Problem 43 - College Algebra and Calculus: An Applied Approach

Question

Graph the solution set of the system of inequalities. $$\left\\{\begin{array}{l}y \leq e^{x} \\ y \geq \ln x \\ x \geq \frac{1}{2} \\\ x \leq 2\end{array}\right.$$

Step-by-Step Solution

Answer

The solution set to the system of inequalities is a region in the xy-plane bounded by the curve of \(y = e^{x}\), the curve of \(y = \ln(x)\), and the vertical lines \(x = 1/2\) and \(x = 2\).

Step 1: Graph y ≤ e^x

Begin by sketching the graph of the function y = e^x. Plot a few points for x and their corresponding y-values. The points below x=0 should not be considered as the third inequality stated x≥1/2. Connect these points in a curve, which will rise steeply as x increases. This line indicates that y is equal to e^x. Continue by shading the area below the line to indicate that y is less than or equal to the value of e^x.

Step 2: Graph y ≥ ln(x)

Next, sketch the graph of the function y = ln(x). This curve will increase as x moves away from 1 towards positive infinity and it will present no values for x less than or equal to zero. Connect the points in a curve. As previous step, ignore any points below x=0.5 due to the third inequality. Then shade the area above the line to show that y is greater than or equal to the value of ln(x).

Step 3: Graph x ≥ 1/2 and x ≤ 2

Plot two vertical lines at x = 1/2 and x = 2 respectively. Shade the area between these two lines to indicate the values of x that are included in the inequality.

Step 4: Combine the Inequalities

The solution set to the system of inequalities will include only those points which satisfy all inequalities taken together. This area will be where all shaded regions from the previous steps intersect.

Key Concepts

Graphing Inequalities

Graphing inequalities is a crucial skill when dealing with systems of inequalities. It involves representing solutions on a coordinate plane. Here's a breakdown of how it's done:

• Graphing a linear inequality like \(y \leq e^x\) means you'll first need to graph the equation \(y = e^x\) as a solid line if it's \(\leq\) or \(\geq\), indicating that points on the line are part of the solution set. Since this is an inequality, you'll also shade the region that satisfies the inequality. For \(y \leq e^x\), you'll shade below the curve.
• For inequalities like \(y \geq \ln x\), start with graphing \(y = \ln x\). Use a solid line and shade above the line for \(y \geq \ln x\).
• You might also need to include inequalities that confine \(x\) to certain intervals, like \(x \geq \frac{1}{2}\) and \(x \leq 2\). This will involve drawing vertical lines and shading the region between them.

After graphing each inequality, find the overlapping shaded areas in each graph. This intersection represents the solution to the system of inequalities. It ensures all conditions from the inequalities are satisfied at once.

Exponential Functions

Exponential functions are a specific type of function that grows or decays at a constant rate, which makes them distinct and powerful. They take the form \(f(x) = a \cdot e^{bx}\), where \(e\) is Euler's number, approximately 2.718.

• The function \(y = e^x\) is a basic exponential function. It is characterized by a smooth curve that never touches the x-axis, known as an asymptote. As \(x\) increases, \(y\) increases quickly, and as \(x\) decreases, \(y\) approaches zero but never reaches it.
• In the context of inequalities like \(y \leq e^x\), it's important to recognize that since \(e^x\) is always positive, the solutions are only valid where \(x\) is defined by the inequality system, e.g., \(x \geq \frac{1}{2}\).

Understanding the behavior of exponential functions helps in predicting the growth pattern and in graphing correctly. They're commonly applicable in modeling population growth, financial interest calculations, and other real-world phenomena.

Logarithmic Functions

Logarithmic functions are the inverse of exponential functions and are mathematical tools that help solve equations involving exponents.

• A logarithmic function such as \(y = \ln(x)\) represents the power to which the base \(e\) must be raised to obtain the value \(x\). In this context, \(ln(x)\) is only defined for \(x > 0\).
• The curve of \(y = \ln(x)\) increases steadily but more slowly than exponential functions like \(e^x\). It intercepts the x-axis at (1,0), showing that the natural log of 1 is 0. As \(x\) approaches zero from the positive side, the logarithmic function descends steeply downward.
• In inequalities such as \(y \geq \ln(x)\), we are interested in the region above the ln curve, indicating that solutions are where the y-values are greater than or equal to the natural logarithm of \(x\).

Logarithmic graphs are an essential component in understanding situations where inputs scale logarithmically, such as in measuring pH in chemistry or sound intensity in decibels.