Engineering Mathematics

K. A. Stroud, Dexter J. Booth

Chapter 23 Partial differentiation 2 - all with Video Answers

Educators

Section 1

Test exercise

01:12

Problem 1

Use partial differentiation to determine expressions for $\frac{\mathrm{d} y}{\mathrm{~d} x}$ in the following cases:
(a) $x^{3}+y^{3}-2 x^{2} y=0$
(b) $e^{x} \cos y=e^{y} \sin x$
(c) $\sin ^{2} x-5 \sin x \cos y+\tan y=0$

Carson Merrill

Numerade Educator

01:04

Problem 2

The base radius of a cone, $r$, is decreasing at the rate of $0-1 \mathrm{~cm} / \mathrm{s}$ while the perpendicular helght, $h$, is increasing at the rate of $0-2 \mathrm{~cm} / \mathrm{s}$. Pind the rate at which the volume, $V$, is changing when $r=2 \mathrm{~cm}$ and $h=3 \mathrm{~cm}$.

Carson Merrill

Numerade Educator

01:05

Problem 3

If $z=2 x y-3 x^{2} y$ and $x$ is increasing at $2 \mathrm{~cm} / 5$, determine at what rate $y$. must be changing in order that $z$ shall be neither increasing nor decreasing at the instant when $x-3 \mathrm{~cm}$ and $y=1 \mathrm{~cm}$.

Carson Merrill

Numerade Educator

01:11

Problem 4

If $z=x^{4}+2 x^{2} y+y^{3}$ and $x=r \cos \theta$ and $y=r \sin \theta$, find $\frac{\partial z}{\partial r}$ and $\frac{\partial z}{\partial 0}$ in their simplest forms.

Carson Merrill

Numerade Educator