If a line is a linear approximation to a function in $1$ variable and a hyperplane is the linear approximation to a function in $2$ variables what is the linear approximation to a function in $3$ variables?

Davon Irwin
2022-07-01
Answered

If a line is a linear approximation to a function in $1$ variable and a hyperplane is the linear approximation to a function in $2$ variables what is the linear approximation to a function in $3$ variables?

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humbast2

Answered 2022-07-02
Author has **21** answers

Your terminology is slightly off.

The linear approximation to a function of 1 variable is indeed a line.

The linear approximation to a function of 2 variables forms a plane.

The linear approximation to a function of any number of variables is a hyperplane.

Unlike the terms "line" and "plane", a hyperplane does not traditionally have a fixed dimension. Instead, the dimensionality depends on context: in ${\mathbb{R}}^{2}$, a hyperplane has one dimension; in ${\mathbb{R}}^{3}$, it has two; and, in general, a hyperplane has one fewer dimension than the ambient space (a nice term for this is "codimension $1$").

In general, we don't bother to describe lines as "hyperplanes in $2D$ space," mainly because one cannot distinguish between dimension $1$ and codimension $1$ when the ambient space has $2$ dimensions. At the same time, it's not wrong.

The distinction is relevant in $3D$ and above, and so we do describe planes as "hyperplanes in $3D$ space."

The linear approximation to a function of 1 variable is indeed a line.

The linear approximation to a function of 2 variables forms a plane.

The linear approximation to a function of any number of variables is a hyperplane.

Unlike the terms "line" and "plane", a hyperplane does not traditionally have a fixed dimension. Instead, the dimensionality depends on context: in ${\mathbb{R}}^{2}$, a hyperplane has one dimension; in ${\mathbb{R}}^{3}$, it has two; and, in general, a hyperplane has one fewer dimension than the ambient space (a nice term for this is "codimension $1$").

In general, we don't bother to describe lines as "hyperplanes in $2D$ space," mainly because one cannot distinguish between dimension $1$ and codimension $1$ when the ambient space has $2$ dimensions. At the same time, it's not wrong.

The distinction is relevant in $3D$ and above, and so we do describe planes as "hyperplanes in $3D$ space."

asked 2022-07-12

Linear approximation of quotient

$\frac{(2.01{)}^{2}}{\sqrt{.95}}$

$\frac{(2.01{)}^{2}}{\sqrt{.95}}$

asked 2022-06-06

You're measuring the velocity of an object by measuring that it takes $1$ second to travel $1.2$ meters. The measurement error is $001$ meters in distance and the error in time is $01$ second.

What is the absolute value of the error in the linear approximation for the velocity?

What is the absolute value of the error in the linear approximation for the velocity?

asked 2022-06-08

Use a linear approximation of

$f(x)=\sqrt[3]{x}$

at

$x=8$

to approximate

$\sqrt[3]{7}$

Express your answer as an exact fraction.

$f(x)=\sqrt[3]{x}$

at

$x=8$

to approximate

$\sqrt[3]{7}$

Express your answer as an exact fraction.

asked 2022-07-11

If $f(x)=g(x)h(x)$

does the linear approximation of $f(x)$ equals the linear approximation of $g(x)$ times the linear approximation of $h(x)$?

is it true for quadratic approximations as well?

does the linear approximation of $f(x)$ equals the linear approximation of $g(x)$ times the linear approximation of $h(x)$?

is it true for quadratic approximations as well?

asked 2022-05-09

Use linear approximation, i.e. the tangent line, to approximate ${11.2}^{2}$ as follows :

Let $f(x)={x}^{2}$ and find the equation of the tangent line to $f(x)$ at $x=11$. Using this, find your approximation for ${11.2}^{2}$.

Let $f(x)={x}^{2}$ and find the equation of the tangent line to $f(x)$ at $x=11$. Using this, find your approximation for ${11.2}^{2}$.

asked 2022-05-25

Linear approximation at $x=0$ to $\mathrm{sin}(6x)$

asked 2022-05-10

Determine how accurate should we measure the side of a cube so that the calculated surface area of the cube lies within $3$% of its true value, using Linear Approximation.

Let $A(x)=TSA$; $x=side$

$A(x)=6{x}^{2}$

Let $A(x)=TSA$; $x=side$

$A(x)=6{x}^{2}$