Difference between revisions of "009A Sample Midterm 2, Problem 2"
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(Created page with "<span class="exam">The function <math style="verticalalign: 5px">f(x)=3x^78x+2</math> is a polynomial and therefore continuous everywhere. <span class="exam">(...") 

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−  Since <math style="verticalalign:   +  Since <math style="verticalalign: 5px">y=0</math> is between <math style="verticalalign: 5px">f(0)=2</math> and <math style="verticalalign: 5px">f(1)=3,</math> 
    
the Intermediate Value Theorem tells us that there is at least one number <math style="verticalalign: 1px">x</math>  the Intermediate Value Theorem tells us that there is at least one number <math style="verticalalign: 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. 