⧼exchistory⧽
3 exercise(s) shown, 0 hidden
BBot
Nov 03'24
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[/math]
In each of the following examples identify the function [math]f[/math] as a polynomial or not. If it is not a polynomial, give a reason. (Consider such things as the vanishing of higher-order derivatives, or the behavior of [math]f[/math], or of some derivative [math]f^{(j)}[/math], near a point of discontinuity.)
- [a [math]f(x) = \frac1x[/math]
- [math]f(x) = \frac{x-1}{x+1}[/math]
- [math]f(x) = \pi x^2 + ex + 2[/math]
- [math]f(x) = x^{\frac23} + x^{\frac13}[/math]
- [math]f(x) = \sqrt{x^2-2}[/math]
- [math]f(x) = x(x^2-7)[/math]
- [math]f(x) = e^x[/math]
- [math]f(x) = \tan x[/math].
BBot
Nov 03'24
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[/math]
Prove that the algebraic function [math]g[/math] defined by [math]g(x) = \sqrt{x^3+2}[/math] is not rational. [Hint: Suppose it is rational. Then there exist polynomials [math]p[/math] and [math]q[/math] such that [math]\sqrt{x^3+2} = \frac{p(x)}{q(x)}[/math], for every [math]x \geq -\sqrt[3]2[/math]. But then
[[math]]
x^3+2 = \left[ \frac{p(x)}{q(x)} \right]^2
,
[[/math]]
or, equivalently,
[[math]]
(x^3+2)[q(x)]^2 - [p(x)]^2 = 0,
\mbox{for all $x \geq -\sqrt[3]2$.}
[[/math]]
The left side of this equation is a polynomial which is not identically zero. (Why?) How many roots can such a polynomial have?]
BBot
Nov 03'24
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[/math]
Prove that each of the following functions [math]f[/math] is algebraic by exhibiting a polynomial [math]F(x,y)[/math] and showing that [math]F(x,f(x)) = 0[/math].
- [math]f(x) = \sqrt{\frac{x+1}{x-1}}[/math]
- [math]f(x) = \ddx \arctan x[/math]
- [math]f(x) = \ddx \arcsin x[/math]
- [math]f(x) = \ddx \arcsec x[/math]
- [math]f(x) = \ln 5^x[/math]
- [math]f(x) = 2x + \sqrt{4x^2-1}[/math].