Difference between revisions of "009A Sample Final 1, Problem 2"

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!Final Answer:    
 
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|'''(a)''' Since <math style="vertical-align: -14px">\lim_{x\rightarrow 3^+}f(x)=\lim_{x\rightarrow 3^-}f(x)=f(3),~f(x)</math>&thinsp; is continuous.
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|&nbsp;&nbsp; '''(a)''' Since <math style="vertical-align: -14px">\lim_{x\rightarrow 3^+}f(x)=\lim_{x\rightarrow 3^-}f(x)=f(3),~f(x)</math>&thinsp; is continuous.
 
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|'''(b)''' Since <math style="vertical-align: -14px">\lim_{h\rightarrow 0^-}\frac{f(3+h)-f(3)}{h}=\lim_{h\rightarrow 0^+}\frac{f(3+h)-f(3)}{h},</math>  
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|&nbsp;&nbsp; '''(b)''' Since <math style="vertical-align: -14px">\lim_{h\rightarrow 0^-}\frac{f(3+h)-f(3)}{h}=\lim_{h\rightarrow 0^+}\frac{f(3+h)-f(3)}{h},</math>  
 
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Revision as of 14:13, 18 April 2016

Consider the following piecewise defined function:

a) Show that is continuous at
b) Using the limit definition of the derivative, and computing the limits from both sides, show that is differentiable at
Foundations:  
Recall:
1.   is continuous at   if
2. The definition of derivative for   is  

Solution:

(a)

Step 1:  
We first calculate We have
Step 2:  
Now, we calculate We have
Step 3:  
Now, we calculate We have
Since   is continuous.

(b)

Step 1:  
We need to use the limit definition of derivative and calculate the limit from both sides.
So, we have
Step 2:  
Now, we have
Step 3:  
Since
  is differentiable at
Final Answer:  
   (a) Since   is continuous.
   (b) Since
  is differentiable at

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