Problem 3 - Calculus of a Single Variable

Question

Determining Concavity In Exercises \(3-14\) , determine the open intervals on which the graph is concave upward or concave downward. $$ y=x^{2}-x-2 $$

Step-by-Step Solution

Answer

The graph of the function \(y = x^2 - x - 2\) is concave upward on the interval (\(-\infty, +\infty\)). So, there are no intervals where it's concave downward.

Step 1: Compute the First Derivative

The first derivative of \(y = x^2 - x - 2\) is \(y' = 2x - 1\).

Step 2: Compute the Second Derivative

The second derivative of \(y = x^2 - x - 2\) is obtained by differentiating the first derivative. This gives \(y'' = 2\).

Step 3: Find Points of Inflection

The points of inflection are found by setting \(y'' = 2\) equal to zero and solving for x. Since the second derivative is a constant, there are no points along the x-axis where the concavity could change.

Step 4: Determine the Concavity

Since our \(y'' = 2\) (which is greater than zero), this indicates that the graph of the function is concave upward for all x on the open intervals (\(-\infty, +\infty\)).

Key Concepts

First Derivative

The first derivative is a fundamental concept in calculus that gives us critical information about the slope of a function at any given point. In this exercise, we start with the function \( y = x^2 - x - 2 \).
The first derivative, denoted as \( y' \), refers to the rate of change or the slope of the tangent to the graph of the function at any point on the x-axis.

• For the function \( y = x^2 - x - 2 \), the first derivative is calculated as \( y' = 2x - 1 \).
• This linear expression indicates that the slope of the function changes with x, but finding the derivative helps us understand these changes.
• In this case, \( y' = 2x - 1 \) means the slope is a linearly increasing function.

Understanding the first derivative is crucial because it forms the basis for later determining critical points and the behavior of the function, such as determining concavity.

Second Derivative

The second derivative provides insight into the curvature or concavity of a function's graph. It is essentially the derivative of the first derivative and reveals whether the graph is curving upwards or downwards at different points.
In our exercise, the first derivative of the function \( y = x^2 - x - 2 \) was \( y' = 2x - 1 \).

• The second derivative, denoted \( y'' \), is computed by differentiating the first derivative, which yields \( y'' = 2 \).
• This constant value indicates that the concavity does not change, as it is independent of x.

The second derivative helps identify points where the graph may change concavity and provides conditions (\( y'' > 0 \) or \( y'' < 0 \)) to determine concavity behavior.

Points of Inflection

Points of inflection are points on the graph where the curvature changes from concave upward to concave downward, or vice versa. These are critically important as they indicate where the function transitions in curvature.
To find these, you typically set the second derivative \( y'' \) equal to zero and solve for x.
However, in this function's case, we find \( y'' = 2 \), a constant, meaning:

• There are no points where the second derivative equals zero.
• Thus, no change in concavity occurs, and no inflection points exist for the function \( y = x^2 - x - 2 \).

When the second derivative does change sign, it indicates potential points of inflection. Here, the constant second derivative indicates no such transitions occur.

Concave Upward

When a graph is described as concave upward, it implies that as you move along the curve, the tangent lines, if drawn, lie below the graph. More intuitively, this means the curve looks like a smile.
A graph is said to be concave upward on intervals where the second derivative \( y'' \) is positive.
For the given function \( y = x^2 - x - 2 \):

• Since \( y'' = 2 \), and 2 is a positive constant, the graph is concave upward for all x-values on the interval \((-\), \(+\infty)\).

Everywhere on this interval, the graph curves in an upward direction, creating that characteristic smile shape.

Concave Downward

Concave downward is the opposite of concave upward. When a graph is concave downward, tangent lines to the curve lie above it, and the graph resembles a frown or an upside-down U-shape.
If the second derivative \( y'' \) is negative, then the graph is concave downward at those intervals.
However, for this exercise with \( y = x^2 - x - 2 \):

• We determined \( y'' = 2 \), which is greater than zero.
• This indicates there are no intervals where the graph is concave downward, as the second derivative is always positive.

Thus, the graph has no sections that curve downwards, which simplifies identifying its behavior as overall always curving upward.