다변수 미분 전체 문제

Let $z=f(x,y)=x{e}^{y}.$ Compute $\Delta z$ from $P(1,2)$ to $Q(1.05,2.1)$ and then find the approximate change in $z$ from point $P$ to point $Q.$ Recall $\Delta z=f(x+\Delta x,y+\Delta y)-f(x,y),$ and $dz$ and $\Delta z$ are approximately equal.

Use the differential $dz$ to approximate the change in $z=\sqrt{4-{x}^{2}-{y}^{2}}$ as $(x,y)$ moves from point $(1,1)$ to point $(1.01,0.97).$ Compare this approximation with the actual change in the function.

Let $z=f(x,y)={x}^{2}+3xy-{y}^{2}.$ Find the exact change in the function and the approximate change in the function as $x$ changes from $2.00\,\text{to}\,2.05$ and $y$ changes from $3.00\,\text{to}\,2.96.$

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 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.)

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.

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

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 $\frac{\partial f}{\partial \theta }.$

For the following exercises, find $\frac{df}{dt}$ using the chain rule and direct substitution.

$f(x,y)={x}^{2}+{y}^{2},$ $x=t,y={t}^{2}$

For the following exercises, find $\frac{df}{dt}$ using the chain rule and direct substitution.

$f(x,y)=\sqrt{{x}^{2}+{y}^{2}},y={t}^{2},x=t$

For the following exercises, find $\frac{df}{dt}$ using the chain rule and direct substitution.

$f(x,y)=xy,x=1-\sqrt{t},y=1+\sqrt{t}$