- Difficulty
- Introductory
- Suggested time
- 3 min
- Marks
- 2
Functions of two variables and partial derivatives practice question 2
Let . Which describes the level curve ?
A level curve is where equals the given value. What algebra would turn into a form from which you can read a circle’s centre and radius?
Complete the square in : . Then compare with the equation of a circle, .
Theory UBC Math 100 §14.3Key results
Worked solution
Strategy
A level curve is the set of points where f takes one fixed value. Set f(x, y) = 5 and complete the square to recognise the curve.
Working
The level curve f = 5: every point where the surface is at height 5.
Complete the square, adding 4 to both sides.
A circle with centre (2, 0) and radius 3. Every level f = c with c > −4 is a circle about the same centre, so the surface is a bowl with its lowest point over (2, 0).
Answer
The answer is The circle with centre (2, 0) and radius 3..
Check
The points (5, 0), (−1, 0) and (2, 3), each 3 from (2, 0), all give f = 5, for instance 4 − 8 + 9 = 5 at (2, 3).
Where this goes wrong
Adding 4 to complete the square on the left but not on the right, which gives (x − 2)² + y² = 5 and the wrong radius √5. Completing the square changes the equation unless both sides get the 4.
Leans on
Completing the square, from Functions, graphs and inverses
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