9.6: Directional Derivatives and the Gradient

1. Find the directional derivative of $f(x, y) = xy$ at point $P(-2, 0)$ in the direction of $\mathbf{v} = \mathbf{i} + \sqrt{3}\,\mathbf{j}$.
    
2. Find the directional derivative of $g(x, y) = e^x\sin y$ at point $Q\left(1, \dfrac{\pi}{4}\right)$ in the direction of $\mathbf{u} = 4\mathbf{i} - 3\mathbf{j}$.
    
3. Find the directional derivative of $h(x, y) = \ln(x^2 + y^2)$ at point $P(1, 2)$ in the direction of $\mathbf{v} = (2, -5)$.
    
4. Find the directional derivative of $f(x, y, z) = y^2 + xz$ at point $P(1, 2, 2)$ in the direction of $\mathbf{u} = (2, -1, 2)$.
    
5. The temperature of a thin plate in the $xy$-plane is $T = x^2 + y^2$. How fast does the temperature change at the point $(1, 5)$ moving in the direction of $\mathbf{v} = (\sqrt{3}, 1)$?
    
6. Suppose the density of a thin plate at the point $(x, y)$ is $\rho(x, y) = \dfrac{1}{\sqrt{1 + x^2 + y^2}}$. Find the rate of change of the density at $(2, 1)$ in the direction $\mathbf{u} = -\mathbf{j}$.
    
7. In what direction does $f(x, y) = x^2 + xy + y^2$ increase most rapidly at the point $P(-5, -4)$? In what direction does $f(x, y)$ decrease most rapidly at $P$?
    
8. In what direction does $f(x, y) = \arctan\left(\dfrac{y}{x}\right)$ increase most rapidly at the point $P(-9, 9)$? In what direction does $f(x, y)$ decrease most rapidly at $P$?
    
9. Suppose the temperature at $(x, y, z)$ is given by $T = xy + \sin(yz)$. In what direction should you go from point $(1, 1, 1)$ to decrease the temperature as quickly as possible?
    
10. A bug is crawling on the surface of a hot plate. The temperature of the plate at point $(x, y)$ is given by $T(x, y) = 100 - x^2 - 3y^3$.
     
    1. If the bug is at the point $(2, 1)$, in what direction should it move to cool off the fastest? How fast will the temperature drop in this direction?
        
    2. If the bug is at the point $(1, 3)$, in what direction should it move in order to maintain its temperature?
        
11. The National Oceanic and Atmospheric Administration (NOAA) measures the atmospheric pressure inside a hurricane. The NOAA finds that the atmospheric pressure of the hurricane at point $(x, y, z)$ can be modeled by the function $\displaystyle P(x, y, z) = 2^{2(55-z)/11}\left(1 - \dfrac{1}{10(1 + x^2 + y^2)}\right)$. At point $(20, 80, 10)$, in what direction does the atmospheric pressure increase the fastest?