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Multivariable Calculus UAF MATH F253X

Dr. Latrice Bowman

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Section Functions of Two Variables

Worksheet FUN-4 Reading

All of the readings for this course come from the OpenStax Calculus Volume 3 textbook, which is available for free online at https://openstax.org/details/books/calculus-volume-3. You can read the textbook online or download a PDF version. The textbook is also available in print from various retailers if you prefer a physical copy.
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For this lesson, you should read sections 4.1 Functions of Several Variables, and 4.2 Limits and Continuity. You should read through the sections carefully, making sure to understand the definitions, examples, and key ideas. You should also work through the exercises at the end of each section to test your understanding and practice applying the concepts. Some of the exercises at the end of the section are assigned in the WP assignments.
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Worksheet FUN-4 Videos

Arc Length and Curvature
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Motion in Space
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Worksheet FUN-4 Written Practice

INFORMATION ABOUT ALL WRITTEN PRACTICE (WP) ASSIGNMENTS:
  • All of the WP problems come from our online textbook: https://openstax.org/details/books/calculus-volume-3
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  • Complete solutions to all WP assignments can be found in Canvas within the section Module.
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  • Links to the text and PDFs of the text can be found in the Canvas module under the Reading and Information.
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THE WP ASSIGNMENTS ARE DESIGNED TO PREPARE YOU FOR THE QUIZZES AND ASSESSMENTS
  • Work on every problem on every assignment.
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  • If you get stuck or do not understand a solution, ask questions.
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  • For each problem, try it once or twice before looking at solutions or asking for help.
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  • Mark the problems you did correctly on the first try.
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  • Highlight problems where you got help (using solutions, videos, a tutor, etc) -These are the problem types that you will need more practice on to be ready for the assessments.
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  • Understand that just copying the solutions instead of working through the problems will greatly reduce your chances of success in this class.
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WP4
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Arc Length and Curvature: Complete problems 103, 105, 107, 109, 112, 115, 117, 125, 133, 135, 139, 147
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Motion in Space: Complete problems 157, 163, 173-177, 181, 183
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Functions of Several Variables: Complete problems 3, 17, 21, 49, 51, 56, 57, 58, 59
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Limits and Continuity: Complete problems 61, 86, 87, 88, 89
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In addition to these problems, on separate paper complete the following exploration. This link Transformations will take you to the exploration. On the right side of the screen, you will be given some instructions and questions to answer. Follow the instructions and write out your answers to each of the questions.
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Worksheet FUN-4 Sample Outcomes

For the quiz on this outcome you would be given 30 minutes to complete 2 problems. Here are samples of the types of problems you will encounter:
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1.

Given \(f(x,y)=\ln(x^2+y^2)\)
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(a)
State the domain of \(f(x,y)\) using correct set notation.
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Hint.
The natural logarithm is only defined for positive arguments.
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Answer.
\(\{(x,y) \in \mathbb{R}^2 : x^2+y^2 > 0\}\)
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Solution.
The argument of the natural logarithm must be positive:
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\(x^2+y^2 > 0\text{.}\)
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This describes all points in the plane except the origin.
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(b)
Evaluate \(f(2e,e)\text{.}\)
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Hint.
To evaluate \(f(2e,e)\text{,}\) substitute \(x=2e\) and \(y=e\) into the function.
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Answer.
\(f(2e,e)=\ln(5)+2\)
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Solution.
To evaluate \(f(2e,e)\text{,}\) substitute \(x=2e\) and \(y=e\) into the function:
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\(f(2e,e)=\ln((2e)^2+e^2)=\ln(4e^2+e^2)=\ln(5e^2)=\ln(5)+\ln(e^2)=\ln(5)+2\text{.}\)
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2.

Given \(f(x,y)=\sqrt{x^2+y^2}\)
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(a)
Find and sketch the level curve of when \(c=0\text{.}\)
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Hint.
Set \(f(x,y)=c\) and solve for \(y\) in terms of \(x\text{.}\)
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Answer.
The level curve is the point \((0,0)\text{.}\)
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Diagram Exploration Keyboard Controls
Key Action
Enter, A Activate keyboard driven exploration
B Activate menu driven exploration
Escape Leave exploration mode
Cursor down Explore next lower level
Cursor up Explore next upper level
Cursor right Explore next element on level
Cursor left Explore previous element on level
X Toggle expert mode
W Extra details if available
Space Repeat speech
M Activate step magnification
Comma Activate direct magnification
N Deactivate magnification
Z Toggle subtitles
C Cycle contrast settings
T Monochrome colours
L Toggle language (if available)
K Kill current sound
Y Stop sound output
O Start and stop sonification
P Repeat sonification output
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Solution.
Setting \(f(x,y)=0\text{:}\)
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\(\sqrt{x^2+y^2}=0\text{.}\)
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This implies \(x^2+y^2=0\text{,}\) which is only satisfied when \(x=0\) and \(y=0\text{.}\)
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(b)
Find and sketch the level curve of when \(c=1\text{.}\)
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Hint.
Set \(f(x,y)=c\) and solve for \(y\) in terms of \(x\text{.}\)
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Answer.
The level curve is the circle \(x^2+y^2=1\text{.}\)
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Diagram Exploration Keyboard Controls
Key Action
Enter, A Activate keyboard driven exploration
B Activate menu driven exploration
Escape Leave exploration mode
Cursor down Explore next lower level
Cursor up Explore next upper level
Cursor right Explore next element on level
Cursor left Explore previous element on level
X Toggle expert mode
W Extra details if available
Space Repeat speech
M Activate step magnification
Comma Activate direct magnification
N Deactivate magnification
Z Toggle subtitles
C Cycle contrast settings
T Monochrome colours
L Toggle language (if available)
K Kill current sound
Y Stop sound output
O Start and stop sonification
P Repeat sonification output
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Solution.
Setting \(f(x,y)=1\text{:}\)
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\(\sqrt{x^2+y^2}=1\text{.}\)
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This implies \(x^2+y^2=1\text{,}\) which describes the unit circle centered at the origin.
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(c)
Find and sketch the level curve of when \(c=4\text{.}\)
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Hint.
Set \(f(x,y)=c\) and solve for \(y\) in terms of \(x\text{.}\)
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Answer.
The level curve is the circle \(x^2+y^2=4\text{.}\)
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Diagram Exploration Keyboard Controls
Key Action
Enter, A Activate keyboard driven exploration
B Activate menu driven exploration
Escape Leave exploration mode
Cursor down Explore next lower level
Cursor up Explore next upper level
Cursor right Explore next element on level
Cursor left Explore previous element on level
X Toggle expert mode
W Extra details if available
Space Repeat speech
M Activate step magnification
Comma Activate direct magnification
N Deactivate magnification
Z Toggle subtitles
C Cycle contrast settings
T Monochrome colours
L Toggle language (if available)
K Kill current sound
Y Stop sound output
O Start and stop sonification
P Repeat sonification output
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Solution.
Setting \(f(x,y)=4\text{:}\)
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\(\sqrt{x^2+y^2}=4\text{.}\)
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This implies \(x^2+y^2=16\text{,}\) which describes the circle centered at the origin with radius 4.
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(d)
Find the limit as \((x,y) \to (3,-4)\) of \(f(x,y)\text{.}\)
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Hint.
To find the limit, since the function is continuous at the point, substitute the point into the function.
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Answer.
\(\lim_{(x,y) \to (3,-4)} f(x,y)=5\)
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Solution.
we can substitute the point into the function since \(f(3,-4)\) is defined:
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\(f(3,-4)=\sqrt{3^2+(-4)^2}=\sqrt{9+16}=\sqrt{25}=5\text{.}\)
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3.

Find the limit as \(\lim_{(x,y) \to (0,0)} f(x,y)=\frac{xy}{x^2+y^4}\text{.}\)
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Hint.
Since the function is indeterminate at the origin, approach along different paths to determine the limit.
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Answer.
\(\lim_{(x,y) \to (0,0)} \frac{xy}{x^2+y^4}\) does not exist.
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Solution.
Approaching the origin along the path \(y=x\text{:}\)
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\(\lim_{x \to 0} \frac{x \cdot x}{x^2+x^4} = \lim_{x \to 0} \frac{x^2}{x^2(1+x^2)} = \lim_{x \to 0} \frac{1}{1+x^2} = 1\text{.}\)
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Approaching the origin along the path \(y=x^2\text{:}\)
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\(\lim_{x \to 0} \frac{x \cdot x^2}{x^2+(x^2)^4} = \lim_{x \to 0} \frac{x^3}{x^2+x^8} = \lim_{x \to 0} \frac{x}{1+x^6} = 0\text{.}\)
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Since the limits along different paths are not equal, the limit does not exist.
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