Question

Use the figure to estimate\({D_x}f(2,2).\)

Short answer

The directional derivative of the function f at the point \((2,2)\) is \({D_u}f(2,2) \approx - 2.83\).

Step-by-step solution

Definition of directional derivative

"The directional derivative of the function\(f(x,y,z)\)at\(f\left( {{x_0},{y_0},{z_0}} \right)\)in the direction of unit vector\(u = \langle a,b,c\rangle \)is\({D_u}f(x,y,z) = |\nabla f(x,y,z)| \cdot |u|(\cos \theta )\), where\(\nabla f(x,y,z) = \left\langle {{f_x},{f_y},{f_z}} \right\rangle = \frac{{\partial f}}{{\partial x}}i + \frac{{\partial f}}{{\partial y}}j + \frac{{\partial f}}{{\partial z}}k\)."

The value of directional derivative

From Figure 1, it can be observed that the value of \(\theta \) is \({225^^\circ }\).

The value of |u| is 1.

From Figure 1, it can be observed that the value of \(|\nabla f(2,2)|\) is approximately \(4\)

Substitute the respective value in \({D_u}f(2,2) = |\nabla f(2,2)| \cdot |u|(\cos \theta )\) and obtain required value.

\({D_u}f(2,2) = |\nabla f(2,2)| \cdot |u|(\cos \theta )\)

\( = (4)(1)\left( {\cos {{225}^^\circ }} \right) \approx 4( - 0.70710678) \approx - 2.83\)

Thus, the required directional derivative is, \({D_u}f(2,2) \approx - 2.83\).