UFCalcSet10 - position relative to the equilibrium point is...

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Exercises UF Calculus Set 10 1. Sketch a continuous function f such that: f (1) = 4 ; lim x →∞ f ( x ) = 2 ; lim x →-∞ f ( x ) = - 2 . 2. Evaluate the limits at infinity: (a) lim x →∞ x 2 - x + 4 3 x 2 + 2 x - 3 (b) lim x →-∞ x + 1 x 3 + 2 x (c) lim x →∞ x + 2 3 x + 1 (d) lim x →∞ ln ± 2 x x + 1 ² (e) lim t →-∞ t 4 t 2 + 1 (f) lim t →∞ t 2 + 2 4 t + 2 (g) lim x →-∞ x x - x 2 + 4 (h) lim t →∞ p t 2 + t - t 3. Write all horizontal and vertical asymptotes for the functions; list any removable discontinu- ities (holes). (a) f ( x ) = x 2 - x x 2 - 1 (b) f ( x ) = x 4 x 2 - 1 (c) f ( x ) = e x 2 - e x (d) f ( x ) = e - 1 /x (e) f ( x ) = 1 ln | x | (f) f ( x ) = tan - 1 ± x 2 x + 1 ² 4. Evaluate the limits at infinity in each direction for the functions: (a) y = 1 - x 2 (b) y = x 3 - x (c) y = x + 1 x (d) y = x 3 / 2 (2 - 5 x ) (e) y = x ln | x | 5. Evaluate the limits at infinity in each direction using squeeze theorem (a) f ( x ) = sin( x ) x (b) f ( x ) = x + sin( x ) x + 1 (c) f ( x ) = xe sin( x ) x 2 + 1
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6. Some applications are given below. Find the limit toward infinity in each case and interpret its importance. (a) When a mass on a spring is pulled back and released in the presence of friction, its
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Unformatted text preview: position relative to the equilibrium point is a function of time t given by Ae-βt sin( αt + c ) for some positive constants A,α,β,c . (b) When an object falls in the presence of air resistence, its velocity is a function of time t given by mg b + Ce-bt/m , where m is the object’s mass, g is the gravitational acceleration near earth, and b is a positive constant. (c) When an object is given an initial velocity v upward from the earth, Newton’s gravita-tional law will show that the maximum height h that it reaches before returning to earth is related to v by r 2 GM ( h-R ) Rh , where G is the gravitational constant, M is the earth’s mass, and R is the earth’s radius. ——————————————————————-...
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This note was uploaded on 05/17/2011 for the course MAC 2311 taught by Professor All during the Spring '08 term at University of Florida.

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UFCalcSet10 - position relative to the equilibrium point is...

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