- #1
cosmos42
- 21
- 1
Homework Statement
ƒ(x,y) = ln(x2+4y2)
Homework Equations
I'm not really sure but I solved for y
The Attempt at a Solution
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Yes, that was a bit confusing in the OP, but judging from the drawing I would say cosmos42 understood about different values for C.HallsofIvy said:"Level curves" are curves on which the expression has a specific value. I would not solve for y. Instead, I would look at the equation [itex]ln(x^2+ 4y^2)= C[/itex] and recognize that, taking the exponential of both sides, [itex]x^2+ 4y^2= e^C[/itex] which is equivalent to [itex]\frac{x^2}{e^C}+ \frac{y^2}{e^C/4}= 1[/itex]. You should immediately see that, for every C, this is an ellipse, centered at the origin, with x intercepts at [itex]\left(-e^{C/2}, 0\right)[/itex] and [itex]\left(e^{C/2}, 0\right)[/itex], y intercepts at [itex]\left(0, -\frac{e^{C/2}}{2}\right)[/itex] and [itex]\left(0, \frac{e^{C/2}}{2}\right)[/itex].
A contour map is a type of graphical representation that displays the shape and elevation of a function by using a series of curves that connect points of equal elevation. Each curve, or contour line, represents a specific elevation level.
A contour map is created by plotting a series of points on a grid and connecting them with smooth curves. The points are chosen based on the function's output values at specific input values, and the curves are drawn to connect points with equal output values. The resulting map shows the function's shape and elevation at different points.
The main purpose of a contour map is to visually represent the shape and elevation of a function. This allows for easier analysis and understanding of the function's behavior, such as identifying areas of steepest ascent or descent, and locating points of maximum or minimum values.
Yes, a contour map can show both positive and negative values. A positive value would be represented by a curve that is higher up on the map, while a negative value would be represented by a curve that is lower down on the map.
Contour maps have a wide range of applications, including in geology, geography, and engineering. They can be used to map out elevation and terrain features, identify areas of high or low pressure in weather patterns, and analyze the flow of fluids or electricity in engineering designs.