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Calculus 3 : Partial Derivatives

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Example Questions

Example Question #331 : Limits

$\begin{align*}&\text{Calculate the limits of the function}\\&\lim_{(x,y)\rightarrow(0-,0-)}\frac{(5xy^{2} + 13y^{3})}{(4xy^{2} + 25x^{3} - 2y^{3})}\\&\text{along the y-axis and the line, }y=x\end{align*}$

Possible Answers:

$\begin{align*}&\text{y-axis: }-\frac{13}{2}\\&y=x:\frac{2}{3}\end{align*}$

$\begin{align*}&\text{y-axis: }-\frac{1}{2}\\&y=x:\frac{1}{18}\end{align*}$

$\begin{align*}&\text{y-axis: }\frac{26}{3}\\&y=x:-52\end{align*}$

$\begin{align*}&\text{y-axis: }-\frac{38}{3}\\&y=x:117\end{align*}$

Correct answer:

$\begin{align*}&\text{y-axis: }-\frac{13}{2}\\&y=x:\frac{2}{3}\end{align*}$

Explanation:

$\begin{align*}&\text{To approach this problem, it is important to understand what}\\ &\text{it is asking. Along the y-axis, }x=0\text{. Let us make that substitution:}\\&\frac{(5(0)y^{2} + 13y^{3})}{(4(0)y^{2} + 25(0)^{3} - 2y^{3})}=\frac{-13y^{3}}{2y^{3}}\\&\lim_{(0,y)\rightarrow(0-,0-)}\frac{-13y^{3}}{2y^{3}}=-\frac{13}{2}\\&\text{We can make a similar substitution for }y=x:\\&\lim_{(x,x)\rightarrow(0-,0-)}\frac{(5x(x)^{2} + 13(x)^{3})}{(4x(x)^{2} + 25x^{3} - 2(x)^{3})}=\frac{2}{3}\\&\text{As these two limits are different, the limit does not exist.}\end{align*}$

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Example Question #332 : Limits

$\begin{align*}&\text{Calculate the limits of the function}\\&\lim_{(x,y)\rightarrow(0-,0-)}-\frac{(5xy^{2} - 17x^{2}y + 6y^{3})}{(17x^{3} - 16y^{3})}\\&\text{along the y-axis and the line, }y=x\end{align*}$

Possible Answers:

$\begin{align*}&\text{y-axis: }-78\\&y=x:-\frac{33}{8}\end{align*}$

$\begin{align*}&\text{y-axis: }\frac{3}{160}\\&y=x:\frac{6}{13}\end{align*}$

$\begin{align*}&\text{y-axis: }\frac{3}{8}\\&y=x:6\end{align*}$

$\begin{align*}&\text{y-axis: }54\\&y=x:\frac{15}{8}\end{align*}$

Correct answer:

$\begin{align*}&\text{y-axis: }\frac{3}{8}\\&y=x:6\end{align*}$

Explanation:

$\begin{align*}&\text{To approach this problem, it is important to understand what}\\ &\text{it is asking. Along the y-axis, }x=0\text{. Let us make that substitution:}\\&-\frac{(5(0)y^{2} - 17(0)^{2}y + 6y^{3})}{(17(0)^{3} - 16y^{3})}=\frac{6y^{3}}{16y^{3}}\\&\lim_{(0,y)\rightarrow(0-,0-)}\frac{6y^{3}}{16y^{3}}=\frac{3}{8}\\&\text{We can make a similar substitution for }y=x:\\&\lim_{(x,x)\rightarrow(0-,0-)}-\frac{(5x(x)^{2} - 17x^{2}(x) + 6(x)^{3})}{(17x^{3} - 16(x)^{3})}=6\\&\text{As these two limits are different, the limit does not exist.}\end{align*}$

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Example Question #333 : Limits

$\begin{align*}&\text{Calculate the limits of the function}\\&\lim_{(x,y)\rightarrow(0-,0+)}\frac{(15x^{3}y^{2} - 3x^{4}y)}{(18x^{4}y - 20x^{2}y^{3} + 10x^{5} + 22y^{5})}\\&\text{along the y-axis and the line, }y=x\end{align*}$

Possible Answers:

$\begin{align*}&\text{y-axis: }-8\\&y=x:0\end{align*}$

$\begin{align*}&\text{y-axis: }0\\&y=x:\frac{2}{5}\end{align*}$

$\begin{align*}&\text{y-axis: }0\\&y=x:\frac{1}{15}\end{align*}$

$\begin{align*}&\text{y-axis: }\frac{18}{5}\\&y=x:0\end{align*}$

Correct answer:

$\begin{align*}&\text{y-axis: }0\\&y=x:\frac{2}{5}\end{align*}$

Explanation:

$\begin{align*}&\text{To approach this problem, it is important to understand what}\\ &\text{it is asking. Along the y-axis, }x=0\text{. Let us make that substitution:}\\&\frac{(15(0)^{3}y^{2} - 3(0)^{4}y)}{(18(0)^{4}y - 20(0)^{2}y^{3} + 10(0)^{5} + 22y^{5})}=\frac{0}{22y^{5}}\\&\lim_{(0,y)\rightarrow(0-,0+)}\frac{0}{22y^{5}}=0\\&\text{We can make a similar substitution for }y=x:\\&\lim_{(x,x)\rightarrow(0-,0-)}\frac{(15x^{3}(x)^{2} - 3x^{4}(x))}{(18x^{4}(x) - 20x^{2}(x)^{3} + 10x^{5} + 22(x)^{5})}=\frac{2}{5}\\&\text{As these two limits are different, the limit does not exist.}\end{align*}$

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Example Question #334 : Limits

$\begin{align*}&\text{Calculate the limits of the function}\\& \lim_{(x,y)\rightarrow(0+,0+)}-\frac{(19x^{5} + 12y^{5})}{(35x^{2}y^{3} + 16xy^{4} - 2x^{5} - 15y^{5})}\\&\text{along the y-axis and the line, }y=x\end{align*}$

Possible Answers:

$\begin{align*}&\text{y-axis: }\frac{4}{45}\\&y=x:-\frac{31}{340}\end{align*}$

$\begin{align*}&\text{y-axis: }\frac{31}{17}\\&y=x:-\frac{64}{5}\end{align*}$

$\begin{align*}&\text{y-axis: }\frac{4}{5}\\&y=x:-\frac{31}{34}\end{align*}$

$\begin{align*}&\text{y-axis: }-\frac{93}{17}\\&y=x:12\end{align*}$

Correct answer:

$\begin{align*}&\text{y-axis: }\frac{4}{5}\\&y=x:-\frac{31}{34}\end{align*}$

Explanation:

$\begin{align*}&\text{To approach this problem, it is important to understand what}\\ &\text{it is asking. Along the y-axis, }x=0\text{. Let us make that substitution:}\\&-\frac{(19(0)^{5} + 12y^{5})}{(35(0)^{2}y^{3} + 16(0)y^{4} - 2(0)^{5} - 15y^{5})}=\frac{12y^{5}}{15y^{5}}\\&\lim_{(0,y)\rightarrow(0+,0+)}\frac{12y^{5}}{15y^{5}}=\frac{4}{5}\\&\text{We can make a similar substitution for }y=x:\\&\lim_{(x,x)\rightarrow(0+,0+)}-\frac{(19x^{5} + 12(x)^{5})}{(35x^{2}(x)^{3} + 16x(x)^{4} - 2x^{5} - 15(x)^{5})}=-\frac{31}{34}\\&\text{As these two limits are different, the limit does not exist.}\end{align*}$

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Example Question #335 : Limits

$\begin{align*}&\text{Calculate the limits of the function}\\&\lim_{(x,y)\rightarrow(0-,0-)}\frac{(18x^{5})}{(18x^{2}y^{3} - 14x^{4}y + 15x^{5} + 2y^{5})}\\&\text{along the y-axis and the line, }y=x\end{align*}$

Possible Answers:

$\begin{align*}&\text{y-axis: }0\\&y=x:\frac{3}{56}\end{align*}$

$\begin{align*}&\text{y-axis: }\frac{120}{7}\\&y=x:0\end{align*}$

$\begin{align*}&\text{y-axis: }-\frac{30}{7}\\&y=x:0\end{align*}$

$\begin{align*}&\text{y-axis: }0\\&y=x:\frac{6}{7}\end{align*}$

Correct answer:

$\begin{align*}&\text{y-axis: }0\\&y=x:\frac{6}{7}\end{align*}$

Explanation:

$\begin{align*}&\text{To approach this problem, it is important to understand what}\\ &\text{it is asking. Along the y-axis, }x=0\text{. Let us make that substitution:}\\&\frac{(18(0)^{5})}{(18(0)^{2}y^{3} - 14(0)^{4}y + 15(0)^{5} + 2y^{5})}=\frac{0}{2y^{5}}\\&\lim_{(0,y)\rightarrow(0-,0-)}\frac{0}{2y^{5}}=0\\&\text{We can make a similar substitution for }y=x:\\&\lim_{(x,x)\rightarrow(0-,0-)}\frac{(18x^{5})}{(18x^{2}(x)^{3} - 14x^{4}(x) + 15x^{5} + 2(x)^{5})}=\frac{6}{7}\\&\text{As these two limits are different, the limit does not exist.}\end{align*}$

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Example Question #336 : Limits

$\begin{align*}&\text{Calculate the limits of the function}\\&\lim_{(x,y)\rightarrow(0+,0+)}\frac{(14x^{6})}{(16x^{3}y^{3} + 19x^{6} + 3y^{6})}\\&\text{along the y-axis and the line, }y=x\end{align*}$

Possible Answers:

$\begin{align*}&\text{y-axis: }\frac{35}{19}\\&y=x:0\end{align*}$

$\begin{align*}&\text{y-axis: }0\\&y=x:\frac{7}{361}\end{align*}$

$\begin{align*}&\text{y-axis: }0\\&y=x:\frac{7}{19}\end{align*}$

$\begin{align*}&\text{y-axis: }-\frac{63}{19}\\&y=x:0\end{align*}$

Correct answer:

$\begin{align*}&\text{y-axis: }0\\&y=x:\frac{7}{19}\end{align*}$

Explanation:

$\begin{align*}&\text{To approach this problem, it is important to understand what}\\ &\text{it is asking. Along the y-axis, }x=0\text{. Let us make that substitution:}\\&\frac{(14(0)^{6})}{(16(0)^{3}y^{3} + 19(0)^{6} + 3y^{6})}=\frac{0}{3y^{6}}\\&\lim_{(0,y)\rightarrow(0+,0+)}\frac{0}{3y^{6}}=0\\&\text{We can make a similar substitution for }y=x:\\&\lim_{(x,x)\rightarrow(0+,0+)}\frac{(14x^{6})}{(16x^{3}(x)^{3} + 19x^{6} + 3(x)^{6})}=\frac{7}{19}\\&\text{As these two limits are different, the limit does not exist.}\end{align*}$

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Example Question #337 : Limits

$\begin{align*}&\text{Calculate the limits of the function}\\&\lim_{(x,y)\rightarrow(0+,0+)}-\frac{(xy^{2} + 5x^{2}y - 6x^{3})}{(x^{3} + 3y^{3})}\\&\text{along the y-axis and the line, }y=x\end{align*}$

Possible Answers:

$\begin{align*}&\text{y-axis: }0\\&y=x:-\frac{31}{16}\end{align*}$

$\begin{align*}&\text{y-axis: }0\\&y=x:0\end{align*}$

$\begin{align*}&\text{y-axis: }-\frac{46}{13}\\&y=x:-\frac{32}{13}\end{align*}$

$\begin{align*}&\text{y-axis: }-\frac{13}{6}\\&y=x:0\end{align*}$

Correct answer:

$\begin{align*}&\text{y-axis: }0\\&y=x:0\end{align*}$

Explanation:

$\begin{align*}&\text{To approach this problem, it is important to understand what}\\ &\text{it is asking. Along the y-axis, }x=0\text{. Let us make that substitution:}\\&-\frac{((0)y^{2} + 5(0)^{2}y - 6(0)^{3})}{((0)^{3} + 3y^{3})}=\frac{0}{3y^{3}}\\&\lim_{(0,y)\rightarrow(0+,0+)}\frac{0}{3y^{3}}=0\\&\text{We can make a similar substitution for }y=x:\\&\lim_{(x,x)\rightarrow(0+,0+)}-\frac{(x(x)^{2} + 5x^{2}(x) - 6x^{3})}{(x^{3} + 3(x)^{3})}=0\\&\text{Since the two limits are equal, it's possible the limit exists.}\\&\text{Testing additional lines could prove or disprove this.}\end{align*}$

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Example Question #338 : Limits

$\begin{align*}&\text{Calculate the limits of the function}\\&\lim_{(x,y)\rightarrow(0+,0+)}-\frac{(3x^{2}y^{2} - 12xy^{3} + 9x^{4})}{(3x^{4} + 2y^{4})}\\&\text{along the y-axis and the line, }y=x\end{align*}$

Possible Answers:

$\begin{align*}&\text{y-axis: }0\\&y=x:0\end{align*}$

$\begin{align*}&\text{y-axis: }-\frac{37}{11}\\&y=x:\frac{5}{2}\end{align*}$

$\begin{align*}&\text{y-axis: }0\\&y=x:-\frac{25}{7}\end{align*}$

$\begin{align*}&\text{y-axis: }-\frac{25}{17}\\&y=x:-\frac{48}{19}\end{align*}$

Correct answer:

$\begin{align*}&\text{y-axis: }0\\&y=x:0\end{align*}$

Explanation:

$\begin{align*}&\text{To approach this problem, it is important to understand what}\\ &\text{it is asking. Along the y-axis, }x=0\text{. Let us make that substitution:}\\&-\frac{(3(0)^{2}y^{2} - 12(0)y^{3} + 9(0)^{4})}{(3(0)^{4} + 2y^{4})}=\frac{0}{2y^{4}}\\&\lim_{(0,y)\rightarrow(0+,0+)}\frac{0}{2y^{4}}=0\\&\text{We can make a similar substitution for }y=x:\\&\lim_{(x,x)\rightarrow(0+,0+)}-\frac{(3x^{2}(x)^{2} - 12x(x)^{3} + 9x^{4})}{(3x^{4} + 2(x)^{4})}=0\\&\text{Since the two limits are equal, it's possible the limit exists.}\\&\text{Testing additional lines could prove or disprove this.}\end{align*}$

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Example Question #339 : Limits

$\begin{align*}&\text{Calculate the limits of the function}\\&\lim_{(x,y)\rightarrow(0+,0-)}-\frac{(11xy^{2} + 8x^{2}y)}{(11x^{2}y + 12x^{3} - 4y^{3})}\\&\text{along the y-axis and the line, }y=x\end{align*}$

Possible Answers:

$\begin{align*}&\text{y-axis: }0\\&y=x:-1\end{align*}$

$\begin{align*}&\text{y-axis: }20\\&y=x:0\end{align*}$

$\begin{align*}&\text{y-axis: }-6\\&y=x:0\end{align*}$

$\begin{align*}&\text{y-axis: }0\\&y=x:-\frac{1}{3}\end{align*}$

Correct answer:

$\begin{align*}&\text{y-axis: }0\\&y=x:-1\end{align*}$

Explanation:

$\begin{align*}&\text{To approach this problem, it is important to understand what}\\ &\text{it is asking. Along the y-axis, }x=0\text{. Let us make that substitution:}\\&-\frac{(11(0)y^{2} + 8(0)^{2}y)}{(11(0)^{2}y + 12(0)^{3} - 4y^{3})}=\frac{0}{4y^{3}}\\&\lim_{(0,y)\rightarrow(0+,0-)}\frac{0}{4y^{3}}=0\\&\text{We can make a similar substitution for }y=x:\\&\lim_{(x,x)\rightarrow(0+,0+)}-\frac{(11x(x)^{2} + 8x^{2}(x))}{(11x^{2}(x) + 12x^{3} - 4(x)^{3})}=-1\\&\text{As these two limits are different, the limit does not exist.}\end{align*}$

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Example Question #340 : Limits

$\begin{align*}&\text{Calculate the limits of the function}\\&\lim_{(x,y)\rightarrow(0+,0+)}\frac{(6xy^{3} + 5x^{4})}{(2x^{4} + 12y^{4})}\\&\text{along the y-axis and the line, }y=x\end{align*}$

Possible Answers:

$\begin{align*}&\text{y-axis: }-\frac{209}{14}\\&y=x:0\end{align*}$

$\begin{align*}&\text{y-axis: }\frac{33}{14}\\&y=x:0\end{align*}$

$\begin{align*}&\text{y-axis: }0\\&y=x:\frac{11}{14}\end{align*}$

$\begin{align*}&\text{y-axis: }0\\&y=x:\frac{11}{196}\end{align*}$

Correct answer:

$\begin{align*}&\text{y-axis: }0\\&y=x:\frac{11}{14}\end{align*}$

Explanation:

$\begin{align*}&\text{To approach this problem, it is important to understand what}\\ &\text{it is asking. Along the y-axis, }x=0\text{. Let us make that substitution:}\\&\frac{(6(0)y^{3} + 5(0)^{4})}{(2(0)^{4} + 12y^{4})}=\frac{0}{12y^{4}}\\&\lim_{(0,y)\rightarrow(0+,0+)}\frac{0}{12y^{4}}=0\\&\text{We can make a similar substitution for }y=x:\\&\lim_{(x,x)\rightarrow(0+,0+)}\frac{(6x(x)^{3} + 5x^{4})}{(2x^{4} + 12(x)^{4})}=\frac{11}{14}\\&\text{As these two limits are different, the limit does not exist.}\end{align*}$

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Calculus 3 : Partial Derivatives

Study concepts, example questions & explanations for Calculus 3

All Calculus 3 Resources

Example Questions

Example Question #331 : Limits

Example Question #332 : Limits

Example Question #333 : Limits

Example Question #334 : Limits

Example Question #335 : Limits

Example Question #336 : Limits

Example Question #337 : Limits

Example Question #338 : Limits

Example Question #339 : Limits

Example Question #340 : Limits

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