Chapter 1: Problem 10
Why do the values of \(\cos ^{-1} x\) lie in the interval \([0, \pi] ?\)
/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none}
Learning Materials
Features
Discover
Chapter 1: Problem 10
Why do the values of \(\cos ^{-1} x\) lie in the interval \([0, \pi] ?\)
All the tools & learning materials you need for study success - in one app.
Get started for free
Use shifts and scalings to graph then given functions. Then check your work with a graphing utility. Be sure to identify an original function on which the shifts and scalings are performed. $$h(x)=-4 x^{2}-4 x+12$$
(Torricelli's law) A cylindrical tank with a cross-sectional area of \(100 \mathrm{cm}^{2}\) is filled to a depth of \(100 \mathrm{cm}\) with water. At \(t=0,\) a drain in the bottom of the tank with an area of \(10 \mathrm{cm}^{2}\) is opened, allowing water to flow out of the tank. The depth of water in the tank at time \(t \geq 0\) is \(d(t)=(10-2.2 t)^{2}\) a. Check that \(d(0)=100,\) as specified. b. At what time is the tank empty? c. What is an appropriate domain for \(d ?\)
The shortest day of the year occurs on the winter solstice (near December 21) and the longest day of the year occurs on the summer solstice (near June 21 ). However, the latest sunrise and the earliest sunset do not occur on the winter solstice, and the earliest sunrise and the latest sunset do not occur on the summer solstice. At latitude \(40^{\circ}\) north, the latest sunrise occurs on January 4 at 7: 25 a.m. ( 14 days after the solstice), and the earliest sunset occurs on December 7 at 4: 37 p.m. ( 14 days before the solstice). Similarly, the earliest sunrise occurs on July 2 at 4: 30 a.m. (14 days after the solstice) and the latest sunset occurs on June 7 at 7: 32 p.m. ( 14 days before the solstice). Using sine functions, devise a function \(s(t)\) that gives the time of sunrise \(t\) days after January 1 and a function \(S(t)\) that gives the time of sunset \(t\) days after January \(1 .\) Assume that \(s\) and \(S\) are measured in minutes and \(s=0\) and \(S=0\) correspond to 4: 00 a.m. Graph the functions. Then graph the length of the day function \(D(t)=S(t)-s(t)\) and show that the longest and shortest days occur on the solstices.
Find a simple function that fits the data in the tables. $$\begin{array}{|r|r|}\hline x & y \\\\\hline-1 & 0 \\\\\hline 0 & 1 \\\\\hline 1 & 2 \\\\\hline 2 & 3 \\\\\hline 3 & 4 \\\\\hline\end{array}$$
Use the following steps to prove that \(\log _{b}(x y)=\log _{b} x+\log _{b} y\). a. Let \(x=b^{p}\) and \(y=b^{q}\). Solve these expressions for \(p\) and \(q\) respectively. b. Use property El for exponents to express \(x y\) in terms of \(b, p\) and \(q\). c. Compute \(\log _{b}(x y)\) and simplify.
What do you think about this solution?
We value your feedback to improve our textbook solutions.