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Homework 4

# Homework 4 - not be accepted for this part(b(5 Points Use...

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INSTRUCTOR: JOS ´ E MANUEL G ´ OMEZ Homework 4. Due date Wednesday October 5, 2011 Please show all your work. (1) (5 Points) Use the intermediate value theorem to show that there exists some x in the interval [0 , π 2 ] for which cos( x ) = 2 x π . (2) (5 Points) Show that the polynomial p ( x ) = x 4 x 2 10 x + 1 has at least one root. (Recall that a root of a polynomial p ( x ) is a number c for which p ( c ) = 0.) (3) (5 Points) Let w ( x ) = x + 1. Compute lim h 0 w ( h + 1) w (1) h . (4) (10 Points) (a) (5 Points) Use the limit deFnition of the derivative to Fnd g (1), if g is the function deFned by g ( x ) = x 2 + 1. Calculations using rules for derivatives will
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Unformatted text preview: not be accepted for this part. (b) (5 Points) Use the information obtained in part (a) to Fnd the equation of the tangent line to the graph of g ( x ) at the point (1 , 2). (5) (10 Points) Consider the function f ( x ) deFned below f ( x ) = b x sin ( 1 x ) if n = 0 , if x = 0 . (a) (5 Points) Show that f is continuous at x = 0. (b) (5 Points) Use the deFnition of the derivative to show that f is not di±erentiable at x = 0; that is, use the limit deFnition of f ′ (0) and show that it does not exist. 1...
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