This preview shows pages 1–2. Sign up to view the full content.
This preview has intentionally blurred sections. Sign up to view the full version.View Full Document
Unformatted text preview: MATH 106 CALCULUS I FOR BIO. & SOC. SCI. FALL 2011 INSTRUCTOR: JOS E MANUEL G OMEZ Solutions Homework 4. Please show all your work. (1) (5 Points) Use the intermediate value theorem to show that there exists some x in the interval [0 , 2 ] for which cos( x ) = 2 x . Solution: Consider the function g ( x ) = cos( x ) 2 x . This is a continuous function on the interval [0 , 2 ]. Note that g (0) = cos(0) 0 = 1 > 0. Also g ( 2 ) = cos ( 2 ) 1 = 1 < 0. By the intermediate value theorem we can find some x in the interval (0 , 2 ) for which g ( x ) = cos( x ) 2 x = 0; that is, cos( x ) = 2 x . (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.) Solution: The function p ( x ) = x 4 x 2 10 x +1 is continuous because it is polynomial....
View Full Document