Chapter 4: Problem 30
\(d\left(8^{x}+x^{8}\right)\)
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Oscillation Show that if \(h>0,\) applying Newton's method to $$f(x)=\left\\{\begin{array}{ll}{\sqrt{x},} & {x \geq 0} \\ {\sqrt{-x},} & {x<0}\end{array}\right.$$ leads to \(x_{2}=-h\) if \(x_{1}=h,\) and to \(x_{2}=h\) if \(x_{1}=-h\) Draw a picture that shows what is going on.
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 \(x=3 ?\)
\(f\) is an even function, continuous on \([-3,3],\) and satisfies the following. (d) What can you conclude about \(f(3)\) and \(f(-3) ?\)
Multiples of \(P i\) Store any number as \(X\) in your calculator. Then enter the command \(X-\tan (X) \rightarrow X\) and press the ENTER key repeatedly until the displayed value stops changing. The result is always an integral multiple of \(\pi .\) Why is this so? [Hint: These are zeros of the sine function.]
Tin Pest When metallic tin is kept below \(13.2^{\circ} \mathrm{C}\) it slowly becomes brittle and crumbles to a gray powder. Tin objects eventually crumble to this gray powder spontaneously if kept in a cold climate for years. The Europeans who saw tin organ pipes in their churches crumble away years ago called the change tin pest because it seemed to be contagious. And indeed it was, for the gray powder is a catalyst for its own formation. A catalyst for a chemical reaction is a substance that controls the rate of reaction without undergoing any permanent change in itself. An autocatalytic reaction is one whose product is a catalyst for its own formation. Such a reaction may proceed slowly at first if the amount of catalyst present is small and slowly again at the end, when most of the original substance is used up. But in between, when both the substance and its catalyst product are abundant, the reaction proceeds at a faster pace. In some cases it is reasonable to assume that the rate \(v=d x / d t\) of the reaction is proportional both to the amount of the original substance present and to the amount of product. That is, \(v\) may be considered to be a function of \(x\) alone, and $$v=k x(a-x)=k a x-k x^{2}$$ where \(\begin{aligned} x &=\text { the amount of product, } \\ a &=\text { the amount of substance at the beginning, } \\ k &=\text { a positive constant. } \end{aligned}\) At what value of \(x\) does the rate \(v\) have a maximum? What is the maximum value of \(v ?\)
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