Zion Wheeler

2022-06-24

What is the arc length of $f(x)=\frac{3x}{\sqrt{x-1}}$ on $x\in [2,6]$?

enfujahl

Beginner2022-06-25Added 20 answers

Writing

$f(x)=3x(x-1{)}^{-\frac{1}{2}}$

then by the product and the chain rule we get

${f}^{\prime}(x)=3(x-1{)}^{-\frac{1}{2}}+3x\ast (-\frac{1}{2})\ast (x-1{)}^{-\frac{3}{2}}$

which simplifies to

${f}^{\prime}(x)=\frac{3(x-2)}{2\ast (x-1{)}^{\frac{3}{2}}}$

so we have to solve

${\int}_{2}^{6}\sqrt{1+(3\frac{x-2}{2\ast (x-1{)}^{\frac{3}{2}}}{)}^{2}}dx$

by a numerical method we get

$\approx 4.51$

$f(x)=3x(x-1{)}^{-\frac{1}{2}}$

then by the product and the chain rule we get

${f}^{\prime}(x)=3(x-1{)}^{-\frac{1}{2}}+3x\ast (-\frac{1}{2})\ast (x-1{)}^{-\frac{3}{2}}$

which simplifies to

${f}^{\prime}(x)=\frac{3(x-2)}{2\ast (x-1{)}^{\frac{3}{2}}}$

so we have to solve

${\int}_{2}^{6}\sqrt{1+(3\frac{x-2}{2\ast (x-1{)}^{\frac{3}{2}}}{)}^{2}}dx$

by a numerical method we get

$\approx 4.51$

What is the derivative of the work function?

How to use implicit differentiation to find $\frac{dy}{dx}$ given $3{x}^{2}+3{y}^{2}=2$?

How to differentiate $y=\mathrm{log}{x}^{2}$?

The solution of a differential equation y′′+3y′+2y=0 is of the form

A) ${c}_{1}{e}^{x}+{c}_{2}{e}^{2x}$

B) ${c}_{1}{e}^{-x}+{c}_{2}{e}^{3x}$

C) ${c}_{1}{e}^{-x}+{c}_{2}{e}^{-2x}$

D) ${c}_{1}{e}^{-2x}+{c}_{2}{2}^{-x}$How to find instantaneous velocity from a position vs. time graph?

How to implicitly differentiate $\sqrt{xy}=x-2y$?

What is 2xy differentiated implicitly?

How to find the sum of the infinite geometric series given $1+\frac{2}{3}+\frac{4}{9}+...$?

Look at this series: 1.5, 2.3, 3.1, 3.9, ... What number should come next?

A. 4.2

B. 4.4

C. 4.7

D. 5.1What is the derivative of $\frac{x+1}{y}$?

How to find the sum of the infinite geometric series 0.9 + 0.09 + 0.009 +…?

How to find the volume of a cone using an integral?

What is the surface area of the solid created by revolving $f\left(x\right)={e}^{2-x},x\in [1,2]$ around the x axis?

How to differentiate ${x}^{\frac{2}{3}}+{y}^{\frac{2}{3}}=4$?

The differential coefficient of $\mathrm{sec}\left({\mathrm{tan}}^{-1}\left(x\right)\right)$.