2.7 Directional Derivatives and the Gradient

$f(x,y)=3x-4y$ at the point $(0,2)$ in the direction $-2\hi$.

$f(x,y,z)=x^{-1}+y^{-1}+z^{-1}$ at $(2,-3,4)$ in the direction $\hi+\hj+\hk$.

$-1$

$-2$

$-3$

Find the gradient vector of $w = f(x,y,z)$ at $P$.

Find the directional derivative of $w = f(x,y,z)$ at $P$ in the direction of the vector $\hi + \hj$

In what direction should the hiker proceed from $P$ to ascend along the steepest path? What is the slope of the path?

Walking north from $P$, will the hiker start to ascend or descend? What is the slope?

In what direction should the hiker walk from $P$ to remain at the same height?

She wishes to stay at the same temperature, but must fly in some initial direction. Find a direction in which the initial rate of change of the temperature is $0$.

If you and another student both get correct answers in part (a), must the directions you give be the same? Why or why not?

What initial direction or directions would suit the mosquito if she wanted to cool down as fast as possible?

A bird passes through $(2,1,3)$ travelling towards $(4,3,4)$ with speed $2$. At what rate does the air temperature it experiences change at this instant?

If instead the bird maintains constant altitude ($z = 3$) as it passes through $(2,1,3)$ while also keeping at a fixed air temperature, $T = 8$, what are its two possible directions of travel?

Find the maximum rate of change of $f(x,y)$ at the point $P\left(1, -\frac{4}{3}\right)$.

Find the directions in which the directional derivative of $f(x,y)$ at the point $P\left(1, -\frac{4}{3}\right)$ has the value $\frac{1}{5}$.

Find the directional derivative of $T$ at $(3, 2, 1)$, in the direction of the point $(0,1,2)$.

At the point $(3, 2, 1)$, in what direction does the temperature decrease most rapidly?

Moving along the curve given by $x=3e^t$, $y = 2 \cos t$, $z= \sqrt{1 + t}$, find $\diff{T}{t}$, the rate of change of temperature with respect to $t$, at $t = 0$.

Suppose $\hi+5\hj+a\hk$ is a vector that is tangent to the temperature level surface $T(x, y, z) = 3$ at $(3, 2, 1)$. What is $a$?

Find the gradients of $f$ and $g$ at $(0,1,-1)$.

A bird at $(0,1,-1)$ flies at speed $6$ in the direction in which $f(x, y, z)$ increases most rapidly. As it passes through $(0,1,-1)$, how quickly does $g(x, y, z)$ appear (to the bird) to be changing?

A bat at $(0,1,-1)$ flies in the direction in which $f (x, y, z)$ and $g(x, y, z)$ do not change, but $z$ increases. Find a vector in this direction.

Find the velocity (vector) of the bee at time $t = 2$.

The temperature $T$ at position $(x, y, z)$ at time $t$ is given by $T = xy - 3x+2yt+z$. Find the rate of change of temperature experienced by the bee at time $t = 2$.

Find the rate of change of temperature at the point $P(1, 2, -1)$ in the direction toward the point $(1, 1, 0)$.

In which direction does the temperature decrease most rapidly?

Find the maximum rate of decrease at $P$.

What is the direction of the steepest ascent at the point $(0, 100, 900)$? (The answer should be in terms of directions of the compass).

What is the slope of the hill at the point $(0, 100, 900)$ in the direction from (a)?

If you ride a bicycle on this hill in the direction of the steepest descent at $5$ m/s, what is the rate of change of your altitude (with respect to time) as you pass through the point (0, 100, 900)?

If the position of an airplane at time $t$ is

find $\diff{}{t} (PT)^2$ at time $t = 0$ as observed from the airplane.

In which direction should a bird at the point $(0,-1,1)$ fly if it wants to keep both $P$ and $T$ constant. (Give one possible direction vector. It does not need to be a unit vector.)

An ant crawls on the surface $z^3 + zx + y^2 = 2$. When the ant is at the point $(0,-1,1)$, in which direction should it go for maximum increase of the temperature $T = 5 + xy - z^2$? Your answer should be a vector $\llt a, b, c\rgt$, not necessarily of unit length. (Note that the ant cannot crawl in the direction of the gradient because that leads off the surface. The direction vector $\llt a, b, c\rgt$ has to be on the tangent plane to the surface.)

Find the unit vector $\vu=\llt u_1,u_2\rgt$ that maximizes

Find a vector $\vv=\llt v_1,v_2,v_3\rgt$ pointing in the direction of the path that the ant could take in order to stay on the same elevation level $e^{-2}$.

Find a vector $\vv=\llt v_1,v_2,v_3\rgt$ pointing in the direction of the path that the ant should take in order to maximize its instantaneous rate of level increase.

A sparrow is flying along the curve $\vr(s)=\big(\frac{1}{3}s^3,2s,s^2\big)$ at a constant speed of $3{\rm ms}^{-1}$. What is the velocity of the sparrow when $s=1$?

At what rate does the sparrow feel the temperature is changing at the point $A\big(\frac{1}{3},2,1\big)$ for which $s=1$.

At the point $A\big(\frac{1}{3},2,1\big)$ in what direction will the temperature be decreasing at maximum rate?

An eagle crosses the path of the sparrow at $A\big(\frac{1}{3},2,1\big)$, is moving at right angles to the path of the sparrow, and is also moving in a direction in which the temperature remains constant. In what directions could the eagle be flying as it passes through the point $A$?

Find the velocity vector $\vv$ of the moth at this moment.

What rate of change of temperature does the moth feel at that moment?

Find the constant of proportionality.

Find the rate of change of $T$ at $(1,2,2)$ in the direction towards the point $(2,1,3)$.

Show that at most points in the ball, the direction of greatest increase is towards the origin.

Sketch the shoreline of the lake in the $xy$-plane.

stay at a constant depth?

increase his depth as rapidly as possible (i.e. be most likely to drown)?

Draw a contour diagram for $T$ showing some isotherms (curves of constant temperature).

In what direction should an ant at position $(2,-1)$ move if it wishes to cool off as quickly as possible?

If the ant moves in that direction at speed $v$ at what rate does its temperature decrease?

What would the rate of decrease of temperature of the ant be if it moved from $(2,-1)$ at speed $v$ in direction $\llt -1,-2\rgt$?

Along what curve through $(2,-1)$ should the ant move to continue experiencing maximum rate of cooling?

Give the direction in which $f$ is increasing the fastest at the point $(1, 0, \pi/2)$.

Give an equation for the plane $T$ tangent to the surface

at the point $(1, 0, \pi/2)$.

Find the distance between $T$ and the point $(0, 1, 0)$.

Find the angle between the plane $T$ and the plane

A bee starts flying at $P$ and flies along the unit vector pointing towards the point $Q = (3,2,2)$. What is the rate of change of $T(x,y,z)$ in this direction?

Use the linear approximation of $T$ at the point $P$ to approximate $T(1.9,1,1.2)$.

Let $S(x,y,z) = x + z$. A bee starts flying at $P$; along which unit vector direction should the bee fly so that the rate of change of $T(x,y,z)$ and of $S(x,y,z)$ are both zero in this direction?

You jump from $P$ to $Q(0.1\,,\,0.9\,,\,1.8)$. Use the linear approximation to determine approximately the amount by which $F$ changes.

You jump from $P$ in the direction along which $G$ increases most rapidly. Will $F$ increase or decrease?

You jump from $P$ in a direction $\llt a\,,\,b\,,\,c\rgt$ along which the rates of change of $F$ and $G$ are both zero. Give an example of such a direction (need not be a unit vector).

Sketch a map of the proposed site clearly marking the car park, a few level curves for the function $f$ and the trail.

In which direction does the trail leave the car park?

Suppose that you walk $3$ meters east and 2 meters north. At your new location, $(3,2)$, in what direction is the sand dune sloping most steeply downward?

If you walk north from the location described in part (a), what is the instantaneous rate of change of height of the sand dune?

If you are standing at $(3,2)$ in what direction should you walk to ensure that you remain at the same height?

Find the equation of the curve through $(3,2)$ that you should move along in order that you are always pointing in a steepest descent direction at each point of this curve.

Show that the gradient vector $\vnabla f$ at $(1,2)$ is $10\hi+5\hj$.

Determine the rate of change of $f$ at $(1,2)$ in the direction of the vector $\hi+2\hj$.

Using the tangent plane approximation, estimate the value of $f(1.01,2.05)$.