Chapter 2 | Limits
193
167. is continuous everywhere and f ( a ), f ( b ) >0, then there is no root of f ( x ) in the interval ⎡ ⎣ a , b ⎤ ⎦ . If f ( x ) [T] The following problems consider the scalar form of Coulomb’s law, which describes the electrostatic force between two point charges, such as electrons. It is given by the equation F ( r ) = k e | q 1 q 2 | r 2 , where k e is Coulomb’s constant, q i are the magnitudes of the charges of the two particles, and r is the distance between the two particles. 168. To simplify the calculation of a model with many interacting particles, after some threshold value r = R , we approximate F as zero. a. Explain the physical reasoning behind this assumption. b. What is the force equation? c. Evaluate the force F using both Coulomb’s law and our approximation, assuming two protons with a charge magnitude of 1.6022×10 −19 coulombs (C), and the Coulomb constant k e =8.988×10 9 Nm 2 /C 2 are 1 m apart. Also, assume R <1m. How much inaccuracy does our approximation generate? Is our approximation reasonable? d. Is there any finite value of R for which this system remains continuous at R ? 169. Instead of making the force 0 at R , instead we let the force be 10 −20 for r ≥ R . Assume two protons, which have a magnitude of charge 1.6022×10 −19 C, and the Coulomb constant k e =8.988×10 9 Nm 2 /C 2 . Is there a value R that can make this system continuous? If so, find it. Recall the discussion on spacecraft from the chapter opener. The following problems consider a rocket launch from Earth’s surface. The force of gravity on the rocket is given by F ( d ) = − mk / d 2 , where m is the mass of the rocket, d is the distance of the rocket from the center of Earth, and k is a constant. 170. [T] Determine the value and units of k given that the mass of the rocket is 3 million kg. ( Hint : The distance from the center of Earth to its surface is 6378 km.)
171. [T] After a certain distance D has passed, the gravitational effect of Earth becomes quite negligible, so we can approximate the force function by F ( d ) = if d < D 10,000 if d ≥ D . Using the value of k found in the previous exercise, find the necessary condition D such that the force function remains continuous. ⎧ ⎩ ⎨ − mk d 2 172. As the rocket travels away from Earth’s surface, there is a distance D where the rocket sheds some of its mass, since it no longer needs the excess fuel storage. We can
write this function as F ( d ) = ⎧
m 1 k d 2 m 2 k d 2
⎩ ⎨ ⎪ ⎪
−
if d < D
. Is there
−
if d ≥ D
a D value such that this function is continuous, assuming m 1 ≠ m 2 ? Prove the following functions are continuous everywhere 173. f ( θ ) = sin θ 174. g ( x ) = | x |
175. Where is f ( x ) = ⎧ ⎩
⎨ 0 if x is irrational 1 if x is rational
continuous?
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