2.9 Calculus of the hyperbolic functions  (Page 4/5)

[T] A chain hangs from two posts $2$ m apart to form a catenary described by the equation $y=2\text{cosh}\left(x\text{/}2\right)-1.$ 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 $y=4\text{cosh}\left(x\text{/}4\right)-3.$ Find the total length of the catenary (arc length).

[T] A high-voltage power line is a catenary described by $y=10\text{cosh}\left(x\text{/}10\right).$ 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 $y=a\text{cosh}\left(x\text{/}a\right).$ 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 $y={\text{sinh}}^{\mathrm{-1}}(x)$ by differentiating $x=\text{sinh}(y).$ ( Hint: Use hyperbolic trigonometric identities.)

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

( Hint: Use hyperbolic trigonometric identities.)

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

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

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

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

The amount of work to pump the water out of a half-full cylinder is half the amount of work to pump the water out of the full cylinder.

If the force is constant, the amount of work to move an object from $x=a$ to $x=b$ is $F\left(b-a\right).$

The disk method can be used in any situation in which the washer method is successful at finding the volume of a solid of revolution.

If the half-life of $\text{seaborgium-}266$ is $360$ ms, then $k=\left(\text{ln}(2)\right)\text{/}360.$

The volume that has a base of the ellipse $x^2\text{/}4+y^2\text{/}9=1$ and cross-sections of an equilateral triangle perpendicular to the $y\text{-axis}\text{.}$ Use the method of slicing.

$y=x^2-x,$ from $x=1\text{to}x=4,$ rotated around the y -axis using the washer method

$x=y^2$ and $x=3y$ rotated around the y -axis using the washer method

$x=2y^2-y^3,x=0,\text{and}y=0$ rotated around the x -axis using cylindrical shells

$y=x^3,x=0,y=0,\text{and}x=2$

$y=x^2-x\text{and}x=0$

[T] $y=\text{ln}\left(x\right)+2\text{and}y=x$

$y=x^2$ and $y=\sqrt{x}$

$y=5+x,$ $y=x^2,$ $x=0,$ and $x=1$

Below $x^2+y^2=1$ and above $y=1-x$

Find the mass of $\rho =e^{\text{−}x}$ on a disk centered at the origin with radius $4.$

Find the center of mass for $\rho ={\text{tan}}^{2}x$ on $x\in \left(-\frac{\pi }{4},\frac{\pi }{4}\right).$

Find the mass and the center of mass of $\rho =1$ on the region bounded by $y=x^5$ and $y=\sqrt{x}.$

The length of $x$ for $y=\text{cosh}\left(x\right)$ from $x=0\text{to}x=2.$

The length of $y$ for $x=3-\sqrt{y}$ from $y=0$ to $y=4$

The shape created by revolving the region between $y=4+x,$ $y=3-x,$ $x=0,$ and $x=2$ rotated around the y -axis.

The loudspeaker created by revolving $y=1\text{/}x$ from $x=1$ to $x=4$ around the x -axis.

Find the total force on the wall of the dam.

You are a crime scene investigator attempting to determine the time of death of a victim. It is noon and $45\text{°}\text{F}$ outside and the temperature of the body is $78\text{°}\text{F}.$ You know the cooling constant is $k=0.00824\text{°}\text{F/min}\text{.}$ When did the victim die, assuming that a human’s temperature is $98\text{°}\text{F}$ ?

[T] The best-fit exponential curve to these data is given by $y=40.71+{1.224}^x.$ Why do you think the gains of the market were unsustainable? Use first and second derivatives to help justify your answer. What would this model predict the Dow Jones industrial average to be in $2014$ ?

Find the volume of the catenoid $y=\text{cosh}\left(x\right)$ from $x=\mathrm{-1}\text{to}x=1$ that is created by rotating this curve around the $x\text{-axis},$ as shown here.

Find surface area of the catenoid $y=\text{cosh}\left(x\right)$ from $x=\mathrm{-1}$ to $x=1$ that is created by rotating this curve around the $x\text{-axis.}$