Mobile Notice
You appear to be on a device with a "narrow" screen width (i.e. you are probably on a mobile phone). Due to the nature of the mathematics on this site it is best viewed in landscape mode. If your device is not in landscape mode many of the equations will run off the side of your device (you should be able to scroll/swipe to see them) and some of the menu items will be cut off due to the narrow screen width.
Assignment Problems Notice
Please do not email me to get solutions and/or answers to these problems. I will not give them out under any circumstances nor will I respond to any requests to do so. The intent of these problems is for instructors to use them for assignments and having solutions/answers easily available defeats that purpose.
If you are looking for some problems with solutions you can find some by clicking on the "Practice Problems" link above.
If you are looking for some problems with solutions you can find some by clicking on the "Practice Problems" link above.
Section 13.7 : Directional Derivatives
For problems 1 – 4 determine the gradient of the given function.
- \(f\left( {x,y} \right) = {y^3}{x^5} + \ln \left( {xy} \right)\)
- \(f\left( {x,y} \right) = {{\bf{e}}^{\frac{x}{y}}} + {y^4}\sin \left( {xy} \right)\)
- \(\displaystyle f\left( {x,y,z} \right) = 4z - \frac{{{y^4}}}{{2{z^3}}} + \sqrt {{x^3}} \left( {z - 1} \right)\)
- \(\displaystyle f\left( {x,y,z} \right) = \cos \left( {\frac{{xy}}{z}} \right) + {z^3}{y^2}x\)
For problems 5 – 8 determine \({D_{\vec u}}f\) for the given function in the indicated direction.
- \(f\left( {x,y} \right) = \ln \left( {2xy} \right) - \sin \left( {{x^2} + {y^2}} \right)\) in the direction of \(\vec v = \left\langle {7, - 3} \right\rangle \)
- \(f\left( {x,y} \right) = 4{x^2}{y^3} - \sqrt {2x + 5y} \) in the direction of \(\vec v = \left\langle { - 1,4} \right\rangle \)
- \(\displaystyle f\left( {x,y,z} \right) = 8x{y^2} - \frac{{5{z^2}}}{x} + {y^4}\) in the direction of \(\vec v = \left\langle { - 4,1,2} \right\rangle \)
- \(\displaystyle f\left( {x,y,z} \right) = \frac{{3x}}{{{y^2} - {z^3}}} + 5{x^2} - 8y\) in the direction of \(\vec v = \left\langle {0,3, - 2} \right\rangle \)
- Determine \({D_{\vec u}}f\left( { - 1,4,6} \right)\) for \(f\left( {x,y,z} \right) = {{\bf{e}}^{x\,y{\,^2}}} + 4z{y^3}\) direction of \(\vec v = \left\langle {2, - 3,6} \right\rangle \).
- Determine \({D_{\vec u}}f\left( {8,1,2} \right)\) for \(\displaystyle f\left( {x,y,z} \right) = \ln \left( {\frac{x}{z}} \right) + \ln \left( {\frac{z}{y}} \right) + {y^2}x\) direction of \(\vec v = \left\langle {1,5,2} \right\rangle \).
For problems 11 – 13 find the maximum rate of change of the function at the indicated point and the direction in which this maximum rate of change occurs.
- \(f\left( {x,y} \right) = {{\bf{e}}^{4x\,y}}\) at \(\left( {6, - 2} \right)\)
- \(f\left( {x,y,z} \right) = {x^2}{y^4} - 3{z^2}x\) at \(\left( {1, - 6,3} \right)\)
- \(\displaystyle f\left( {x,y,z} \right) = \ln \left( {\frac{{2x + 3y}}{z}} \right)\) at \(\left( {2,7,4} \right)\)
- Given \(\displaystyle \vec u = \left\langle { - \frac{3}{5}, - \frac{4}{5}} \right\rangle \), \(\displaystyle \vec v = \left\langle {\frac{4}{{\sqrt {20} }},\frac{2}{{\sqrt {20} }}} \right\rangle \), \(\displaystyle \vec w = \left\langle { - \frac{3}{{\sqrt {11} }}, - \frac{2}{{\sqrt {11} }}} \right\rangle \), \(\displaystyle {D_{\vec u}}\left( { - 1,4} \right) = \frac{{14}}{5}\) and \(\displaystyle {D_{\vec v}}\left( { - 1,4} \right) = - \frac{{22}}{{\sqrt {20} }}\) determine the value of \({D_{\vec w}}\left( { - 1,4} \right)\).
- Given \(\displaystyle \vec u = \left\langle {\frac{1}{{\sqrt {15} }},\frac{4}{{\sqrt {15} }}} \right\rangle \), \(\displaystyle \vec v = \left\langle { - \frac{3}{{\sqrt {34} }}, - \frac{5}{{\sqrt {34} }}} \right\rangle \), \(\displaystyle \vec w = \left\langle { - \frac{1}{{\sqrt 2 }},\frac{1}{{\sqrt 2 }}} \right\rangle \), \(\displaystyle {D_{\vec u}}\left( {0,1} \right) = \frac{{18}}{{\sqrt {15} }}\) and \(\displaystyle {D_{\vec v}}\left( {0,1} \right) = - \frac{{40}}{{\sqrt {34} }}\) determine the value of \({D_{\vec w}}\left( {0,1} \right)\).