Difference between revisions of "009A Sample Midterm 2, Problem 2"
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!Foundations: | !Foundations: | ||
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| − | |What is a zero of the function <math>f(x)?</math> | + | |What is a zero of the function <math style="vertical-align: -5px">f(x)?</math> |
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| − | | A zero is a value <math>c</math> such that <math>f(c)=0.</math> | + | | A zero is a value <math style="vertical-align: -1px">c</math> such that <math style="vertical-align: -5px">f(c)=0.</math> |
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| If <math style="vertical-align: -5px">f(x)</math>  is continuous on a closed interval <math style="vertical-align: -5px">[a,b]</math> | | If <math style="vertical-align: -5px">f(x)</math>  is continuous on a closed interval <math style="vertical-align: -5px">[a,b]</math> | ||
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| − | | and <math style="vertical-align: 0px">c</math> is any number between <math style="vertical-align: -5px">f(a)</math>  and <math style="vertical-align: -5px">f(b)</math> | + | | and <math style="vertical-align: 0px">c</math> is any number between <math style="vertical-align: -5px">f(a)</math>  and <math style="vertical-align: -5px">f(b),</math> |
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!Step 1: | !Step 1: | ||
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| − | |First, <math>f(x)</math> is continuous on the interval <math>[0,1]</math> since <math>f(x)</math> is continuous everywhere. | + | |First, <math style="vertical-align: -5px">f(x)</math> is continuous on the interval <math style="vertical-align: -5px">[0,1]</math> since <math style="vertical-align: -5px">f(x)</math> is continuous everywhere. |
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|Also, | |Also, | ||
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!Step 2: | !Step 2: | ||
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| − | |Since <math>0</math> is between <math>f(0)=2</math> and <math>f(1)=-3,</math> | + | |Since <math style="vertical-align: -1px">0</math> is between <math style="vertical-align: -5px">f(0)=2</math> and <math style="vertical-align: -5px">f(1)=-3,</math> |
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| − | |the Intermediate Value Theorem tells us that there is at least one number <math>x</math> | + | |the Intermediate Value Theorem tells us that there is at least one number <math style="vertical-align: -1px">x</math> |
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| − | |such that <math>f(x)=0.</math> | + | |such that <math style="vertical-align: -5px">f(x)=0.</math> |
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| − | |This means that <math>f(x)</math> has a zero in the interval <math>[0,1].</math> | + | |This means that <math style="vertical-align: -5px">f(x)</math> has a zero in the interval <math style="vertical-align: -5px">[0,1].</math> |
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Revision as of 16:15, 18 February 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
![{\displaystyle [0,1].}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8786b5ef9daedb24adb59e7825c4096d99a99648)
| Foundations: |
|---|
| What is a zero of the function |
| A zero is a value such that |
Solution:
(a)
| Step 1: |
|---|
| 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 |
![{\displaystyle [a,b]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9c4b788fc5c637e26ee98b45f89a5c08c85f7935)
(b)
| Step 1: |
|---|
| First, is continuous on the interval since is continuous everywhere. |
| Also, |
|
|
| and
. |
![{\displaystyle [0,1]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/738f7d23bb2d9642bab520020873cccbef49768d)
| 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. |