Problem 15 - Calculus for Biology and Medicine

Question

Find the largest possible domain and the corresponding range of each function. Determine the equation of the level curves \(f(x, y)=c\), together with the possible values of \(c .\) \(f(x, y)=\ln \left(y-x^{2}\right)\)

Step-by-Step Solution

Answer

Domain: \(y > x^2\); Range: all real numbers; Level curves: \(y = x^2 + e^c\), \(c\) can be any real number.

Step 1: Understanding the Function's Domain

To find the largest possible domain of the function \(f(x, y) = \ln(y - x^2)\), we need the argument of the logarithm \((y - x^2)\) to be positive. This implies that \(y - x^2 > 0\). Therefore, the domain of \(f\) is all points \((x, y)\) where \(y > x^2\).

Step 2: Determining the Range of the Function

The range of the function \(f(x, y) = \ln(y - x^2)\) is all real numbers. Since \(y - x^2 > 0\), \(\ln(y - x^2)\) can take any real value by appropriately choosing the value of \(y\) relative to \(x\) to vary \(y - x^2\).

Step 3: Finding the Level Curves Equation

The level curves of the function are given by setting the function equal to a constant \(c\): \(f(x, y) = c\), which means \(\ln(y - x^2) = c\). This equation simplifies to \(y - x^2 = e^c\). Each level curve will be an upward-facing parabola given by \(y = x^2 + e^c\).

Step 4: Identifying Possible Values for the Constant

For \(y = x^2 + e^c\) to be meaningful, \(e^c\) must be positive (since it is an exponential function). Therefore, \(c\) can be any real number, because \(e^c\) is always positive for any real value of \(c\).

Key Concepts

Domain and Range

The concepts of domain and range are essential in understanding functions. The domain refers to all the possible input values for which the function is defined. Here, we deal with the function \( f(x, y) = \ln(y - x^2) \). For a logarithmic function of form \( \ln(a) \), the argument \( a \) must be greater than zero. This condition ensures that \( (y - x^2) > 0 \), meaning

• \( y > x^2 \)

This inequality describes the domain of the function as all coordinate points \( (x, y) \) where \( y \) surpasses \( x^2 \). Think of this as an area above the parabola formed by \( y = x^2 \).

The range is the set of all possible output values the function can yield. Because a logarithmic function can approach any real number as its argument changes, the range of \( f(x, y) = \ln(y - x^2) \) spans all real numbers. By selecting appropriate \( y \) values larger than \( x^2 \), the expression \( y - x^2 \) can cover all positive values, allowing \( \ln(y - x^2) \) to output any real number.

Logarithmic Functions

Logarithmic functions provide a unique insight into mathematical relationships, particularly through their ability to represent inverse exponential behavior. The function we are dealing with is \( f(x, y) = \ln(y - x^2) \), and this is a natural logarithmic function.

A key characteristic of logarithmic functions is that they can only be defined with positive arguments. This is because the logarithm of a non-positive number is undefined in the realm of real numbers. For our function, the requirement of \( (y - x^2) > 0 \) must hold for \( \ln(y - x^2) \) to be valid.

• This implies that the function reflects scenarios where \( y \) significantly exceeds \( x^2 \).

Another feature of logarithmic functions is their capacity to cover all real numbers as output. As \( y - x^2 \) takes increasingly larger values, \( \ln(y - x^2) \) similarly grows, extending infinitely in the positive direction. Conversely, as \( y \) nears \( x^2 \), it draws \( \ln(y - x^2) \) toward negative infinity.

These properties are why logarithmic functions are adept at describing phenomena with wide-ranging scales, including in fields like acoustics and earthquake intensity, where real-world measurements are converted via a logarithm for better interpretability and manipulation.

Level Curves

Level curves give a snapshot of a function's behavior at specific output values by holding the function constant. For our function \( f(x, y) = \ln(y - x^2) \), a level curve is defined by fixing the function's output, say \( c \). Thus, \( \ln(y - x^2) = c \) implies:

• \( y - x^2 = e^c \)
• or rearranged, \( y = x^2 + e^c \)

By observing this equation, each level curve takes the shape of a parabola opening upwards, shifted along the \( y \)-axis by \( e^c \).

Since \( e^c \) is always positive, the shift ensures the curves lie above the parabola formed by \( y = x^2 \). The constant \( c \) can vary over all real numbers, adjusting the elevation of the level curves across the plane.

Understanding level curves allows one to visualize how the function's output changes with \( x \) and \( y \). Each curve signifies a uniform functional value and aids in grasping the nature of a 3D surface, providing insights into gradients and topographical features akin to contour lines found on geographic maps.