Chapter 4: Problem 43
Writing to Learn Is the function \(f(x)=x^{2}-x+1\) ever negative? Explain
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True or False A continuous function on a closed interval must attain a maximum value on that interval. Justify your answer.
Frictionless Cart A small frictionless cart, attached to the wall by a spring, is pulled 10 cm from its rest position and released at time \(t=0\) to roll back and forth for 4 sec. Its position at time \(t\) is \(s=10 \cos \pi t .\) (a) What is the cart's maximum speed? When is the cart moving that fast? Where is it then? What is the magnitude of the acceleration then? (b) Where is the cart when the magnitude of the acceleration is greatest? What is the cart's speed then?
True or False If \(f^{\prime}(c)=0\) and \(f(c)\) is not a local maximum, then \(f(c)\) is a local minimum. Justify your answer.
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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