Problem 46
Radioactive Decay The half-life of a certain radioactive substance is 12 hours. There are 8 grams present initially. (a) Express the amount of substance remaining as a function of time $t . (b) When will there be 1 gram remaining?
Problem 46
In Exercises 45 and \(46,\) a parametrization is given for a curve.
(a) Graph the curve. What are the initial and terminal points, if any?
Indicate the direction in which the curve is traced.
(b) Find a Cartesian equation for a curve that contains the parametrized
curve. What portion of the graph of the Cartesian equation is traced by the
parametrized curve?
$$x=\tan t, \quad y=-2 \sec t, \quad-\pi / 2
Problem 46
Multiple Choice Which of the following gives the best approximation for the zero of \(f(x)=4-e^{x} ?\) (A) \(x=-1.386 \quad\) (B) \(x=0.386 \quad\) (C) \(x=1.386\) (D) \(x=3 \quad\) (E) there are no zeros
Problem 46
Even-Odd (a) Show that \(\csc x\) is an odd function of \(x\) . (b) Show that the reciprocal of an odd function is odd.
Problem 47
extending the idea The Witch of Agnesi The bell-shaped witch of Agnesi can be constructed as follows. Start with the circle of radius \(1,\) centered at the point \((0,1)\) as shown in the figure Choose a point \(A\) on the line \(y=2,\) and connect it to the origin with a line segment. Call the point where the segment crosses the circle \(B .\) Let \(P\) be the point where the vertical line through \(A\) crosses the horizontal line through \(B\) . The witch is the curve traced by \(P\) as \(A\) moves along the line \(y=2\) .Find a parametrization for the witch by expressing the coordinates of \(P\) in terms of \(t\) , the radian measure of the angle that segment OA makes with the positive \(x\) -axis. The following equalities (which you may assume) will help: (i) \(x=A Q \quad\) (ii) \(y=2-A B \sin t \quad\) (iii) \(A B \cdot A O=(A Q)^{2}\)
Problem 47
Doubling Your Money Determine how much time is required for a \(\$ 500\) investment to double in value if interest is earned at the rate of 4.75\(\%\) compounded annually.
Problem 47
True or False The slope of a vertical line is zero. Justify your answer.
Problem 47
Let \(y_{1}=x^{2}\) and \(y_{2}=2^{x}\) . (a) Graph \(y_{1}\) and \(y_{2}\) in \([-5,5]\) by \([-2,10] .\) How many times do you think the two graphs cross? (b) Compare the corresponding changes in \(y_{1}\) and \(y_{2}\) as \(x\) changes from 1 to \(2,2\) to \(3,\) and so on. How large must \(x\) be for the changes in \(y_{2}\) to overtake the changes in \(y_{1} ?\) (c) Solve for \(x : x^{2}=2^{x}\) . \(\quad\) (d) Solve for \(x : x^{2}<2^{x}\)
Problem 47
Even-Odd Show that the product of an even function and an odd function is an odd function.
Problem 48
Population Growth The population of Glenbrook is \(375,000\) and is increasing at the rate of 2.25\(\%\) per year. Predict when the population will be 1 million. In Exercises 49 and \(50,\) let \(x=0\) represent \(1990, x=1\) represent \(1991,\) and so forth.
