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Chapter 12: Q. 21. (page 964)

Use Theorem 12.32 to find the indicated derivatives in Exercises
21鈥�26. Express your answers as functions of a single variable
$\frac{d z}{d t}$ when $z = \sin x \cos y, x = e^{t} a n d y = t^{3}$

Short Answer

Expert verified

The required single variable function is

$\frac{d z}{d t} = (\cos (e^{t}) \cos (t^{3})) (e^{t}) + (- \sin (e^{t}) \sin (t^{3})) (3 t^{2})$

Step by step solution

01

Given information

Think about the following function.

$z = \sin x \cos y, x = e^{t} a n d y = t^{3}$

02

The objective is to find out the single variable function.

By using the chain rule,

$\frac{d z}{d t} = \frac{d z d x}{d x d t} + \frac{d z d y}{d y d t} z = \sin x \cos y \frac{d z}{d x} = \cos x \cos y \frac{d z}{d y} = - \sin x \sin y$

Before moving on to the next stage,

$x = e^{t} 鈬� \frac{d x}{d t} = e^{t} y = t^{3} 鈬� \frac{d y}{d t} = 3 t^{2}$

Then,

$\frac{d z}{d t} = \frac{d z d x}{d x d t} + \frac{d z d y}{d y d t} \frac{d z}{d t} = (\cos x \cos y) (e^{t}) + (- \sin x \sin y) (3 t^{2})$

The single variable function is

$\frac{d z}{d t} = (\cos (e^{t}) \cos (t^{3})) (e^{t}) + (- \sin (e^{t}) \sin (t^{3})) (3 t^{2})$

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Most popular questions from this chapter

$Extrema : Find the local maxima, local minima, and saddle points of the given functions f (x, y) = x^{3} + y^{3} 鈭� 6 x^{2} + 3 y^{2} - 4 .$

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.

$f (x, y, z) = \sqrt{\frac{x y}{z}}, P = (2, 3, 1), v = 鉄� 2, 1, 鈭� 2 鉄�$

In Exercises, by considering the function $f (x, y) = x^{2} y$ subject to the constraint $x + y = 0,$ you will explore a situation in which the method of Lagrange multipliers does not provide an extremum of a function.

Explain why $(0, 0)$ is not an extremum of $f (x, y) = x^{2} y$ subject to the constraint $x + y = 0 .$

Evaluate the following limits, or explain why the limit does not exist.

$\lim_{(x, y) 鈫� (0, 0)} \frac{x + y}{x^{2} + y^{2}}$

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 a point on the boundary of a triangle $T$ in the xy-plane.

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