접평면과 선형근사

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.

T/F: If $f(x,y)$ is differentiable on $S$, the $f$ is continuous on $S$.

T/F: If $f_x$ and $f_y$ are continuous on $S$, then $f$ is differentiable on $S$.

T/F: If $z=f(x,y)$ is differentiable, then the change in $z$ over small changes $dx$ and $dy$ in $x$ and $y$ is approximately $dz$.

Finish the sentence: “The new $z$-value is approximately the old $z$-value plus the approximate _____.”

In Exercises 5–8, find the total differential $dz$.

$z = x\sin y + x^2$

In Exercises 5–8, find the total differential $dz$.

$z = (2x^2+3y)^2$

In Exercises 5–8, find the total differential $dz$.

$z = 5x-7y$

In Exercises 5–8, find the total differential $dz$.

$z = xe^{x+y}$