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Section 2-7 : Directional Derivatives

For problems 1 & 2 determine the gradient of the given function.

  1. \(\displaystyle f\left( {x,y} \right) = {x^2}\sec \left( {3x} \right) - \frac{{{x^2}}}{{{y^3}}}\) Solution
  2. \(f\left( {x,y,z} \right) = x\cos \left( {xy} \right) + {z^2}{y^4} - 7xz\) Solution

For problems 3 & 4 determine \({D_{\vec u}}f\) for the given function in the indicated direction.

  1. \(\displaystyle f\left( {x,y} \right) = \cos \left( {\frac{x}{y}} \right)\) in the direction of \(\vec v = \left\langle {3, - 4} \right\rangle \) Solution
  2. \(f\left( {x,y,z} \right) = {x^2}{y^3} - 4xz\) in the direction of \(\vec v = \left\langle { - 1,2,0} \right\rangle \) Solution
  3. Determine \({D_{\vec u}}f\left( {3, - 1,0} \right)\) for \(f\left( {x,y,z} \right) = 4x - {y^2}{{\bf{e}}^{3x\,z}}\) direction of \(\vec v = \left\langle { - 1,4,2} \right\rangle \). Solution

For problems 6 & 7 find the maximum rate of change of the function at the indicated point and the direction in which this maximum rate of change occurs.

  1. \(f\left( {x,y} \right) = \sqrt {{x^2} + {y^4}} \) at \(\left( { - 2,3} \right)\) Solution
  2. \(f\left( {x,y,z} \right) = {{\bf{e}}^{2x}}\cos \left( {y - 2z} \right)\) at \(\left( {4, - 2,0} \right)\) Solution