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The second midterm exam for math 201, held on october 2008. The exam covers various topics in calculus, including newton's law of gravitation, rates of change, differentiation, and tangent lines. Students are required to find derivatives, prove differentiation rules, and find equations of tangent lines.
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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.