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Let <span class="nolink">«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi»f«/mi»«mo»:«/mo»«msup»«mi»§#8477;«/mi»«mn»2«/mn»«/msup»«mo»§#8594;«/mo»«mi»§#8477;«/mi»«/math»</span> be the function defined by<br /><br /><span class="nolink">«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi»f«/mi»«mo»(«/mo»«mi»x«/mi»«mo»,«/mo»«mi»y«/mi»«mo»)«/mo»«mo»=«/mo»«mfenced close=¨¨ open=¨{¨»«mtable»«mtr»«mtd»«mi»#g«/mi»«mo»§nbsp;«/mo»«mo»§nbsp;«/mo»«mo»§nbsp;«/mo»«mi»y«/mi»«mo»§#8800;«/mo»«mn»0«/mn»«/mtd»«/mtr»«mtr»«mtd»«mn»0«/mn»«mo»§nbsp;«/mo»«mo»§nbsp;«/mo»«mo»§nbsp;«/mo»«mo»§nbsp;«/mo»«mi»y«/mi»«mo»=«/mo»«mn»0«/mn»«/mtd»«/mtr»«/mtable»«/mfenced»«/math»</span><br /><br />and the vector <span class="nolink">«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi»v«/mi»«mo»=«/mo»«mfenced»«mrow»«mi»u«/mi»«mo»,«/mo»«mi»w«/mi»«/mrow»«/mfenced»«mo»,«/mo»«mo»§nbsp;«/mo»«mo»§nbsp;«/mo»«mo»§nbsp;«/mo»«mi»u«/mi»«mo»,«/mo»«mi»w«/mi»«mo»§#8712;«/mo»«mi»§#8477;«/mi»«mo»,«/mo»«mo»§nbsp;«/mo»«msqrt»«mrow»«msup»«mi»u«/mi»«mn»2«/mn»«/msup»«mo»+«/mo»«msup»«mi»w«/mi»«mn»2«/mn»«/msup»«/mrow»«/msqrt»«mo»=«/mo»«mn»1«/mn»«/math»</span><br />Choose the true statement<br />
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Choice 1
Answer
Since «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«munder»«mo largeop=¨true¨»lim«/mo»«mtable»«mtr»«mtd»«mo»(«/mo»«mi»x«/mi»«mo»,«/mo»«mi»y«/mi»«mo»)«/mo»«mo»§#8594;«/mo»«mo»(«/mo»«mn»0«/mn»«mo»,«/mo»«mn»0«/mn»«mo»)«/mo»«/mtd»«/mtr»«mtr»«mtd»«mi»y«/mi»«mo»=«/mo»«mn»0«/mn»«/mtd»«/mtr»«/mtable»«/munder»«mi»f«/mi»«mo»(«/mo»«mi»x«/mi»«mo»,«/mo»«mi»y«/mi»«mo»)«/mo»«mo»=«/mo»«munder»«mo largeop=¨true¨»lim«/mo»«mtable»«mtr»«mtd»«mo»(«/mo»«mi»x«/mi»«mo»,«/mo»«mi»y«/mi»«mo»)«/mo»«mo»§#8594;«/mo»«mo»(«/mo»«mn»0«/mn»«mo»,«/mo»«mn»0«/mn»«mo»)«/mo»«/mtd»«/mtr»«mtr»«mtd»«mi»y«/mi»«mo»=«/mo»«mi»kx«/mi»«/mtd»«/mtr»«/mtable»«/munder»«mi»f«/mi»«mo»(«/mo»«mi»x«/mi»«mo»,«/mo»«mi»y«/mi»«mo»)«/mo»«mo»=«/mo»«mn»0«/mn»«/math» the function is continuous at the origin
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Choice 2
Answer
The function is diferentiable since the partial derivatives at the origin are continuous
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Choice 3
Answer
The directional derivative of «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi»f«/mi»«mo»(«/mo»«mi»x«/mi»«mo»,«/mo»«mi»y«/mi»«mo»)«/mo»«/math» in the direction of «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi»v«/mi»«mo»=«/mo»«mfenced»«mrow»«mi»u«/mi»«mo»,«/mo»«mi»w«/mi»«/mrow»«/mfenced»«/math» at the origin is #sol
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Choice 4
Answer
The function is continuous at the origin because there exists the directional derivative at the origin in any direction
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Choice 5
Answer
The function is not continuous at the origin despite having directional derivatives in any direction
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Choice 6
Answer
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Choice 7
Answer
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Choice 8
Answer
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Feedback
Combined feedback
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Created / last saved
Created
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Admin User
on
Thursday, 10 December 2015, 10:53 AM
Last saved
by
Admin User
on
Thursday, 10 December 2015, 10:53 AM
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