91Ó°ÊÓ

Estimate the indicated value without using a calculator. $$ \ln 0.993 $$

Estimate the indicated value without using a calculator. $$ \ln 0.9996 $$

Find two choices for \(t\) such that the distance between (3,-2) and \((1, t)\) equals 5 .

Find the area of this trapezoid, whose vertices are (2,1),(6,1),(8,4) and (1,4) .

Find the equation of the circle centered at the origin in the \(u v\) -plane that has twice the circumference of the circle whose equation equals $$ u^{2}+v^{2}=10 $$

Explain why a square yard contains 9 square feet.

Explain why a square foot contains 144 square inches.

The functions cosh and \(\sinh\) are defined by $$ \cosh x=\frac{e^{x}+e^{-x}}{2} \text { and } \sinh x=\frac{e^{x}-e^{-x}}{2} $$ for every real number \(x .\) For reasons that do not concern us here, these functions are called the hyperbolic cosine and hyperbolic sine; they are useful in engineering. Show that cosh is an even function.

The functions cosh and \(\sinh\) are defined by $$ \cosh x=\frac{e^{x}+e^{-x}}{2} \text { and } \sinh x=\frac{e^{x}-e^{-x}}{2} $$ for every real number \(x .\) For reasons that do not concern us here, these functions are called the hyperbolic cosine and hyperbolic sine; they are useful in engineering. Show that $$ (\cosh x)^{2}-(\sinh x)^{2}=1 $$ for every real number \(x\).

In ancient China and Babylonia, the area inside a circle was said to be one- half the radius times the circumference. Show that this formula agrees with our formula for the area inside a circle.