다변수 미분 전체 문제

The radius $r$ and height $h$ of a right circular cylinder are measured with possible errors of $4\%\,\text{and}\,5\%,$ respectively. Approximate the maximum possible percentage error in measuring the volume (Recall that the percentage error is the ratio of the amount of error over the original amount. So, in this case, the percentage error in $V$ is given by $\frac{dV}{V}.)$

The base radius and height of a right circular cone are measured as $10$ in. and $25$ in., respectively, with a possible error in measurement of as much as $0.1$ in. each. Use differentials to estimate the maximum error in the calculated volume of the cone.

The electrical resistance $R$ produced by wiring resistors ${R}_{1}$ and ${R}_{2}$ in parallel can be calculated from the formula $\frac{1}{R}=\frac{1}{{R}_{1}}+\frac{1}{{R}_{2}}.$ If ${R}_{1}$ and ${R}_{2}$ are measured to be $7\Omega$ and $6\Omega ,$ respectively, and if these measurements are accurate to within $0.05\Omega ,$ estimate the maximum possible error in computing $R.$ (The symbol $\Omega$ represents an ohm, the unit of electrical resistance.)

The area of an ellipse with axes of length $2a$ and $2b$ is given by the formula

$A=\pi ab.$ Approximate the percent change in the area when $a$ increases by $2\%$ and $b$ increases by $1.5\%.$

The period $T$ of a simple pendulum with small oscillations is calculated from the formula $T=2\pi \sqrt{\frac{L}{g}},$ where $L$ is the length of the pendulum and $g$ is the acceleration resulting from gravity. Suppose that $L$ and $g$ have errors of, at most, $0.5\%$ and $0.1\%,$ respectively. Use differentials to approximate the maximum percentage error in the calculated value of $T.$

Electrical power $P$ is given by $P=\frac{{V}^{2}}{R},$ where $V$ is the voltage and $R$ is the resistance. Approximate the maximum percentage error in calculating power if $120$ $V$ is applied to a $2000-\Omega$ resistor and the possible percent errors in measuring $V$ and $R$ are $3\%$ and $4\%,$ respectively.

For the following exercises, find the linear approximation of each function at the indicated point.

$f(x,y)=x\sqrt{y},\quad P(1,4)$

For the following exercises, find the linear approximation of each function at the indicated point.

$f(x,y)={e}^{x}\cos \,y;P(0,0)$

For the following exercises, find the linear approximation of each function at the indicated point.

$f(x,y)=\arctan (x+2y),P(1,0)$

For the following exercises, find the linear approximation of each function at the indicated point.

$f(x,y)=\sqrt{20-{x}^{2}-7{y}^{2}},\quad P(2,1)$

For the following exercises, find the linear approximation of each function at the indicated point.

$f(x,y,z)=\sqrt{{x}^{2}+{y}^{2}+{z}^{2}},\quad P(3,2,6)$

[T] Find an equation of the tangent plane to the surface $f(x,y)={x}^{2}+{y}^{2}$ at point $(1,2,5),$ and graph the surface and the tangent plane at the point.

[T] Find the equation for the tangent plane to the surface at the indicated point, and graph the surface and the tangent plane: $z=\ln (10{x}^{2}+2{y}^{2}+1),P(0,0,0).$

[T] Find an equation of the tangent plane to the surface $z=f(x,y)=\sin (x+{y}^{2})$ at point $\left(\frac{\pi }{4},0,\frac{\sqrt{2}}{2}\right),$ and graph the surface and the tangent plane.

For the following exercises, use the information provided to solve the problem.

Let $w(x,y,z)=xy\,\cos \,z,$ where $x=t,y={t}^{2},$ and $z=\arcsin \,t.$ Find $\frac{dw}{dt}.$

For the following exercises, use the information provided to solve the problem.

Let $w(t,v)={e}^{tv}$ where $t=r+s$ and $v=rs.$ Find $\frac{\partial w}{\partial r}$ and $\frac{\partial w}{\partial s}.$

For the following exercises, use the information provided to solve the problem.

If $w=5{x}^{2}+2{y}^{2},x=-3s+t,$ and $y=s-4t,$ find $\frac{\partial w}{\partial s}$ and $\frac{\partial w}{\partial t}.$

For the following exercises, use the information provided to solve the problem.

If $w=x{y}^{2},x=5\,\cos (2t),$ and $y=5\,\sin (2t),$ find $\frac{dw}{dt}.$

If $f(x,y)=xy,x=r\,\cos \,\theta ,$ and $y=r\,\sin \,\theta ,$ find $\frac{\partial f}{\partial r}$ and express the answer in terms of $r$ and $\theta .$

For the following exercises, use the information provided to solve the problem.

Suppose $f(x,y)=x+y,$ where $x=r\,\cos \,\theta$ and $y=r\,\sin \,\theta .$ Find $\frac{\partial f}{\partial \theta }.$