쌍곡선함수의 미적분

For the following exercises, use this scenario: A cable hanging under its own weight has a slope $S=dy/dx$ that satisfies $dS/dx=c\sqrt{1+{S}^{2}}.$ The constant $c$ is the ratio of cable density to tension.

Show that $S=\sinh (cx)$ satisfies this equation.

For the following exercises, use this scenario: A cable hanging under its own weight has a slope $S=dy/dx$ that satisfies $dS/dx=c\sqrt{1+{S}^{2}}.$ The constant $c$ is the ratio of cable density to tension.

Integrate $dy/dx=\sinh (cx)$ to find the cable height $y(x)$ if $y(0)=1/c.$

Sketch the cable and determine how far down it sags at $x=0.$

[T] A chain hangs from two posts $2$ m apart to form a catenary described by the equation $y=2\,\cosh (x/2)-1.$ Find the slope of the catenary at the left fence post.

[T] A chain hangs from two posts four meters apart to form a catenary described by the equation $y=4\,\cosh (x/4)-3.$ Find the total length of the catenary (arc length).

[T] A high-voltage power line is a catenary described by $y=10\,\cosh (x/10).$ Find the ratio of the area under the catenary to its arc length. What do you notice?

A telephone line is a catenary described by $y=a\,\cosh (x/a).$ Find the ratio of the area under the catenary to its arc length. Does this confirm your answer for the previous question?

Prove the formula for the derivative of $y={\sinh }^{-1}(x)$ by differentiating $x=\sinh (y).$ (Hint: Use hyperbolic trigonometric identities.)

Prove the formula for the derivative of $y={\cosh }^{-1}(x)$ by differentiating $x=\cosh (y).$

(Hint: Use hyperbolic trigonometric identities.)

For the following exercises, solve each problem.

Prove that ${(\cosh (x)+\sinh (x))}^{n}=\cosh (nx)+\sinh (nx).$

Prove the expression for ${\sinh }^{-1}(x).$ Multiply $x=\sinh (y)=(1/2)({e}^{y}-{e}^{-y})$ by $2{e}^{y}$ and solve for $y.$ Does your expression match the textbook?

Prove the expression for ${\cosh }^{-1}(x).$ Multiply $x=\cosh (y)=(1/2)({e}^{y}+{e}^{-y})$ by $2{e}^{y}$ and solve for $y.$ Does your expression match the textbook?

List the different indeterminate forms described in this section.

T/F: l'Hopital's Rule provides a faster method of computing derivatives.

T/F: l'Hopital's Rule states that $\displaystyle \frac{d}{dx}\left[\frac{f(x)}{g(x)}\right] = \frac{f\,'(x)}{g'(x)}.$

Explain what the indeterminate form “$1^\infty$” means.

Fill in the blanks:

The Quotient Rule is applied to $\displaystyle \frac{f(x)}{g(x)}$ when taking _____;

l'Hopital's Rule is applied to $\displaystyle \frac{f(x)}{g(x)}$ when taking certain _____.

Create (but do not evaluate!) a limit that returns “$\infty^0$”.

Create a function $f(x)$ such that $\displaystyle \lim_{x\to 1}f(x)$ returns “$0^0$”.

Create a function $f(x)$ such that $\displaystyle \lim_{x\to \infty}f(x)$ returns “$0\cdot\infty$”.