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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 differentiable there; b) Differentiable at x = c but not continuous there; c) Not continuous at x = 0 but continuous for all x = 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 ( 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 first fact will show why we switched from c to a 2 ): (i) s ( t ) = a (the acceleration is constant). (ii) [ s ( 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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