Problem 27 - Calculus 3

Question

For the following exercises, find the level curves of each function at the indicated value of \(c\) to visualize the given function. $$ g(x, y)=\ln \left(\frac{y}{x^{2}}\right) ; c=-2,0,2 $$

Step-by-Step Solution

Answer

The level curves are \( y = x^{2}e^{-2} \), \( y = x^{2} \), and \( y = x^{2}e^{2} \).

Step 1: Understanding Level Curves

Level curves are obtained by setting a function equal to a constant value. For the given function \( g(x, y) = \ln \left(\frac{y}{x^{2}}\right) \), we find the level curves by solving \( \ln \left(\frac{y}{x^{2}}\right) = c \) for the specified values of \(c\).

Step 2: Solve for \(c = -2\)

Set \( \ln \left(\frac{y}{x^{2}}\right) = -2 \). To eliminate the natural logarithm, exponentiate both sides: \( \frac{y}{x^{2}} = e^{-2} \). Thus, the level curve at \(c = -2\) is \( y = x^{2}e^{-2} \).

Step 3: Solve for \(c = 0\)

Set \( \ln \left(\frac{y}{x^{2}}\right) = 0 \). Exponentiating both sides results in \( \frac{y}{x^{2}} = e^{0} \). Since \(e^{0} = 1\), the level curve at \(c = 0\) is \( y = x^{2} \).

Step 4: Solve for \(c = 2\)

Set \( \ln \left(\frac{y}{x^{2}}\right) = 2 \). Exponentiate both sides to get \( \frac{y}{x^{2}} = e^{2} \). The level curve at \(c = 2\) is \( y = x^{2}e^{2} \).

Step 5: Compile Level Curves

Collect the solutions from the previous steps neatly. The level curves for \(g(x, y) = \ln \left(\frac{y}{x^{2}}\right)\) at \(c = -2, 0, 2\) are \( y = x^{2}e^{-2} \), \( y = x^{2} \), and \( y = x^{2}e^{2} \), respectively.

Key Concepts

Natural Logarithm

The natural logarithm, often denoted as \( \ln \), is a mathematical function that plays a crucial role in many areas of mathematics and science. Unlike regular logarithms that can have any base, the natural logarithm specifically uses the base \( e \), where \( e \approx 2.71828 \). This number \( e \) is an irrational constant that is foundational to the concept of continuous growth or decay in natural processes.

Understanding natural logarithms is essential because they simplify complex multiplicative processes and transform them into additive ones. For example:

• Converting multiplication problems into simpler addition.
• Solving equations where the unknown is an exponent.
• Modeling natural phenomena in fields like biology and economics.

In practical terms, the natural logarithm \( \ln(x) \) can be interpreted as the amount of time it takes for an investment to grow to \( x \) under continuous compounding at a unit interest rate. This property makes it especially useful in finance and calculus problems.

Exponentiation

Exponentiation is the mathematical operation where a number (the base) is raised to the power of an exponent. This operation is denoted by expressions such as \( a^b \), where \( a \) is the base and \( b \) is the exponent. If the exponent is positive, it represents repeated multiplication of the base. For example, \( 3^2 = 3 \times 3 = 9 \).

When it comes to natural logs and exponentiation, these two functions are inverses of each other. If you exponentiate an expression with a natural logarithm, it cancels out, and vice versa:

• \( e^{\ln(x)} = x \)
• \( \ln(e^x) = x \)

This property allows us to move between logarithmic and exponential forms easily, which is particularly useful in solving equations. In the context of the exercise, this concept helps simplify the given equations to find level curves of functions by eliminating the logarithm through exponentiation.

Functions of Two Variables

A function of two variables, denoted as \( f(x, y) \), assigns a unique value to every pair \((x, y)\) in its domain. These functions are pivotal in multivariable calculus since they help in modeling scenarios where two factors impact outcomes simultaneously. Graphically, functions of two variables can be envisioned as surfaces in a three-dimensional space, with \( x \) and \( y \) forming the plane and function value \( f(x, y) \) determining the height or depth of the surface.

Level curves are a key concept when analyzing these functions. A level curve is formed by slicing the surface at particular heights and projecting these slices onto the \( x-y \) plane. For a function \( g(x, y) = c \), the level curve consists of all points \((x, y)\) where the function equals the constant \( c \). This visualization helps understand the topology of a surface and how changes in one variable affect another.To illustrate, in the original exercise, the level curves of \( g(x, y) = \ln \left( \frac{y}{x^2} \right) \) are determined for specific \( c \) values (-2, 0, 2), portraying distinct horizontal slices of the surface where the function 'levels' at those specific \( c \) values, thus enabling a deeper analysis of the function's behavior.