Difference between revisions of "009A Sample Midterm 2, Problem 2"

From Math Wiki
Jump to navigation Jump to search
(Created page with "<span class="exam">The function  <math style="vertical-align: -5px">f(x)=3x^7-8x+2</math>  is a polynomial and therefore continuous everywhere. <span class="exam">(...")
 
Line 48: Line 48:
 
!Step 2: &nbsp;
 
!Step 2: &nbsp;
 
|-
 
|-
|Since &nbsp;<math style="vertical-align: -1px">0</math>&nbsp; is between &nbsp;<math style="vertical-align: -5px">f(0)=2</math>&nbsp; and &nbsp;<math style="vertical-align: -5px">f(1)=-3,</math>
+
|Since &nbsp;<math style="vertical-align: -5px">y=0</math>&nbsp; is between &nbsp;<math style="vertical-align: -5px">f(0)=2</math>&nbsp; and &nbsp;<math style="vertical-align: -5px">f(1)=-3,</math>
 
|-
 
|-
 
|the Intermediate Value Theorem tells us that there is at least one number &nbsp;<math style="vertical-align: -1px">x</math>
 
|the Intermediate Value Theorem tells us that there is at least one number &nbsp;<math style="vertical-align: -1px">x</math>

Revision as of 19:33, 13 April 2017

The function    is a polynomial and therefore continuous everywhere.

(a) State the Intermediate Value Theorem.

(b) Use the Intermediate Value Theorem to show that    has a zero in the interval  


Foundations:  
What is a zero of the function  
        A zero is a value    such that  


Solution:

(a)  
Intermediate Value Theorem
        If    is continuous on a closed interval  
        and    is any number between    and  

        then there is at least one number    in the closed interval such that  

(b)

Step 1:  
First,    is continuous on the interval    since    is continuous everywhere.
Also,

       

and

        .

Step 2:  
Since    is between    and  
the Intermediate Value Theorem tells us that there is at least one number  
such that  
This means that    has a zero in the interval  


Final Answer:  
    (a)     See solution above.
    (b)     See solution above.

Return to Sample Exam