The contours of the function $f(x,y) = e^{-x^2-y^2}$ consist of

Question

BlackTom AI · Accepted Answer

Level sets are defined by f(x,y) = constant. Set e^{-x^2 - y^2} = k (with 0 < k ≤ 1 because e^{-x^2-y^2} ranges from 0 (exclusive) to 1).

1) Relevant concepts
- A contour (level curve) of f is the set {(x,y) : f(x,y) = k} for fixed k.
- Algebraic manipulation of exponentials and logarithms converts the level condition to a familiar geometric equation (circle).

2) Step-by-step reasoning
Start with the level condition:
e^{-x^2 - y^2} = k.

3) Calculations and intermediate steps
Take natural logarithms of both sides (valid because k>0):
-ln(e^{x^2+y^2})? (better to write directly)
- x^2 - y^2 = ln k.
Multiply by -1:
x^2 + y^2 = - ln k.

Since 0 < k ≤ 1, ln k ≤ 0, so -ln k ≥ 0. Define R^2 = -ln k, so R = sqrt(-ln k) (R ≥ 0). Thus the level set is
x^2 + y^2 = R^2,
which is a circle of radius R centered at the origin. Varying k produces different radii R, all centered at the origin, hence concentric circles. Note the special case k = 1 gives R = 0 (the single point at the origin); as k → 0+, R → ∞.

4) Why the correct answer is correct
The algebra yields the equation of a circle centered at the origin for every admissible k, so contours are circles sharing the same center (concentric).

5) Why other options are incorrect
- b (parallel lines with gradient 1): level sets are not linear; the equation is quadratic x^2 + y^2 = const, not of the form y = x + c.
- c (exponential curves): after taking logs the relation is polynomial (circle), not an exponential graph.
- d (lines parallel to the x-axis): would require y = constant; the actual relation ties x and y symmetrically as x^2 + y^2 = constant.

6) Conclusion
Therefore the contours consist of concentric circles, matching answer a.

The contours of the function \(f(x,y) = e^{-x^2-y^2}\) consist of单项选择题

类似问题

The contours of the function \(f(x,y) = e^{-x^2-y^2}\) consist of

The contours of the function \(f(x,y) = e^{-x^2-y^2}\) consist of

The contours of the function [math: f(x,y)=e−x2−y2] consist of

You own a bond that pays $64 in interest annually. The face value is $1,000 and the current market price is $1,021.61. The bond matures in 11 years. What is the yield to maturity?

Which of the following statements is CORRECT?

更多留学生实用工具

智能学习助手

风格化写作助手

论文查重助手

文献引用助手

课堂转译助手

课堂笔记助手

Quiz搜索助手

学校历年真题

智能刷题助手

智能匹配练习

希望你的学习变得更简单