Problem 28 - Calculus

Question

In each case, graph a smooth curve whose slope meets the condition. (a) Everywhere positive and increasing gradually. (b) Everywhere positive and decreasing gradually. (c) Everywhere negative and increasing gradually (becoming less negative). (d) Everywhere negative and decreasing gradually (becoming more negative).

Step-by-Step Solution

Answer

(a) \( f(x) = e^x \); (b) \( f(x) = \ln(x) \); (c) \( f(x) = -e^{-x} \); (d) \( f(x) = -x^2 \).

Step 1: Identify the Type of Function for Part (a)

For a curve with a slope that is everywhere positive and increasing gradually, we need a function where both the function and its derivative are positive and increasing. An example of such a function could be an exponential growth function like \( f(x) = e^x \).

Step 2: Graph for Part (a)

Graph the function \( f(x) = e^x \). It is an upward-sloping curve that gets steeper as \( x \) increases, satisfying the condition of an increasing and positive slope everywhere.

Step 3: Identify the Type of Function for Part (b)

For a curve where the slope is everywhere positive but decreasing gradually, we need a function whose derivative is positive but decreasing to zero at infinity. A suitable function is a logarithmic function like \( f(x) = \ln(x) \) for \( x > 0 \).

Step 4: Graph for Part (b)

Graph the function \( f(x) = \ln(x) \). It rises above the x-axis while becoming less steep as \( x \) increases, satisfying the condition.

Step 5: Identify the Type of Function for Part (c)

For a curve where the slope is everywhere negative and increasing gradually (less negative), choose a function with a derivative that starts very negative but approaches zero. Consider the function \( f(x) = -e^{-x} \).

Step 6: Graph for Part (c)

Graph the function \( f(x) = -e^{-x} \). It is a downward curve that becomes less steep as \( x \) increases, which matches the description of a slope that is becoming less negative.

Step 7: Identify the Type of Function for Part (d)

For a curve with a slope everywhere negative but decreasing gradually (more negative), choose a function with a derivative that becomes more negative. A function like \( f(x) = -x^2 \) has these properties.

Step 8: Graph for Part (d)

Graph the function \( f(x) = -x^2 \). It is a parabola opening downwards whose slope gets steeper (more negative) as \( x \) moves away from the origin, fitting the criteria.

Key Concepts

Slope Analysis

Slope analysis is a crucial concept when graphing functions, as it provides information about the direction and steepness of the curve at various points. The slope of a function, represented by its derivative, tells us whether the function is increasing or decreasing at any given point.

• A positive slope indicates that the function is increasing.
• A negative slope indicates that the function is decreasing.

Understanding how the slope changes is vital for analyzing the behavior of a function. For example, if a slope is everywhere positive and increases gradually, it means the function not only increases, but the rate of increase becomes faster. When graphing, shapes of curves such as exponential or polynomial functions can be anticipated by examining their slope behavior.

Exponential Functions

Exponential functions are a type of mathematical function in which a constant is raised to the power of a variable. They're extremely important, especially in fields like finance, biology, and physics, due to their unique growth patterns. For instance, the function \(f(x) = e^x\) is an exponential function where the base \(e\) (approximately equal to 2.718) is constant. These functions are characterized by having a slope that is positive and increases rapidly.

• Exponential growth: increases quickly and can model diverse real-world phenomena such as population growth or compound interest.
• Graphing \(f(x) = e^x\): yields an upward-sloping curve that becomes steeper with increasing \(x\).

The essence of exponential functions is their accelerating slope, making them useful for describing processes that become faster over time.

Logarithmic Functions

Logarithmic functions are the inverse of exponential functions. They are of the form \(f(x) = \ln(x)\) or \(f(x) = \log_b(x)\), representing the logarithm of \(x\) with a certain base. These functions are essential in scenarios where growth happens rapidly at first and then slows down, such as the spread of sound or certain chemical reactions.A key property of logarithmic functions is a slope that is positive, but which decreases as \(x\) increases. This means:

• The function rises quickly at first, but the rate of increase slows down with bigger \(x\) values.

When graphing \(f(x) = \ln(x)\), visualize a curve that is steep near the origin and becomes flatter as it extends along the x-axis. This property of slowing growth curve makes logarithmic functions useful for modeling phenomena where initial rapid changes slow down over time.

Polynomial Functions

Polynomial functions involve sums of powers of \(x\) with coefficients, like \(f(x) = ax^n + bx^{n-1} + \,\ldots\, + k\), and are among the most versatile in mathematics. They can represent a wide variety of shapes depending on their degree and coefficients.For example, the function \(f(x) = -x^2\) is a quadratic polynomial that leads to a parabola. This particular function features a negative slope that decreases, meaning:

• The parabola opens downward because \(x^2\) is negative.
• The slope starts less negative (less steep) and becomes more negative (steeper) as \(x\) increases away from the vertex of the parabola.

This decreasing slope behavior is typical of functions with higher degree negative terms and can describe various physical phenomena, such as projectile motion where force or speed reduces over time.