Problem 3 - Active Calculus Multivariable

Question

Match the surfaces with the verbal description of the level curves by placing the letter of the verbal description to the left of the number of the surface. (a) \(z=\sqrt{\left(x^{2}+y^{2}\right)}\) (b) \(z=2 x+3 y\) (c) \(z=2 x^{2}+3 y^{2}\) (d) \(z=x^{2}+y^{2}\) (e) \(z=x y\) (f) \(z=\frac{1}{x-1}\) \((\mathrm{g}) \quad z=\sqrt{\left(25-x^{2}-y^{2}\right)}\) A. a collection of equally spaced concentric circles B. a collection of unequally spaced concentric circles C. two straight lines and a collection of hyperbolas D. a collection of unequally spaced parallel lines E. a collection of concentric ellipses F. a collection of equally spaced parallel lines

Step-by-Step Solution

Answer

1(a) - A 2(b) - F 3(c) - E 4(d) - A 5(e) - C 6(f) - D 7(g) - B

Step 1: 1. z = sqrt(x^2 + y^2)

Replace z with constant c: c = \( \sqrt{x^{2}+y^{2}} \) Square both sides: \(c^{2} = x^{2} + y^{2}\) This captures the shape of concentric circles. #Conclusion# 1: 1(a) matches A.

Step 2: 2. z = 2x + 3y

Replace z with constant c: c = 2x + 3y This captures parallel lines with regular spacing between them. #Conclusion# 2: 2(b) matches F.

Step 3: 3. z = 2x^2 + 3y^2

Replace z with constant c: c = 2x^2 + 3y^2 This captures concentric ellipses with varying distances between them. #Conclusion# 3: 3(c) matches E.

Step 4: 4. z = x^2 + y^2

Replace z with constant c: c = x^2 + y^2 This captures concentric circles with equal distances between them. #Conclusion# 4: 4(d) matches A.

Step 5: 5. z = xy

Replace z with constant c: c = xy This captures a combination of straight lines and hyperbolas. #Conclusion# 5: 5(e) matches C.

Step 6: 6. z = 1 / (x - 1)

Replace z with constant c: c = 1 / (x - 1) This captures parallel lines with unequal distances between them. #Conclusion# 6: 6(f) matches D.

Step 7: 7. z = sqrt(25 - x^2 - y^2)

Replace z with constant c: c = \(\sqrt{25 - x^{2} - y^{2}}\) Square both sides: \(c^{2} = 25 - x^{2} - y^{2}\) This captures concentric circles with unequal distances between them. #Conclusion# 7: 7(g) matches B. #Final Matchings#: 1(a) - A 2(b) - F 3(c) - E 4(d) - A 5(e) - C 6(f) - D 7(g) - B

Key Concepts

Level Curves

Level curves are an essential tool in multivariable calculus for visualizing the topography of a surface defined by a function of two variables, like temperature on a map or elevation on a topographical map. When we hold the output of a function constant, say at a value of 'c', and plot all the points (x, y) that satisfy this condition, we create a curve called a level curve. For example, the equation for a circle, c^2 = x^2 + y^2, generates a level curve that represents concentric circles when 'c' takes on various constant values. Each circle corresponds to a different 'height' of 'z' on the three-dimensional surface z = \(x^2 + y^2\). This concept is beautifully illustrated in the provided exercise, where different surfaces correspond to different arrangements of level curves, such as equally spaced concentric circles for equal values of 'z'.

Conic Sections

Conic sections are the curves obtained by intersecting a cone with a plane. These shapes include ellipses, circles, parabolas, and hyperbolas, and they are prevalent in various fields, from astronomy to physics. In the context of multivariable calculus, when we represent conic sections as level curves, we obtain a visual representation of how a function behaves in three dimensions. A great instance of this is the function z = 2x^2 + 3y^2. By setting z = c, we get a family of curves that represent concentric ellipses, which are a type of conic section. The varying distances between these ellipses give us insight into the rate at which the function's value changes, which is crucial when understanding the topography of the surface it represents.

Concentric Circles

Imagine tossing a stone into a still pond and observing the ripples of waves emanating from the point of impact. These ripples are similar to concentric circles, which are circles with a common center but different radii. In multivariable calculus, concentric circles can describe level curves of a function like z = \(x^2 + y^2\), especially when these level curves are for constant values of 'z'. In our exercise, both functions z = \(x^2 + y^2\) and z = \sqrt{25 - x^2 - y^2\} display collections of concentric circles, with the former illustrating equally spaced circles, while the latter depicts circles that are not equidistant from each other, providing a rich context to understand how the function's output changes with distance from the origin.

Parallel Lines

Parallel lines are straight lines in the same plane that never meet, no matter how far they are extended. This characteristic is particularly useful in the context of functions involving multiple variables. When setting a function equal to a constant, z = c, which represents a specific 'height' or value of the function, the resulting level curves can be parallel lines. For instance, the function z = 2x + 3y produces equally spaced parallel lines as its level curves, signifying that the change in 'z' occurs uniformly along those lines. Conversely, the function z = 1 / (x - 1) showcases parallel lines that are not equally spaced, suggesting that the rate of change in 'z' is not constant. Recognizing the arrangement of these parallel lines aids in visualizing the slope of the surface and, by extension, the gradient of the function.