Practices on Gradients and etc.

  1. Find the graident vector for the following functions:

    1. MATH MATH

    2. MATH

    3. MATH

  2. Find the graident vector for the following functions:

    1. MATH at $(2,3).$ [answer: $(-1,18)$ ]

    2. MATH at $(3,1).$ [answer: $\frac{5}{2}]$

    3. MATH at MATH [answer: (-1,0)]

    4. MATH at $(1,0,\frac{1}{2}).$ [answer: $(0,0,-\pi )]$

  3. Find the directional derivative at the given point and given direction for the following functions:

    1. MATH at $(0,0)$ in the direction of $(2,1).$ [answer: $\sqrt{5}]$

    2. MATH at $(1,0)$ in the direction of $(3,4).$ [answer: $\dfrac{7-4e}{5}]$

    3. $f(x,y,z)=xy+yz+zx$ at $(1,-1,1)$ in the direction of $(1,2,1)$ [answer: MATH

    4. MATH at $(1,-1,1)$ in the direction of $i+j.$ [answer: $-3\sqrt{2}]$

  4. Find the direction derivative for MATH in the direction from $(x,y)$ toward the origin. [answer: MATH

  5. Find the direction derivative for MATH at $(0,1)$ toward $(-1,3).$ [answer: MATH

  6. If MATH

    1. Find the direction where the direction derivative changes the most at $(2,-1,1).$ [answer: $(2,-6,9)]$

    2. Find the largest change for the direction derivative at $(2,-1,1).$ [answer: 11]

  7. Let MATH is a path on the level surface of f(x,y,z)=0, and $P=(1,2,-2)$ is a point on $r(t).$Find the directional derivative for $f$ at point $P$ when following the path of $r(t).$ [hint: MATH thus MATH $u;$ MATH MATH

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