Math_137_Winter_2010_Solution_4

Math_137_Winter_2010_Solution_4 - Math 137 Winter 2010...

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Math 137 Winter 2010 Assignment 4 Due Friday, February 12 All solutions must be clearly stated and fully justified. Use the format given on UW-Ace under Content, in the folder Assignments; it is the file Math 137 Assignment Templates . Text problems: Section 2.5: 40, 52 Section 2.7: 14, 20, 22, 32 Section 2.8: 12, 26 Section 3.1: 22, 30, 36, 60 Section 3.2: 10, 18, 52 Section 3.3: 10, 14, 34 Section 2.5: 40. The gravitational force exerted by the earth on a unit mass at a distance r from the center of the planet is F(r) = GMr/R 3 if f < R, and GM/r 2 if r R, where M is Earth’s mass, R its radius and G the gravitational constant. Is F a continuous function of r? 52. a) Prove that ln x = 3 – 2x has at least one real root, and b) use your calculator to find an interval of length 0.01 that contains a root. a) Let f(x) = ln x – 3 + 2x; f(1) = – 1 < 0 and f(2) = ln 2 + 1 1.7 > 0. Since f(x) is continuous on [1,2] because ln x and any polynomial are, by the IVT there is a real number c (1, 2) such that f(c) = 0. This is a root of ln x = 3 – 2x. b) f(1.34) – 0.33 < 0 and f(1.35) 0.0001 > 0, so (1.34, 1.35) is an interval of length 0.01 that contains a root. Section 2.7: 14. If a rock is thrown upward on the planet Mars with a velocity of 10 m/sec, its height in meters after t seconds is given by H(t) = 10t – 1.86t 2 . a) Find the velocity of the rock after one second. h h h h h h h H v h h 86 . 1 10 86 . 1 72 . 3 86 . 1 10 10 lim 86 . 1 10 ) 1 ( 86 . 1 ) 1 ( 10 lim ) 1 ( ) 1 ( 2 0 2 0 + + = + + + = = 28 . 6 86 . 1 28 . 6 lim ) 86 . 1 28 . 6 ( lim 86 . 1 72 . 3 10 lim 0 0 2 0 = = = = h h h h h h h h h h h m / s b) Find the velocity of the rock when t = a.
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h a a h ah a h a h a a h a h a a v h h 2 2 2 0 2 2 0 86 . 1 10 86 . 1 72 . 3 86 . 1 10 10 lim 86 . 1 10 ) ( 86 . 1 ) ( 10 lim ) ( + + = + + + = a h a h h a h h h ah h h h h 72 . 3 10 86 . 1 72 . 3 10 lim ) 86 . 1 72 . 3 10 ( lim 86 . 1 72 . 3 10 lim 0 0 2 0 = = = = m/s c) When will the rock hit the surface? It hits when H(t) = 10t – 1.86t 2 = 0 t(10 – 1.86t) = 0 t = 0 or 10/1.86 5.4 sec. The rock was launched at t = 0 and it hit at approximately t = 5.4 sec. d) With what velocity will the rock hit the surface?
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This note was uploaded on 02/21/2010 for the course MATH 137 taught by Professor Speziale during the Spring '08 term at Waterloo.

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Math_137_Winter_2010_Solution_4 - Math 137 Winter 2010...

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