Difference between revisions of "009A Sample Final 2, Problem 7"
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{| class="mw-collapsible mw-collapsed" style = "text-align:left;" | {| class="mw-collapsible mw-collapsed" style = "text-align:left;" | ||
!Foundations: | !Foundations: | ||
| + | |- | ||
| + | |'''1.''' '''Intermediate Value Theorem''' | ||
| + | |- | ||
| + | | If <math style="vertical-align: -5px">f(x)</math> is continuous on a closed interval <math style="vertical-align: -5px">[a,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> then there is at least one number <math style="vertical-align: 0px">x</math> in the closed interval such that <math style="vertical-align: -5px">f(x)=c.</math> | ||
| + | |- | ||
| + | |'''2.''' '''Mean Value Theorem''' | ||
| + | |- | ||
| + | | Suppose <math style="vertical-align: -5px">f(x)</math> is a function that satisfies the following: | ||
|- | |- | ||
| | | | ||
| + | <math style="vertical-align: -5px">f(x)</math> is continuous on the closed interval <math style="vertical-align: -5px">[a,b].</math> | ||
|- | |- | ||
| | | | ||
| + | <math style="vertical-align: -5px">f(x)</math> is differentiable on the open interval <math style="vertical-align: -5px">(a,b).</math> | ||
|- | |- | ||
| | | | ||
| + | Then, there is a number <math style="vertical-align: 0px">c</math> such that <math style="vertical-align: 0px">a<c<b</math> and <math style="vertical-align: -14px">f'(c)=\frac{f(b)-f(a)}{b-a}.</math> | ||
|} | |} | ||
| Line 19: | Line 31: | ||
!Step 1: | !Step 1: | ||
|- | |- | ||
| − | | | + | |First, we note that |
| + | |- | ||
| + | | <math>f(0)=-2.</math> | ||
| + | |- | ||
| + | |Also, | ||
| + | |- | ||
| + | | <math>f(1)=1.</math> | ||
| + | |- | ||
| + | |Since <math style="vertical-align: -5px">f(0)<0</math> and <math style="vertical-align: -5px">f(1)>0,</math> | ||
| + | |- | ||
| + | |there exists <math style="vertical-align: 0px">x</math> with <math style="vertical-align: -1px">0<x<1</math> such that | ||
|- | |- | ||
| − | | | + | | <math style="vertical-align: -5px">f(x)=0</math> |
|- | |- | ||
| − | | | + | |by the Intermediate Value Theorem. |
|- | |- | ||
| − | | | + | |Hence, <math style="vertical-align: -5px">f(x)</math> has at least one zero. |
|} | |} | ||
| Line 31: | Line 53: | ||
!Step 2: | !Step 2: | ||
|- | |- | ||
| − | | | + | |Suppose that <math style="vertical-align: -5px">f(x)</math> has more than one zero. |
| + | |- | ||
| + | |So, there exist <math style="vertical-align: -4px">a,b</math> such that | ||
| + | |- | ||
| + | | <math style="vertical-align: -5px">f(a)=f(b)=0.</math> | ||
| + | |- | ||
| + | |Then, by the Mean Value Theorem, there exists <math style="vertical-align: 0px">c</math> with <math style="vertical-align: 0px">a<c<b</math> such that | ||
| + | |- | ||
| + | | <math style="vertical-align: -5px">f'(c)=0.</math> | ||
| + | |- | ||
| + | |We have <math style="vertical-align: -5px">f'(x)=3x^2+2.</math> | ||
| + | |- | ||
| + | |Since <math style="vertical-align: -5px">x^2\ge 0,</math> | ||
| + | |- | ||
| + | | <math style="vertical-align: -5px"> f'(x) \ge 2.</math> | ||
|- | |- | ||
| − | | | + | |Therefore, it is impossible for <math style="vertical-align: -5px">f'(c)=0.</math> Hence, <math style="vertical-align: -5px">f(x)</math> has at most one zero. |
|} | |} | ||
| Line 40: | Line 76: | ||
!Final Answer: | !Final Answer: | ||
|- | |- | ||
| − | | | + | | See solution above. |
|} | |} | ||
[[009A_Sample_Final_2|'''<u>Return to Sample Exam</u>''']] | [[009A_Sample_Final_2|'''<u>Return to Sample Exam</u>''']] | ||
Revision as of 15:28, 7 March 2017
Show that the equation Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x^3+2x-2=0} has exactly one real root.
| Foundations: |
|---|
| 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 |
| 2. Mean Value Theorem |
| Suppose is a function that satisfies the following: |
|
is continuous on the closed interval |
|
is differentiable on the open interval Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (a,b).} |
|
Then, there is a number Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle c} such that Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a<c<b} and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f'(c)=\frac{f(b)-f(a)}{b-a}.} |
![{\displaystyle [a,b]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9c4b788fc5c637e26ee98b45f89a5c08c85f7935)
![{\displaystyle [a,b].}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3ba5cb29655f824ce80a0b6a32d9326d0e8742cd)
Solution:
| Step 1: |
|---|
| First, we note that |
| Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f(0)=-2.} |
| Also, |
| Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f(1)=1.} |
| Since Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f(0)<0} and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f(1)>0,} |
| there exists Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x} with Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 0<x<1} such that |
| Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f(x)=0} |
| by the Intermediate Value Theorem. |
| Hence, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f(x)} has at least one zero. |
| Step 2: |
|---|
| Suppose that Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f(x)} has more than one zero. |
| So, there exist Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a,b} such that |
| Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f(a)=f(b)=0.} |
| Then, by the Mean Value Theorem, there exists Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle c} with Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a<c<b} such that |
| Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f'(c)=0.} |
| We have Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f'(x)=3x^2+2.} |
| Since Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x^2\ge 0,} |
| Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f'(x) \ge 2.} |
| Therefore, it is impossible for Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f'(c)=0.} Hence, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f(x)} has at most one zero. |
| Final Answer: |
|---|
| See solution above. |