ws08 - be going when it hits you? Where was the chandelier...

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WORKSHEET 8 - Fall 1995 1. Give examples of functions satisfying the following conditions. If no such function exists, explain why. a) Continuous at x = c but not diFerentiable there; b) DiFerentiable at x = c but not continuous there; c) Not continuous at x = 0 but continuous for all x 6 =0; d) Continuous at x = 0 but not continuous at any other x . e) Continuous at each irrational number and not continuous at any rational numbers. 2. Prove that f ( x )= x 3 3 x + c never has two roots in [0 , 1] no matter what c is. 3. a) Prove that Galileo was wrong: If a body falls a distance s ( t )in t seconds, and s 0 ( t ) is proportional to s ,then s cannot be a function of the form s ( t )= ct 2 . b) Prove that the following facts are true about s if s ( t )=( a 2 ) t 2 ( the frst Fact will show why we switched From c to a 2 ): (i) s 0 ( t )= a (the acceleration is constant). (ii) [ s 0 ( t )] 2 =2 as ( t ). c) If s is measured in feet, the value a is 32. How many seconds do you have to get out of the way of a chandelier which falls from a 400-foot ceiling? If you don’t make it, how fast will the chandelier
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Unformatted text preview: be going when it hits you? Where was the chandelier when it was moving with half that speed? 4. Derive the derivatives of tan x , cot x , sec x , and csc x when given only that d dx sin x = cos x d dx cos x = sin x. 5. a) Show that lim h f ( a + h ) f ( a ) h = lim x a f ( x ) f ( a ) x a . b) Use the above fact to calculate: i) lim x 8 x 4 / 3 16 x 8 iv) lim x 1 x 2 + x ( x + 2)( x + 1) ii) lim x 2 (6 + x ) 2 / 3 4 x 2 iii) lim x 1 x 100 1 x 1 v) lim x 1 x 3 2 x 2 + 2 x 1 x 1 6. If f + g is diFerentiable at a , are f and g necessarily diFerentiable at a ? If f g and f are diFerentiable at a , what conditions on f imply that g is diFerentiable at a ?...
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