Question

Sketch the graph of the function. $f(x, y)=e^{-y}$

   Sketch the graph of the function.
 $f(x, y)=e^{-y}$

Calculus

James Stewart 7th Edition

Chapter 14, Problem 26

Instant Answer

Solved by Expert Lucas Finney

08/13/2021

Step 1

The function $f(x, y) = e^{-y}$ is a function of two variables, which means its graph will be a surface in three-dimensional space. Show more…

Show all steps

lock

Thanks for your feedback!

Sketch the graph of the function. $f(x, y)=e^{-y}$

Play audio

Feedback

Powered by NumerAI

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

Graph of a Function of Two Variables

A graph of a function in two variables represents a surface in three-dimensional space defined by z = f(x, y). This visualization helps in understanding how the function assigns a real number (the height z) to every pair (x, y) in its domain.

Exponential Function and Decay

The exponential function is characterized by a constant relative growth (or decay) rate. In the function f(x, y) = e^(-y), the output exhibits exponential decay as the variable y increases since e^(-y) rapidly decreases toward zero for large positive values of y.

Independence from One Variable

In the given function, the variable x does not appear in the expression, which means the function's value is independent of x. As a result, for any fixed y, varying x does not change the output, leading to a graph that forms a surface constant in the x-direction.

Level Curves and Traces

Level curves are sets of points in the domain where the function takes on a constant value. For f(x, y) = e^(-y), these curves are horizontal lines in the x-y plane, indicating that the function remains constant along lines parallel to the x-axis, a reflection of the absence of x in the function.

Recommended Videos

Transcript

Ace is your personal tutor. It breaks down any question with clear steps so you can learn.

Start Using Ace

Continue solving this problem

Create a free account to:

View full step-by-step solution
Ask follow-up questions with Ace AI
Save progress and study later