Problem 47 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

Determine whether the statement is true or false. If it is true, explain why it is true. If it is false, give an example to show why it is false. If \(x

Step-by-Step Solution

Answer

The statement "If \(x

Step 1: Understand the exponential function

The exponential function with base \(e\) is denoted as \(f(x) = e^x\). This function is both continuous and strictly increasing. This means that as the value of the input increases, the value of the output also increases. Since the exponential function is strictly increasing, it means that if \(x

Step 2: Determine the truth of the statement

Since we know that the exponential function is strictly increasing, it follows that the given statement "If \(x

Step 3: Explain why the statement is true

The statement is true because the exponential function with base \(e\) is strictly increasing. This means that as the input value (in this case, \(x\)) increases, the output value (in this case, \(e^x\)) also increases. Since the statement "If \(x

Key Concepts

Continuous Functions

The concept of continuous functions is fundamental in mathematics, particularly when dealing with functions like the exponential function. A continuous function is one where small changes in the input result in small changes in the output. In simpler terms, you can draw the graph of a continuous function without lifting your pencil off the paper. For the exponential function, represented as \( f(x) = e^x \), continuity ensures that there are no sudden jumps or gaps in its graph.

This means that even as the inputs change slightly, the outputs change smoothly without abrupt alterations. Such functions behave consistently, making predictions and calculations more reliable.

• Continuous functions are predictable and smooth.
• For \( e^x \), this means there's a smooth curve escalating upwards as \( x \) increases.
• Continuity is crucial for understanding limits and derivatives.

By grasping continuity, you can better understand how functions like \( e^x \) behave over different ranges of \( x \), knowing that each incremental increase or decrease in \( x \) will steadily affect \( e^x \) accordingly.

Strictly Increasing Functions

Strictly increasing functions have a crucial property: as one input increases, the output also increases. This "strict" nature means that if \( x < y \), then \( f(x) < f(y) \). The exponential function \( e^x \) is a classic example of a strictly increasing function. No matter which two distinct numbers you choose for \( x \) and \( y \) with \( x < y \), you will always find that \( e^x < e^y \).

Key characteristics include:

• The slope of the curve is always positive; the graph never decreases.
• For every consecutive pair of points, the output value strictly rises.

In practical terms, this increasing property means that exponential functions are predictably larger in value as you move right along the x-axis on a graph.
Understanding strictly increasing behavior helps in comparing different values of exponential functions, enabling us to make confident claims like "if \( x < y \), then \( e^x < e^y \)." This property makes strictly increasing functions ideal for modeling growth and decay, such as population growth or radioactive decay.

Inequalities in Mathematics

Inequalities are mathematical expressions that show the relationship between expressions that are not equal. They use symbols like \(<\), \(>\), \( \leq \), and \( \geq \). In the original exercise statement "If \( x < y \), then \( e^x < e^y \)", we see an example of inequality involving exponential functions.

Inequalities allow us to describe bounds and relationships without needing exact values. Here are some basic aspects to consider:

• They can show that something is bigger, smaller, or potentially equal to something else.
• They form the basis for defining ranges and domains in mathematics.
• Working with inequalities requires understanding rules for operations like adding or multiplying both sides.

The statement "\( e^x < e^y \)" too is an inequality. It straightforwardly emanates from the property of the exponential function being strictly increasing. This type of inequality is critical in solving equations and understanding functions. It helps students establish logical reasoning chains, determining how one quantity relates to another under various conditions.

Previous

Problem 47

Next

Problem 49

Other exercises in this chapter

Problem 46

Determine whether the statement is true or false. If it is true, explain why it is true. If it is false, give an example to show why it is false. $$e^{x y}=e^{x

View solution

Problem 47

Sketch the graphs of the equations on the same coordinate axes. \(y=2^{x}\) and \(y=\log _{2} x\)

View solution

Problem 49

Use logarithms to solve the equation for \(t\). $$e^{0.4 t}=8$$

View solution

Problem 50

Use logarithms to solve the equation for \(t\). $$\frac{1}{3} e^{-3 t}=0.9$$

View solution

Key Concepts

Continuous Functions

The concept of continuous functions is fundamental in mathematics, particularly when dealing with functions like the exponential function. A continuous function is one where small changes in the input result in small changes in the output. In simpler terms, you can draw the graph of a continuous function without lifting your pencil off the paper. For the exponential function, represented as \( f(x) = e^x \), continuity ensures that there are no sudden jumps or gaps in its graph.

This means that even as the inputs change slightly, the outputs change smoothly without abrupt alterations. Such functions behave consistently, making predictions and calculations more reliable.

• Continuous functions are predictable and smooth.
• For \( e^x \), this means there's a smooth curve escalating upwards as \( x \) increases.
• Continuity is crucial for understanding limits and derivatives.

By grasping continuity, you can better understand how functions like \( e^x \) behave over different ranges of \( x \), knowing that each incremental increase or decrease in \( x \) will steadily affect \( e^x \) accordingly.

Strictly Increasing Functions

Strictly increasing functions have a crucial property: as one input increases, the output also increases. This "strict" nature means that if \( x < y \), then \( f(x) < f(y) \). The exponential function \( e^x \) is a classic example of a strictly increasing function. No matter which two distinct numbers you choose for \( x \) and \( y \) with \( x < y \), you will always find that \( e^x < e^y \).

Key characteristics include:

• The slope of the curve is always positive; the graph never decreases.
• For every consecutive pair of points, the output value strictly rises.

In practical terms, this increasing property means that exponential functions are predictably larger in value as you move right along the x-axis on a graph.
Understanding strictly increasing behavior helps in comparing different values of exponential functions, enabling us to make confident claims like "if \( x < y \), then \( e^x < e^y \)." This property makes strictly increasing functions ideal for modeling growth and decay, such as population growth or radioactive decay.

Inequalities in Mathematics

Inequalities are mathematical expressions that show the relationship between expressions that are not equal. They use symbols like \(<\), \(>\), \( \leq \), and \( \geq \). In the original exercise statement "If \( x < y \), then \( e^x < e^y \)", we see an example of inequality involving exponential functions.

Inequalities allow us to describe bounds and relationships without needing exact values. Here are some basic aspects to consider:

• They can show that something is bigger, smaller, or potentially equal to something else.
• They form the basis for defining ranges and domains in mathematics.
• Working with inequalities requires understanding rules for operations like adding or multiplying both sides.

The statement "\( e^x < e^y \)" too is an inequality. It straightforwardly emanates from the property of the exponential function being strictly increasing. This type of inequality is critical in solving equations and understanding functions. It helps students establish logical reasoning chains, determining how one quantity relates to another under various conditions.