The following diagram shows some level curves of the...

Name: Problem 120, Chapter 15: Finite Mathematics and Applied Calculus
Uploaded: 2021-10-19T12:53:22-08:00
Duration: 1 min 6 s
Channel: Harshita Goel
Description: The following diagram shows some level curves of the altitude of a mountain valley, as well as the location, on the 2,000 -ft curve, of a hiker. The hiker is currently moving at the greatest possible rate of descent. In which of the three directions shown is he moving? Explain your answer.

The following diagram shows some level curves of the altitude of a mountain valley, as well as the location, on the 2,000 -ft curve, of a hiker. The hiker is currently moving at the greatest possible rate of descent. In which of the three directions shown is he moving? Explain your answer.

Labs

Want to see this concept in action?

NEW

Explore this concept interactively to see how it behaves as you change inputs.

View Labs

Key Concepts

Gradient

The gradient of a function is a vector that points in the direction of the greatest rate of increase. It is composed of the partial derivatives of the function with respect to each variable, and its magnitude indicates how steep the climb is. In problems involving altitude, the gradient shows the direction in which the height increases most rapidly.

Level Curves

Level curves, or contour lines, are curves along which the function (in this case, altitude) remains constant. They are useful in visualizing a function in two dimensions and help identify areas of equal altitude. The gradient is always perpendicular to the level curves, which is key in determining directions such as steepest ascent or descent.

Steepest Descent Direction

The direction of steepest descent is exactly opposite to the gradient. Since the gradient indicates the direction of greatest increase, moving opposite to it guarantees the maximum rate of decrease. This means that when descending a mountain valley, a hiker following the steepest descent will be moving perpendicular to the level curves but in the opposite direction to the gradient.

Transcript

Please give Ace some feedback