Suppose you are climbing a hill whose shape is given by the...

Suppose you are climbing a hill whose shape is given by the equation $z=1000-0.005 x^{2}-0.01 y^{2},$ where $x, y,$ and $z$ are measured in meters, and you are standing at a point with coordinates $(60,40,966) .$ The positive $x$ -axis points east and the positive $y$ -axis points north. (a) If you walk due south, will you start to ascend or descend? At what rate? (b) If you walk northwest, will you start to ascend or descend? At what rate? (c) In which direction is the slope largest? What is the rate of ascent in that direction? At what angle above the hor- izontal does the path in that direction begin?

Suppose you are climbing a hill whose shape is given by the
equation $z=1000-0.005 x^{2}-0.01 y^{2},$ where $x, y,$ and $z$
are measured in meters, and you are standing at a point with
coordinates $(60,40,966) .$ The positive $x$ -axis points east
and the positive $y$ -axis points north.
(a) If you walk due south, will you start to ascend or
descend? At what rate?
(b) If you walk northwest, will you start to ascend or
descend? At what rate?
(c) In which direction is the slope largest? What is the rate
of ascent in that direction? At what angle above the hor-
izontal does the path in that direction begin?

Key Concepts

Gradient Vector

The gradient vector of a scalar function is a vector composed of its partial derivatives. It points in the direction of the steepest ascent and its magnitude represents the rate of maximum increase. In multivariable calculus, it is used to determine how the function changes at a given point on a surface.

Directional Derivative

The directional derivative measures the rate at which a function changes in a specified direction. It is computed as the dot product of the gradient vector with a unit vector in the desired direction. This concept is essential for understanding how the function’s value increases or decreases as one moves along a particular path.

Maximum Rate of Ascent

The maximum rate of ascent is given by the magnitude of the gradient vector. This value indicates the steepest incline at a particular point on the surface. The corresponding direction is provided by the gradient vector, which shows the path where the function increases most rapidly.

Angle of Inclination

The angle of inclination refers to the angle formed between the direction of the greatest slope (ascend) and the horizontal plane. When the maximum rate of ascent is known, this angle can be determined using trigonometric relationships, typically by taking the arctan of the maximum slope, which is significant in applications that involve elevation or decline along a path.

Transcript

00:01 In this problem, we are given the shown information, and we are asked to first pretend that we are walking in the southern direction, due south, in fact, and to determine whether that means we are going up a hill or down a hill.

00:19 And so to do that, we need first our gradient vector for the function.

00:24 And so we need to take z with respect to x, and that is going to be equal to negative, 0 .01x and the partial of z with respect to y is going to be equal to and again this this was the partial of z with respect to x and so the partial of z with respect to y is going to be equal to negative 0 .0 y now we're going to evaluate these at our given point now we really only need the 60 and 40 in this case because z is what or z is a function of x and y so what we're just going to use the 60 and 40 so i'm going to evaluate at this partial at the point 60 40 and i get negative 0 .01 times 60 and that is equal to negative 0 .6 and then i evaluate the negative 0 .0 2 y at the point 60 40 and so i get negative 0 .02 times 40 and so i get it, negative 0 .8.

01:38 And so that is our gradient vector, negative 0 .6, and negative 0 .8.

01:46 Now we need our direction vector.

01:48 Since they're telling us that we're walking due south, that means that we're not going south, or we're going straight down south, which means that our direction vector is going to be zero in the x direction and negative 1 in the y direction.

02:03 And because we know this and we have to make the length of this vector 1, we're going to divide it by the square root of the sum of each of its components squared.

02:11 So 0 squared plus negative 1 squared.

02:14 0 squared is 0.

02:16 Negative 1 squared is 1.

02:17 So we just get squared of 1, which is 1, which we are going to divide into each component and just get 0 and negative 1 back.

02:28 And now we are going to take the dot product of these two vectors.

02:34 And when we do, we get negative .6 comma negative 0 .8 .coma .coma 0 .0 .6 times 0 is 0 .9 .8 times negative 0 .8.

02:50 And so our answer is going to be 0 .8.

02:56 And all it asks us if we're is if we're starting to ascend or descend.

03:02 And now we know that our label for this is going to be 0 .8 meters per every meter we travel forward.

03:16 Meaning we go up 0 .8 meters for every meter we go forward, and therefore we know that we are ascending, or we're going uphill.

03:34 And so that is the answer to part a, and our answer to part b.

03:40 It asks us about walking northwest.

03:44 And so if we go northwest, that's going to be in this direction, that is going to have a directional vector of negative 1 ,000.

03:57 Because we're going negative 1 in the x direction and 1 in the y direction because that's how much we're going up and we've already found our gradient vector up here and so we're going to take the dot product of that with our new vector once we make its length of 1 and so i'm going to divide it by the square root of the sound of each of its component squared so negative 1 squared plus 1 squared negative 1 squared is 1 1 squared is 1 so the square root of 2 and so i get negative 1 over root 2 comma 1 over root 2.

04:32 And again, i'm going to take the dot product of this vector with that new vector, so negative 0 .6, negative 0 .8.

04:46 And i'm going to write those in fractions.

04:48 So negative 0 .6 is negative 3 fifths...

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