Assignment 4

# Assignment 4 - -f t iv Since part iii can be rewritten as f...

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MATH 137, Calculus 1 for Honours Math, Winter 2008 ASSIGNMENT 4 Due date: February 8, 2008, 10:00 a.m. Solve all of the following FROM THE TEXTBOOK Section 2.5: #16, #17, #41, #60 Section 3.1: #49, #70 Section 3.3: #41, #46 Section 3.4: #23, #54 PLUS THE FOLLOWING PROBLEM. P1. The height in metres of a growing tree after t weeks is given by f ( t ) = 6 / (0 . 2 + 5 e - t/ 2 ) m . i. How tall was the tree, and how fast was it growing, at t = 0 weeks? ii. Show that the tree never grows taller than 30 m in height. iii. Show that the function f satisﬁes the equation f 0 ( t ) = 1 60 f ( t )[30
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Unformatted text preview: -f ( t )] iv. Since part iii. can be rewritten as f ( t ) = 1 60 { 15 2-[ f ( t )-15] 2 } , we see that the maximum of f ( t ) occurs at t = 15 m. Find the time t at which this occurs. v. Use the info in i.iv. to sketch the graph of y = f ( t ). BONUS PROBLEM. Prove the following xed point result. If f : [0 , 1]- [0 , 1] is continuous then it must have a xed point. (i. e. c [0 , 1] s.t. f ( c ) = c ) Hint: apply IVT to g : [0 , 1]- [0 , 1] ,g ( x ) = f ( x )-x ....
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