Math 201 - Second Midterm Exam, October 2008, Exams of Calculus

Download Math 201 - Second Midterm Exam, October 2008 and more Exams Calculus in PDF only on Docsity!

Math 201

21 October 2008 Second Midterm

NAME (Print!): Check one: (1pm): (2pm):

Problem Points Score

1 20 2 20

3 30

4 20

5 10

Total 100

Problem 1 (20 points): Newton’s Law of Gravitation states that the magnitude F of the force exerted by a body of mass m on a body of mass M is F =

GM m r^2 where G is the gravitational constant and r is the distance between the two bodies. (a) Find dFdr and explain its meaning. What does the minus sign indicate? (b) Suppose that it is known Earth attracts an object with a force that decreases at the rate of 2 N/km when r = 20, 000 km. How fast does this force change when r = 10, 000 km.

Problem 3 (30 points): Find dydx for each of the following: (a) tan(x − y) = (^) 1+yx 2

(b) y = 2^3

x^2

(c) y = xe

x

Problem 4 (20 points): Prove the following differentiation rules: (a) Using the limit definition of the derivative, prove (^) dxd

x = 2 √^1 x.

(b) Show that for any real number n we have (^) dxd xn^ = nxn−^1.

(THIS PAGE INTENTIONALLY BLANK)