Problem 5 - Advanced Engineering Mathematics

Question

Sketch some of the level curves associated with the given function. $$ f(x, y)=e^{y-x^{2}} $$

Step-by-Step Solution

Answer

The level curves are parabolas: \(y = x^2 + \ln(c)\) for different \(c\).

Step 1: Understand Level Curves

Level curves are the curves where the function maintains constant values. For a given function \(f(x, y)\), these curves can be defined by the equation \(f(x, y) = c\), where \(c\) is a constant.

Step 2: Set Up the Level Curve Equation

Begin by setting \(f(x, y) = e^{y-x^{2}} = c\). This means \(e^{y-x^{2}} = c\). To find \(y\) in terms of \(x\) and \(c\), we can take the natural logarithm.

Step 3: Solve for y

Take the natural logarithm of both sides: \(y - x^2 = \ln(c)\). Now, solve for \(y\): \(y = x^2 + \ln(c)\). This is the equation of the level curves.

Step 4: Choose Values for c

Select several values for \(c\) to sketch the level curves. Common choices might include \(c = 1, e, e^2\), as these values simplify logarithms as \(\ln(1) = 0, \ln(e) = 1, \ln(e^2) = 2\).

Step 5: Plot the Level Curves

For each chosen \(c\), plot the corresponding \(y = x^2 + \ln(c)\) on a graph. These are simple parabolas shifted vertically depending on \(\ln(c)\). For instance:- For \(c = 1\), \(y = x^2 + 0 = x^2\).- For \(c = e\), \(y = x^2 + 1\).- For \(c = e^2\), \(y = x^2 + 2\).Sketch these parabolas on the coordinate plane.

Key Concepts

Function Analysis

To better understand the behavior of mathematical functions, we often perform a function analysis. This process involves exploring various characteristics and behaviors, including finding level curves. Level curves allow us to see where the function takes on constant values in a two-dimensional space.

For the function given by \( f(x, y) = e^{y-x^2} \), we aim to find lines where the function maintains the same value.

By setting \( f(x, y) = c \) where \( c \) is a constant, we essentially analyze how changing \( x \) impacts \( y \) to preserve this constant function value. This lays the groundwork for drawing and understanding level curves, which represent pockets of unchanging function values in our visual graph of \( f(x, y) \).

In function analysis, it's important to:

• Identify constraints and properties of the function.
• Apply meaningful transformations, like taking logarithms, to simplify the analysis.
• Visualize these transformed expressions as graphs or curves.

Hence, understanding level curves requires recognizing constant sets within the domain of our function and analyzing how changes in one variable influence the outcomes in another.

Natural Logarithm

The natural logarithm, denoted by \( \ln \), is a fundamental concept when analyzing exponential functions. It acts as the inverse of the exponential function, like \( e^{x} \). In our exercise, we take advantage of the natural logarithm to simplify the given exponential equation.

For the function \( f(x, y) = e^{y-x^2} \), to determine the shape and location of the level curves, we express \( y - x^2 = \ln(c) \) by taking the logarithm of both sides.

This operation helps translate an exponential relationship into an easier-to-understand linear one. Resulting in \( y = x^2 + \ln(c) \), where every value of \( c \) corresponds to a distinctive curve.

Key concepts to remember about natural logarithms include:

• They transform multiplicative relationships into additive ones, simplifying complex calculations.
• Logarithmic properties are crucial for handling exponential equations, as they convert powers into straightforward algebra.
• Each logarithmic calculation can plot specific vertical shifts (like \( \ln(c) \)) onto our level curves.

Thus, the natural logarithm is pivotal for dissecting exponential-related level curves and converting them into more manageable visual formats.

Graphical Representation

Converting mathematical expressions into graphical representations is a powerful way to comprehend functions better. In this problem, our task is to sketch level curves of the function \( f(x, y) = e^{y-x^2} \), effectively visualizing them as parabolas defined by \( y = x^2 + \ln(c) \).

By selecting distinct values of \( c \) like \( c = 1, e, \text{and } e^2 \), we get simplified equations \( y = x^2, y = x^2 + 1, \text{and } y = x^2 + 2 \) respectively.

These translate into familiar shifts in the vertical axis of the parabola centered along the \( y = x^2 \) curve on a graph. Here’s what you need to do:

• Each curve is a parabola whose vertex shifts upwards by \( \ln(c) \) each time.
• Sketch these curves in a coordinate system, noting how the vertical positions differ with varied constant values.

This graphical approach helps students not only visualize but intuitively grasp how changing parameter values affect equation-based images.

By plotting and comparing these level curves, students can solidify their understanding of parameter impacts, graphical transformations, and the essence of analyzing and visualizing mathematical functions.