Chapter 4: Problem 6
\(f(x)=\cos ^{-1} x, \quad a=0\)
Over 30 million students worldwide already upgrade their learning with Vaia!
All the tools & learning materials you need for study success - in one app.
Get started for free
In Exercises 62 and \(63,\) feel free to use a CAS (computer algebra system), if you have one, to solve the problem. Logistic Functions Let \(f(x)=c /\left(1+a e^{-h x}\right)\) with \(a>0\) \(a b c \neq 0\) (a) Show that \(f\) is increasing on the interval \((-\infty, \infty)\) if \(a b c>0\) and decreasing if \(a b c<0\) . (b) Show that the point of inflection of \(f\) occurs at \(x=(\ln |a|) / b\)
Vertical Motion Two masses hanging side by side from springs have positions \(s_{1}=2 \sin t\) and \(s_{2}=\sin 2 t\) respectively, with \(s_{1}\) and \(s_{2}\) in meters and \(t\) in seconds. (a) At what times in the interval \(t>0\) do the masses pass each other? [Hint: \(\sin 2 t=2 \sin t \cos t ]\) (b) When in the interval \(0 \leq t \leq 2 \pi\) is the vertical distance between the masses the greatest? What is this distance? (Hint: \(\cos 2 t=2 \cos ^{2} t-1 . )\)
Multiple Choice What is the maximum area of a right triangle with hypotenuse 10? \(\begin{array}{llll}{\text { (A) } 24} & {\text { (B) } 25} & {\text { (C) } 25 \sqrt{2}} & {\text { (D) } 48} & {\text { (E) } 50}\end{array}\)
Moving Ships Two ships are steaming away from a point \(O\) along routes that make a \(120^{\circ}\) angle. Ship \(A\) moves at 14 knots (nautical miles per hour; a nautical mile is 2000 yards). Ship \(B\) moves at 21 knots. How fast are the ships moving apart when \(O A=5\) and \(O B=3\) nautical miles? 29.5 knots
Particle Motion A particle moves along the parabola \(y=x^{2}\) in the first quadrant in such a way that its \(x\) -coordinate (in meters) increases at a constant rate of 10 \(\mathrm{m} / \mathrm{sec} .\) How fast is the angle of inclination \(\theta\) of theline joining the particle to the origin changing when \(x=3 ?\)
What do you think about this solution?
We value your feedback to improve our textbook solutions.
