Problem 21 - Calculus for Biology and Medicine

Question

Sketch the graph of (a) a function that is increasing at an accelerating rate; and (b) a function that is increasing at a decelerating rate. (c) Assume that your functions in (a) and (b) are twice differentiable. Explain in each case how you could check the respective properties by using the first and the second derivatives. Which of the functions is concave up, and which is concave down?

Step-by-Step Solution

Answer

The function in (a) is concave up, and the function in (b) is concave down.

Step 1: Identify the Type of Rate Change in (a)

For a function to increase at an accelerating rate, its first derivative must be positive, and the second derivative must also be positive. This means both the slope of the function and the rate of change of the slope are increasing.

Step 2: Select an Example for Part (a)

Consider the function \( f(x) = x^3 \). The first derivative is \( f'(x) = 3x^2 \), which is positive for \( x > 0 \). The second derivative is \( f''(x) = 6x \), which is also positive for \( x > 0 \). This indicates that the function is increasing at an accelerating rate in the interval \( x > 0 \).

Step 3: Check Concavity for Part (a)

Since \( f''(x) > 0 \) for \( x > 0 \), the function \( f(x) = x^3 \) is concave up in this interval.

Step 4: Identify the Type of Rate Change in (b)

For a function to increase at a decelerating rate, its first derivative must be positive, but the second derivative must be negative. This means the slope is increasing but the rate of change of the slope is decreasing.

Step 5: Select an Example for Part (b)

Consider the function \( g(x) = \sqrt{x} \). The first derivative is \( g'(x) = \frac{1}{2\sqrt{x}} \), which is positive for \( x > 0 \). The second derivative is \( g''(x) = -\frac{1}{4x^{3/2}} \), which is negative for \( x > 0 \). This indicates that the function is increasing at a decelerating rate in the interval \( x > 0 \).

Step 6: Check Concavity for Part (b)

Since \( g''(x) < 0 \) for \( x > 0 \), the function \( g(x) = \sqrt{x} \) is concave down in this interval.

Key Concepts

First Derivative

The first derivative of a function, denoted as \( f'(x) \), is a powerful tool in calculus. It tells us how the function is changing at a specific point. In simple terms, it gives us the slope of the tangent line to the function at any particular point. If \( f'(x) > 0 \), the function is increasing at that point because the slope of the tangent line is positive. This means as you move along the function, the values are going up.
However, if \( f'(x) < 0 \), the function is decreasing; here, the slope is negative, so the function's values are going down as you move along it.

• Positive first derivative: Function is increasing.
• Negative first derivative: Function is decreasing.

Understanding the behavior of \( f'(x) \) is crucial for analyzing the rate at which something changes, whether we are looking at simple linear growth or more complex cases of acceleration and deceleration.

Second Derivative

The second derivative, denoted as \( f''(x) \), provides additional insight into the function's behavior by describing how the rate of change is itself changing. It measures the curvature or the concavity of the function. If the second derivative is positive, \( f''(x) > 0 \), the rate of change of the function is increasing, and thus, the function is said to be concave up.
On the other hand, if \( f''(x) < 0 \), this means that the rate of change of the function is decreasing, and the function is concave down.

• Positive second derivative: Function is concave up (like a smile).
• Negative second derivative: Function is concave down (like a frown).

These properties help determine the nature of the changes in the function, such as whether it is accelerating or decelerating.

Concavity

The concept of concavity relates directly to the second derivative and describes the shape of the graph of the function. When discussing concavity:

• If a function is concave up, it means that the tangent lines to the curve are below the graph in that interval - the function looks like an upward-facing bowl.
• Conversely, if a function is concave down, the tangent lines to the curve are above the graph, resembling a downward-facing bowl.

For the exercise, function \( f(x) = x^3 \) has a positive second derivative for \( x > 0 \), indicating concave up, while \( g(x) = \sqrt{x} \) has a negative second derivative, indicating concave down. This property is crucial when sketching graphs, as it determines how the curve leans or bends in different sections.

Function Graphing

Graphing a function involves plotting the points that the function takes and understanding the overall shape of the curve based on derivatives. The first derivative helps determine where the function is increasing or decreasing, and the second derivative helps identify the intervals of concavity.
For instance, in the problem provided:

• The function \( f(x) = x^3 \) is increasing at an accelerating rate where both derivatives are positive, making the graph curve upward with increasing steepness.
• The function \( g(x) = \sqrt{x} \) is increasing at a decelerating rate, meaning the graph rises but gradually flattens out, reflecting the negative second derivative.

Understanding these concepts makes it easier to accurately sketch function graphs, predict behavior, and identify significant landmarks like turning points and inflection points.