Difference between revisions of "009A Sample Final 3, Problem 4"

From Grad Wiki
Jump to navigation Jump to search
Line 26: Line 26:
 
!Step 1:    
 
!Step 1:    
 
|-
 
|-
−
|
+
|We first calculate &nbsp;<math style="vertical-align: -14px">\lim_{x\rightarrow 0^+}f(x).</math>&nbsp; We have
 
|-
 
|-
 
|
 
|
 +
&nbsp; &nbsp; &nbsp; &nbsp;<math>\begin{array}{rcl}
 +
\displaystyle{\lim_{x\rightarrow 0^+}f(x)} & = & \displaystyle{\lim_{x\rightarrow 0^+} x-\cos x}\\
 +
&&\\
 +
& = & \displaystyle{0-\cos(0)}\\
 +
&&\\
 +
& = & \displaystyle{-1.}
 +
\end{array}</math>
 +
|}
 +
 +
{| class="mw-collapsible mw-collapsed" style = "text-align:left;"
 +
!Step 2: &nbsp;
 
|-
 
|-
−
|
+
|Now, we calculate &nbsp;<math style="vertical-align: -14px">\lim_{x\rightarrow 0^-}f(x).</math>&nbsp; We have
 
|-
 
|-
 
|
 
|
 +
&nbsp; &nbsp; &nbsp; &nbsp;<math>\begin{array}{rcl}
 +
\displaystyle{\lim_{x\rightarrow 0^-}f(x)} & = & \displaystyle{\lim_{x\rightarrow 0^-} \frac{x}{|x|}}\\
 +
&&\\
 +
& = & \displaystyle{\lim_{x\rightarrow 0^-} \frac{x}{-x}}\\
 +
&&\\
 +
& = & \displaystyle{\lim_{x\rightarrow 0^-} -1}\\
 +
&&\\
 +
& = & \displaystyle{-1.}
 +
\end{array}</math>
 
|}
 
|}
  
 
{| class="mw-collapsible mw-collapsed" style = "text-align:left;"
 
{| class="mw-collapsible mw-collapsed" style = "text-align:left;"
−
!Step 2: &nbsp;
+
!Step 3: &nbsp;
 +
|-
 +
|Since
 
|-
 
|-
 
|
 
|
 +
&nbsp; &nbsp; &nbsp; &nbsp; <math style="vertical-align: -15px">\lim_{x\rightarrow 3^+}f(x)=\lim_{x\rightarrow 3^-}f(x)=-1,</math>
 
|-
 
|-
−
|
+
|we have
 +
|-
 +
|&nbsp; &nbsp; &nbsp; &nbsp; <math>\lim_{x\rightarrow 3} f(x)=-1.</math>
 +
|-
 +
|But,
 +
|-
 +
|&nbsp; &nbsp; &nbsp; &nbsp; <math>f(0)=0\ne \lim_{x\rightarrow 3} f(x).</math>
 +
|-
 +
|Thus, <math style="vertical-align: -5px">f(x)</math>&nbsp; is not continuous.
 +
|-
 +
|It is a jump discontinuity.
 
|}
 
|}
  
Line 47: Line 80:
 
!Final Answer: &nbsp;  
 
!Final Answer: &nbsp;  
 
|-
 
|-
−
|
+
|&nbsp; &nbsp; &nbsp; &nbsp;<math style="vertical-align: -5px">f(x)</math>&nbsp; is not continuous. It is a jump discontinuity.
 
|}
 
|}
 
[[009A_Sample_Final_3|'''<u>Return to Sample Exam</u>''']]
 
[[009A_Sample_Final_3|'''<u>Return to Sample Exam</u>''']]

Revision as of 10:48, 6 March 2017

Discuss, without graphing, if the following function is continuous at  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=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) = \left\{ \begin{array}{lr} \frac{x}{|x|} & \text{if }x < 0\\ 0 & \text{if }x = 0\\ x-\cos x & \text{if }x > 0 \end{array} \right. }

If you think  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}   is not continuous at  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=0,}   what kind of discontinuity is it?

Foundations:  
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)}   is continuous at    if
       


Solution:

Step 1:  
We first calculate  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 \lim_{x\rightarrow 0^+}f(x).}   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 \begin{array}{rcl} \displaystyle{\lim_{x\rightarrow 0^+}f(x)} & = & \displaystyle{\lim_{x\rightarrow 0^+} x-\cos x}\\ &&\\ & = & \displaystyle{0-\cos(0)}\\ &&\\ & = & \displaystyle{-1.} \end{array}}

Step 2:  
Now, we calculate  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 \lim_{x\rightarrow 0^-}f(x).}   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 \begin{array}{rcl} \displaystyle{\lim_{x\rightarrow 0^-}f(x)} & = & \displaystyle{\lim_{x\rightarrow 0^-} \frac{x}{|x|}}\\ &&\\ & = & \displaystyle{\lim_{x\rightarrow 0^-} \frac{x}{-x}}\\ &&\\ & = & \displaystyle{\lim_{x\rightarrow 0^-} -1}\\ &&\\ & = & \displaystyle{-1.} \end{array}}

Step 3:  
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 \lim_{x\rightarrow 3^+}f(x)=\lim_{x\rightarrow 3^-}f(x)=-1,}

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 \lim_{x\rightarrow 3} f(x)=-1.}
But,
        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\ne \lim_{x\rightarrow 3} f(x).}
Thus, 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)}   is not continuous.
It is a jump discontinuity.


Final Answer:  
       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)}   is not continuous. It is a jump discontinuity.

Return to Sample Exam

Retrieved from "https://gradwiki.math.ucr.edu/index.php?title=009A_Sample_Final_3,_Problem_4&oldid=3144"