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Unformatted text preview: 180 CHAPTER 2 . . Differentiation 2-36 set f (0) = 0. You may want to use your CAS to solve the equations.] Graph the resulting function; does it look right? Suppose that airline regulations prohibit a derivative of 2 10 or larger. Why might such a regulation exist? Show that the flight path you found is illegal. Argue that in fact all flight paths meeting the four requirements are illegal. Therefore, the de- scent needs to start farther away than 10 miles. Find a flight path with descent starting at 20 miles away that meets all re- quirements. 2. We discuss a graphical interpretation of the second derivative in Chapter 3. You can discover the most important aspects of that here. For f ( x ) = x 4 2 x 2 1, solve the equations f ( x ) = and f ( x ) = 0. What do the solutions of the equation f ( x ) = represent graphically? The solutions of the equation f ( x ) = are a little harder to interpret. Looking at the graph of f ( x ) near x = 0, would you say that the graph is curving up or curving down? Compute f (0). Looking at the graph near x = 2 and x = 2, is the graph curving up or down? Compute f (2) and f ( 2). Where does the graph change from curving up to curving down and vice versa? Hypothesize a relationship between f ( x ) and the curving of the graph of y = f ( x ). Test your hypothesis on a variety of functions. (Try y = x 4 4 x 3 .) 3. In the enjoyable book Surely Youre Joking Mr. Feynman, physicist Richard Feynman tells the story of a contest he had pitting his brain against the technology of the day (an abacus). The contest was to compute the cube root of 1729.03. Feynman came up with 12.002 before the abacus expert gave up. Feynman admits to some luck in the choice of the number 1729.03: he knew that a cubic foot contains 1728 cubic inches. Explain why this told Feynman that the answer is slightly greater than 12. How did he get three digits of accuracy? I had learned in calculus that for small fractions, the cube roots excess is one-third of the numbers excess. The excess, 1.03, is only one part in nearly 2000. So all I had to do is find the frac- tion 1/1728, divide by 3 and multiply by 12. To see what he did, find an equation of the tangent line to y = x 1 / 3 at x = 1728 and find the y-coordinate of the tangent line at x = 1729 . 03. 4. Suppose that you want to find solutions of the equation x 3 4 x 2 + 2 = 0. Show graphically that there is a solution between x = 0 and x = 1. We will approximate this solu- tion in stages. First, find an equation of the tangent line to y = x 3 4 x 2 + 2 at x = 1. Then, determine where this tangent line crosses the x-axis. Show graphically that the x-intercept is considerably closer to the solution than is x = 1....
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- Spring '10