2.6 Linear Approximations and Error

Let $x_0$ and $y_0$ be constants and let $m$ and $n$ be integers. If $m<0$ assume that $x_0\ne 0$, and if $n<0$ assume that $y_0\ne 0$. Define $P(x,y) = x^m y^n$.

Consider the following work.

We compute, approximately, the $y$-coordinate of the point whose polar coordinates are $r=0.9$ and $\theta=2^\circ$. In general, the $y$-coordinate of the point whose polar coordinates are $r$ and $\theta$ is $Y(r,\theta) = r\sin\theta$. The partial derivatives

So the linear approximation to $Y(r_0+\De r,\theta_0+\De\theta)$ with $r_0=1$ and $\theta_0=0$ is

Applying this with $\De r=-0.1$ and $\De\theta=2$ gives the (approximate) $y$-coordinate

This conclusion is ridiculous. We're saying that the $y$-coordinate is more than twice the distance from the point to the origin. What was the mistake?

Find an approximate value for $f(x,y)=\sin(\pi xy+\ln y)$ at $(0.01,1.05)$ without using a calculator or computer.

Let $\displaystyle f(x,y)=\frac{x^2y}{x^4+2y^2}$. Find an approximate value for $f (-0.9\,,\, 1.1)$ without using a calculator or computer.

Four numbers, each at least zero and each at most 50, are rounded to the first decimal place and then multiplied together. Estimate the maximum possible error in the computed product.

One side of a right triangle is measured to be $3$ with a maximum possible error of $\pm 0.1$, and the other side is measured to be $4$ with a maximum possible error of $\pm 0.2$. Use the linear approximation to estimate the maximum possible error in calculating the length of the hypotenuse of the right triangle.

If two resistors of resistance $R_1$ and $R_2$ are wired in parallel, then the resulting resistance R satisfies the equation $\frac{1}{R} =\frac{1}{R_1}+\frac{1}{R_2}$. Use the linear approximation to estimate the change in $R$ if $R_1$ decreases from $2$ to $1.9$ ohms and $R_2$ increases from 8 to 8.1 ohms.

The total resistance $R$ of three resistors, $R_1$, $R_2$, $R_3$, connected in parallel is determined by

If the resistances, measured in Ohms, are $R_1=25\Om$, $R_2=40\Om$ and $R_3=50\Om$, with a possible error of 0.5% in each case, estimate the maximum error in the calculated value of $R$.

The specific gravity $S$ of an object is given by $S=\frac{A}{A-W}$ where $A$ is the weight of the object in air and $W$ is the weight of the object in water. If $A=20\pm .01$ and $W=12\pm.02$ find the approximate percentage error in calculating $S$ from the given measurements.

The pressure in a solid is given by

where $s$ is the specific heat and $r$ is the density. We expect to measure $(s,r)$ to be approximately $(2,2)$ and would like to have the most accurate value for $P$. There are two different ways to measure $s$ and $r$. Method $1$ has an error in $s$ of $\pm 0.01$ and an error in $r$ of $\pm 0.1$, while method 2 has an error of $\pm 0.02$ for both $s$ and $r$.

Should we use method 1 or method 2? Explain your reasoning carefully.

A rectangular beam that is supported at its two ends and is subjected to a uniform load sags by an amount

where $p={\rm load}$, $\ell={\rm length}$, $h={\rm height}$, $w={\rm width}$ and $C$ is a constant. Suppose $p\approx 100$, $\ell\approx 4$, $w\approx .1$ and $h\approx.2$. Will the sag of the beam be more sensitive to changes in the height of the beam or to changes in the width of the beam.

Let $z=f(x,y)=\frac{2y}{x^2+y^2}$. Find an approximate value for $f(-0.8,2.1)$.

Suppose that a function $z = f (x, y)$ is implicitly defined by an equation:

Let $z = f(x,y)$ be given implicitly by

Two sides and the enclosed angle of a triangle are measured to be $3\pm.1$m, $4\pm.1$m and $90\pm 1^\circ$ respectively. The length of the third side is then computed using the cosine law $C^2=A^2+B^2-2AB\cos\theta$. What is the approximate maximum error in the computed value of $C$?

Use differentials to find a reasonable approximation to the value of $f(x,y)=xy\sqrt{x^2+y^2}$ at $x=3.02$, $y=3.96$. Note that $3.02\approx 3$ and $3.96\approx 4$.

Use differentials to estimate the volume of metal in a closed metal can with diameter 8cm and height 12cm if the metal is 0.04cm thick.

Let $z$ be a function of $x$, $y$ such that

Consider the surface given by:

Consider the function $f$ that maps each point $(x, y)$ in $\bbbr^2$ to $ye^{-x}$.

A surface is defined implicitly by $z^4 - xy^2 z^2 + y = 0$.

A surface $z = f (x, y)$ has derivatives $\pdiff{f}{x}=3$ and $\pdiff{f}{y}=-2$ at $(x, y, z) = (1, 3, 1)$.

According to van der Waal's equation, a gas satisfies the equation

where $p$, $V$ and $T$ denote pressure, volume and temperature respectively. Suppose the gas is now at pressure $1$, volume $2$ and temperature $5$. Find the approximate change in its volume if $p$ is increased by $0.2$ and $T$ is increased by $0.3$.

Consider the function $f(x, y) = e^{-x^2 +4y^2}$.

Let $z = f (x, y) = \ln(4x^2 + y^2)$.

The surface $x^4+y^4+z^4+xyz=17$ passes through $(0,1,2)$, and near this point the surface determines $x$ as a function, $x=F(y,z)$, of $y$ and $z$.