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Chapter 12: Q. 33 (page 944)

Find the first-order partial derivatives for the functions in Exercises 27–36.
$f (r, θ) = r \sin θ$

Short Answer

Expert verified

The first-order partial derivatives are $\frac{\partial f}{\partial r} = \sin θ and \frac{\partial f}{\partial θ} = r \cos θ$

Step by step solution

01

Given

The given function is: $f (r, θ) = r \sin θ$

02

To find

We have to find the first-order partial derivatives.

03

Calculation

$f (r, θ) = r \sin θ \frac{\partial f}{\partial r} = \frac{\partial}{\partial r} (r \sin θ) \frac{\partial f}{\partial r} = \sin θ f (r, θ) = r \sin θ \frac{\partial f}{\partial θ} = \frac{\partial}{\partial θ} (r \sin θ) \frac{\partial f}{\partial θ} = r \cos θ Hence, \frac{\partial f}{\partial r} = \sin θ and \frac{\partial f}{\partial θ} = r \cos θ$

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Most popular questions from this chapter

When you use the method of Lagrange multipliers to find the maximum and minimum of $f (x, y) = x + y$ subject to the constraint $x y = 1,$ you obtain two points. Is there a relative maximum at one of the points and a relative minimum at the other? Which is which?

Use Theorem 12.32 to find the indicated derivatives in Exercises 21–26. Express your answers as functions of a single variable.

$\frac{d θ}{d t} w h e n θ = \tan^{- 1} \frac{y}{x}, x = e^{t}, a n d y = e^{2 t}$

$Recall that the volume, V, of a cylinder is given by V (r, h) = π r^{2} h . Find \frac{\partial V}{\partial r} and \frac{\partial V}{\partial h} . What are the physical interpretations of these partial derivatives ?$

Find the gradient of the given function, and find the direction in which the function increases most rapidly at the specified point P.

$f (x, y) = \tan^{- 1} (\frac{y}{x}), P = (- 2, 2)$

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 on the boundary of the rectangle $R$ defined by $a \leq x \leq b and c \leq y \leq d .$

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