{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

Solutions Homework 4 - MATH 106 CALCULUS I FOR BIO SOC SCI...

Info icon This preview shows pages 1–2. Sign up to view the full content.

View Full Document Right Arrow Icon
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 , π 2 ] for which cos( x ) = 2 x π . Solution: Consider the function g ( x ) = cos( x ) 2 x π . This is a continuous function on the interval [0 , π 2 ]. Note that g (0) = cos(0) 0 = 1 > 0. Also g ( π 2 ) = cos ( π 2 ) 1 = 1 < 0. By the intermediate value theorem we can find some x in the interval (0 , π 2 ) for which g ( x ) = cos( x ) 2 x π = 0; that is, 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.) Solution: The function p ( x ) = x 4 x 2 10 x +1 is continuous because it is polynomial. Note that p (0) = 1 > 0 and p (1) = 1 1 10 + 1 = 9 < 0. By the intermediate value theorem we can find some number c (0 , 1) such that p ( c ) = 0.
Image of page 1

Info icon This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
Image of page 2
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

What students are saying

  • Left Quote Icon

    As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

    Student Picture

    Kiran Temple University Fox School of Business ‘17, Course Hero Intern

  • Left Quote Icon

    I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

    Student Picture

    Dana University of Pennsylvania ‘17, Course Hero Intern

  • Left Quote Icon

    The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

    Student Picture

    Jill Tulane University ‘16, Course Hero Intern