s \cos t, and y = s \sin t . Find \frac{\partial x}{\partial s} and \frac{\partial x}{\partial t} .$