Chapter 4: Problem 43
The volume of a cube with side 15\(\pm 0.2 \mathrm{cm}\)
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Tolerance The height and radius of a right circular cylinder are equal, so the cylinder's volume is \(V=\pi h^{3} .\) The volume is to be calculated with an error of no more than 1\(\%\) of the true value. Find approximately the greatest error that can be tolerated in the measurement of \(h,\) expressed as a percentage of \(h .\)
Free Fall On the moon, the acceleration due to gravity is 1.6 \(\mathrm{m} / \mathrm{sec}^{2} .\) (a) If a rock is dropped into a crevasse, how fast will it be going just before it hits bottom 30 sec later? (b) How far below the point of release is the bottom of the crevasse? (c) If instead of being released from rest, the rock is thrown into the crevasse from the same point with a downward velocity of \(4 \mathrm{m} / \mathrm{sec},\) when will it hit the bottom and how fast will it be going when it does?
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 \(x=(\ln |a|) / b\)
Multiple Choice Which of the following conditions would enable you to conclude that the graph of \(f\) has a point of inflection at \(x=c ?\) (A) There is a local maximum of \(f^{\prime}\) at \(x=c\) . (B) \(f^{\prime \prime}(c)=0 .\) (C) \(f^{\prime \prime}(c)\) does not exist. (D) The sign of \(f^{\prime}\) changes at \(x=c\) . (E) \(f\) is a cubic polynomial and \(c=0\)
Newton's Method Suppose your first guess in using Newton's method is lucky in the sense that \(x_{1}\) is a root of \(f(x)=0 .\) What happens to \(x_{2}\) and later approximations?
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