 ## MATH1111 Quizzes

Local Linearity and the Differential Quiz
Web resources available Questions

This quiz tests the work covered in lecture on local linearity and the differential and corresponds to Section 14.3 of the textbook Calculus: Single and Multivariable (Hughes-Hallett, Gleason, McCallum et al.).
There is a useful applet at http://www.slu.edu/classes/maymk/banchoff/TangentPlane.html - take some time to read the instructions and add your own functions.

There are more web quizzes at Wiley, select Section 3. This quiz has 10 questions.

Suppose $f\left(3,2\right)=4\phantom{\rule{0.3em}{0ex}},\phantom{\rule{1em}{0ex}}{f}_{x}\left(3,2\right)=-2$ and ${f}_{y}\left(3,2\right)=3$ for some surface $z=f\left(x,y\right)\phantom{\rule{0.3em}{0ex}}.$
Which of the following is the tangent plane to the surface at $\left(3,2,4\right)\phantom{\rule{0.3em}{0ex}}?$ Exactly one option must be correct)
 a) $4z=-2\left(x-3\right)+3\left(y-2\right)$ b) $z=4-2\left(x-3\right)+3\left(y-2\right)$ c) $z+4=2\left(x+3\right)+3\left(y+2\right)$ d) $4z=3\left(x+3\right)-2\left(y+2\right)$

Choice (a) is incorrect
Try again, check the formula for the tangent plane.
Choice (b) is correct!
The tangent at the point $\left(a,b\right)$ on the surface is $z=f\left(a,b\right)+{f}_{x}\left(a,b\right)\left(x-a\right)+{f}_{\left(}a,b\right)\left(y-b\right)$ so the above equation is correct.
Choice (c) is incorrect
Try again, check the formula for the tangent plane.
Choice (d) is incorrect
Try again, check the formula for the tangent plane.
Is the plane $z=12+8\left(x-1\right)+7\left(y-2\right)$ the tangent plane to the surface, $f\left(x,y\right)={x}^{2}+3xy+{y}^{2}-1$ at $\left(1,2\right)\phantom{\rule{0.3em}{0ex}}?$ Exactly one option must be correct)
 a) Yes. b) No

Choice (a) is incorrect
$f\left(1,2\right)=10$ and $z=12$ at $\left(1,2\right)$ so the plane does not touch the surface.
Choice (b) is correct!
${f}_{x}\left(x,y\right)=2x+3y$ so ${f}_{x}\left(1,2\right)=2+6=8$
${f}_{y}\left(x,y\right)=3x+2y$ so ${f}_{y}\left(1,2\right)=3+4=7$
$f\left(1,2\right)=10$ so the tangent plane is $z=10+8\left(x-1\right)+7\left(y-2\right)\phantom{\rule{0.3em}{0ex}}.$
Which of the following is the tangent plane to the surface $f\left(x,y\right)={x}^{2}-2xy-3{y}^{2}$ at the point $\left(-2,1,5\right)\phantom{\rule{0.3em}{0ex}}?$ Exactly one option must be correct)
 a) $z+6x+2y+15=0$ b) $z-6x-2y+5=0$ c) $z+6x+2y+5=0\phantom{\rule{0.3em}{0ex}}.$ d) None of the above, since $\left(-2,1,5\right)$ is not on the surface.

Choice (a) is incorrect
Try again, look carefully at the signs of the constant terms.
Choice (b) is incorrect
Try again, carefully rearrange your equation.
Choice (c) is correct!
${f}_{x}\left(x,y\right)=2x-2y$ so ${f}_{x}\left(-2,1\right)=-4-2=-6$
${f}_{y}\left(x,y\right)=-2x-6y$ so ${f}_{y}\left(-2,1\right)=4-6=-2$
$f\left(-2,1\right)=5$ so the tangent plane is $z=5-6\left(x+2\right)-2\left(y-1\right)⇒z+6x+2y+5=0$ as required.
Choice (d) is incorrect
Try again, the point is on the surface.
Which of the following is the differential of $f\left(x,y\right)=sinxy\phantom{\rule{0.3em}{0ex}}{e}^{xy}\phantom{\rule{0.3em}{0ex}}?$ Exactly one option must be correct)
 a) $df=cosxy\phantom{\rule{0.3em}{0ex}}{e}^{xy}\left({y}^{2}\phantom{\rule{0.3em}{0ex}}dx+{x}^{2}\phantom{\rule{0.3em}{0ex}}dy\right)$ b) $df={e}^{xy}\left(cosxy+sinxy\right)\left(x\phantom{\rule{0.3em}{0ex}}dx+y\phantom{\rule{0.3em}{0ex}}dy\right)$ c) $df={e}^{xy}\left(-cosxy+sinxy\right)\left(y\phantom{\rule{0.3em}{0ex}}dx+x\phantom{\rule{0.3em}{0ex}}dy\right)$ d) $df={e}^{xy}\left(cosxy+sinxy\right)\left(y\phantom{\rule{0.3em}{0ex}}dx+x\phantom{\rule{0.3em}{0ex}}dy\right)$

Choice (a) is incorrect
Try again, you must use the product rule to differentiate $f\left(x,y\right)\phantom{\rule{0.3em}{0ex}}.$
Choice (b) is incorrect
Try again, you have not differentiated $f\left(x,y\right)$ correctly.
Choice (c) is incorrect
Try again, you have not differentiated $sinxy$ correctly.
Choice (d) is correct!
${f}_{x}\left(x,y\right)=ycosxy\phantom{\rule{0.3em}{0ex}}{e}^{xy}+sinxy\left(y{e}^{xy}\right)=y{e}^{xy}\left(cosxy+sinxy\right)$ using the product rule, and

${f}_{y}\left(x,y\right)=xcosxy\phantom{\rule{0.3em}{0ex}}{e}^{xy}+sinxy\left(x{e}^{xy}\right)=x{e}^{xy}\left(cosxy+sinxy\right)$ using the product rule, so

$df=y{e}^{xy}\left(cosxy+sinxy\right)\phantom{\rule{0.3em}{0ex}}dx+x{e}^{xy}\left(cosxy+sinxy\right)\phantom{\rule{0.3em}{0ex}}dy={e}^{xy}\left(cosxy+sinxy\right)\left(y\phantom{\rule{0.3em}{0ex}}dx+x\phantom{\rule{0.3em}{0ex}}dy\right)\phantom{\rule{0.3em}{0ex}}.$