Level curves of a function of two variables: their equations, and the one through a given point. 60 problems with worked solutions; in 60 of them every
equation is proved by a computer algebra system.
20 worked problems
Try each one before opening the solution. Every line is labeled with what checked it. How we verify
Find the equation of the level curve of \( \displaystyle f(x, y) = 3 x + 2 y \) that passes through \( \displaystyle (2, 1) \), and describe it.
A level curve is f(x, y) = c; the point fixes c.
Reviewed
\[ \left. 3 x + 2 y \right|_{\substack{ x=2\\ y=1 }} = 8 \]
c = f(2, 1).✓ Proved
So the level curve is Eq(3*x + 2*y, 8), a line.
Reviewed
Answer \( 3 x + 2 y = 8\ (\text{a line}) \)
A contour map of f(x, y) = 3x + 2y: each curve joins points where f has the same value. The level curve through (2, 1) is highlighted.
✓ Nihil obstat Lines: 1 proved, 2 reviewed. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. The full receipt ·
How we verify
Find the equation of the level curve of \( \displaystyle f(x, y) = x y \) that passes through \( \displaystyle (-1, 3) \), and describe it.
A level curve is f(x, y) = c; the point fixes c.
Reviewed
\[ \left. x y \right|_{\substack{ x=-1\\ y=3 }} = -3 \]
c = f(-1, 3).✓ Proved
So the level curve is Eq(x*y, -3), a hyperbola.
Reviewed
Answer \( x y = -3\ (\text{a hyperbola}) \)
A contour map of f(x, y) = x·y: each curve joins points where f has the same value. The level curve through (-1, 3) is highlighted.
✓ Nihil obstat Lines: 1 proved, 2 reviewed. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. The full receipt ·
How we verify
A contour map of f(x, y) = -2x² + y: each curve joins points where f has the same value. The level curve through (1, 1) is highlighted.
✓ Nihil obstat Lines: 1 proved, 2 reviewed. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. The full receipt ·
How we verify
A contour map of f(x, y) = -x² + y²: each curve joins points where f has the same value. The level curve through (2, 1) is highlighted.
✓ Nihil obstat Lines: 1 proved, 2 reviewed. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. The full receipt ·
How we verify
Find the equation of the level curve of \( \displaystyle f(x, y) = 3 x^{2} + 4 y^{2} \) that passes through \( \displaystyle (2, 1) \), and describe it.
A contour map of f(x, y) = 3x² + 4y²: each curve joins points where f has the same value. The level curve through (2, 1) is highlighted.
✓ Nihil obstat Lines: 1 proved, 2 reviewed. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. The full receipt ·
How we verify
Find the equation of the level curve of \( \displaystyle f(x, y) = x y \) that passes through \( \displaystyle (2, 1) \), and describe it.
A level curve is f(x, y) = c; the point fixes c.
Reviewed
\[ \left. x y \right|_{\substack{ x=2\\ y=1 }} = 2 \]
c = f(2, 1).✓ Proved
So the level curve is Eq(x*y, 2), a hyperbola.
Reviewed
Answer \( x y = 2\ (\text{a hyperbola}) \)
A contour map of f(x, y) = x·y: each curve joins points where f has the same value. The level curve through (2, 1) is highlighted.
✓ Nihil obstat Lines: 1 proved, 2 reviewed. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. The full receipt ·
How we verify
Find the equation of the level curve of \( \displaystyle f(x, y) = - x^{2} + y^{2} \) that passes through \( \displaystyle (2, -1) \), and describe it.
A contour map of f(x, y) = -x² + y²: each curve joins points where f has the same value. The level curve through (2, -1) is highlighted.
✓ Nihil obstat Lines: 1 proved, 2 reviewed. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. The full receipt ·
How we verify
Find the equation of the level curve of \( \displaystyle f(x, y) = 3 x + 3 y \) that passes through \( \displaystyle (2, -1) \), and describe it.
A level curve is f(x, y) = c; the point fixes c.
Reviewed
\[ \left. 3 x + 3 y \right|_{\substack{ x=2\\ y=-1 }} = 3 \]
c = f(2, -1).✓ Proved
So the level curve is Eq(3*x + 3*y, 3), a line.
Reviewed
Answer \( 3 x + 3 y = 3\ (\text{a line}) \)
A contour map of f(x, y) = 3x + 3y: each curve joins points where f has the same value. The level curve through (2, -1) is highlighted.
✓ Nihil obstat Lines: 1 proved, 2 reviewed. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. The full receipt ·
How we verify
Find the equation of the level curve of \( \displaystyle f(x, y) = - 4 x^{2} - 3 y^{2} + 20 \) that passes through \( \displaystyle (-1, 3) \), and describe it.
A contour map of f(x, y) = -4x² - 3y² + 20: each curve joins points where f has the same value. The level curve through (-1, 3) is highlighted.
✓ Nihil obstat Lines: 1 proved, 2 reviewed. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. The full receipt ·
How we verify
Find the equation of the level curve of \( \displaystyle f(x, y) = x + 3 y \) that passes through \( \displaystyle (-1, 3) \), and describe it.
A level curve is f(x, y) = c; the point fixes c.
Reviewed
\[ \left. x + 3 y \right|_{\substack{ x=-1\\ y=3 }} = 8 \]
c = f(-1, 3).✓ Proved
So the level curve is Eq(x + 3*y, 8), a line.
Reviewed
Answer \( x + 3 y = 8\ (\text{a line}) \)
A contour map of f(x, y) = x + 3y: each curve joins points where f has the same value. The level curve through (-1, 3) is highlighted.
✓ Nihil obstat Lines: 1 proved, 2 reviewed. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. The full receipt ·
How we verify
Find the equation of the level curve of \( \displaystyle f(x, y) = - 3 x^{2} - y^{2} + 16 \) that passes through \( \displaystyle (2, -1) \), and describe it.
A contour map of f(x, y) = -3x² - y² + 16: each curve joins points where f has the same value. The level curve through (2, -1) is highlighted.
✓ Nihil obstat Lines: 1 proved, 2 reviewed. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. The full receipt ·
How we verify
A contour map of f(x, y) = -2x² + y: each curve joins points where f has the same value. The level curve through (-1, 3) is highlighted.
✓ Nihil obstat Lines: 1 proved, 2 reviewed. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. The full receipt ·
How we verify
A contour map of f(x, y) = x² + 3y²: each curve joins points where f has the same value. The level curve through (1, 1) is highlighted.
✓ Nihil obstat Lines: 1 proved, 2 reviewed. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. The full receipt ·
How we verify
Find the equation of the level curve of \( \displaystyle f(x, y) = - x^{2} + y^{2} \) that passes through \( \displaystyle (-1, 3) \), and describe it.
A contour map of f(x, y) = -x² + y²: each curve joins points where f has the same value. The level curve through (-1, 3) is highlighted.
✓ Nihil obstat Lines: 1 proved, 2 reviewed. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. The full receipt ·
How we verify
Find the equation of the level curve of \( \displaystyle f(x, y) = x + 2 y \) that passes through \( \displaystyle (0, 1) \), and describe it.
A level curve is f(x, y) = c; the point fixes c.
Reviewed
\[ \left. x + 2 y \right|_{\substack{ x=0\\ y=1 }} = 2 \]
c = f(0, 1).✓ Proved
So the level curve is Eq(x + 2*y, 2), a line.
Reviewed
Answer \( x + 2 y = 2\ (\text{a line}) \)
A contour map of f(x, y) = x + 2y: each curve joins points where f has the same value. The level curve through (0, 1) is highlighted.
✓ Nihil obstat Lines: 1 proved, 2 reviewed. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. The full receipt ·
How we verify
Find the equation of the level curve of \( \displaystyle f(x, y) = 3 x + 3 y \) that passes through \( \displaystyle (-1, 3) \), and describe it.
A level curve is f(x, y) = c; the point fixes c.
Reviewed
\[ \left. 3 x + 3 y \right|_{\substack{ x=-1\\ y=3 }} = 6 \]
c = f(-1, 3).✓ Proved
So the level curve is Eq(3*x + 3*y, 6), a line.
Reviewed
Answer \( 3 x + 3 y = 6\ (\text{a line}) \)
A contour map of f(x, y) = 3x + 3y: each curve joins points where f has the same value. The level curve through (-1, 3) is highlighted.
✓ Nihil obstat Lines: 1 proved, 2 reviewed. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. The full receipt ·
How we verify
Find the equation of the level curve of \( \displaystyle f(x, y) = 4 x + y \) that passes through \( \displaystyle (0, 1) \), and describe it.
A level curve is f(x, y) = c; the point fixes c.
Reviewed
\[ \left. 4 x + y \right|_{\substack{ x=0\\ y=1 }} = 1 \]
c = f(0, 1).✓ Proved
So the level curve is Eq(4*x + y, 1), a line.
Reviewed
Answer \( 4 x + y = 1\ (\text{a line}) \)
A contour map of f(x, y) = 4x + y: each curve joins points where f has the same value. The level curve through (0, 1) is highlighted.
✓ Nihil obstat Lines: 1 proved, 2 reviewed. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. The full receipt ·
How we verify
Find the equation of the level curve of \( \displaystyle f(x, y) = x y \) that passes through \( \displaystyle (2, -1) \), and describe it.
A level curve is f(x, y) = c; the point fixes c.
Reviewed
\[ \left. x y \right|_{\substack{ x=2\\ y=-1 }} = -2 \]
c = f(2, -1).✓ Proved
So the level curve is Eq(x*y, -2), a hyperbola.
Reviewed
Answer \( x y = -2\ (\text{a hyperbola}) \)
A contour map of f(x, y) = x·y: each curve joins points where f has the same value. The level curve through (2, -1) is highlighted.
✓ Nihil obstat Lines: 1 proved, 2 reviewed. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. The full receipt ·
How we verify
Find the equation of the level curve of \( \displaystyle f(x, y) = 4 x + y \) that passes through \( \displaystyle (2, -1) \), and describe it.
A level curve is f(x, y) = c; the point fixes c.
Reviewed
\[ \left. 4 x + y \right|_{\substack{ x=2\\ y=-1 }} = 7 \]
c = f(2, -1).✓ Proved
So the level curve is Eq(4*x + y, 7), a line.
Reviewed
Answer \( 4 x + y = 7\ (\text{a line}) \)
A contour map of f(x, y) = 4x + y: each curve joins points where f has the same value. The level curve through (2, -1) is highlighted.
✓ Nihil obstat Lines: 1 proved, 2 reviewed. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. The full receipt ·
How we verify
So the level curve is Eq(-x**2 + y, 1), a parabola.
Reviewed
Answer \( - x^{2} + y = 1\ (\text{a parabola}) \)
A contour map of f(x, y) = -x² + y: each curve joins points where f has the same value. The level curve through (0, 1) is highlighted.
✓ Nihil obstat Lines: 1 proved, 2 reviewed. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. The full receipt ·
How we verify