Analyzing Continuity and Differentiability on a Graph of g(x)

Math Problem Statement

Use the graph of g in the figure to do the following. a. Find the values of x in left parenthesis negative 4 comma 4 right parenthesis at which g is not continuous. b. Find the values of x in left parenthesis negative 4 comma 4 right parenthesis at which g is not differentiable. c. Sketch a graph of g prime. -4 -2 2 4 6 8 10 -2 2 4 6 8 10 x y

A coordinate system has a horizontal x-axis labeled from negative 5 to 10 in increments of 1 and a vertical y-axis labeled from negative 2 to 10 in increments of 1. A curve consists of two pieces. The first piece is a downward opening parabola that starts at (negative 4, 4), rises to a vertex at (negative 2, 6), and then falls ending at (0, 4). The second piece is another downward opening parabola that starts at (0, 4), rises to a vertex at (2, 5), and then falls ending at (4, 4). There is a hole in the graph at (negative 2, 6). . . . Question content area right Part 1 a. In the interval left parenthesis negative 4 comma 4 right parenthesis, g is not continuous at xequals negative 2. (Use a comma to separate answers as needed.) Part 2 b. In the interval left parenthesis negative 4 comma 4 right parenthesis, g is not differentiable at xequals negative 2 comma 0. (Use a comma to separate answers as needed.) Part 3 c. Choose the correct graph. A. -8 8 x y

A coordinate system has a horizontal x-axis labeled from negative 8 to 8 in increments of 2 and an unlabeled vertical y-axis. A graph consists of two branches. The first branch is a line segment that extends from an endpoint above the x-axis at x equals negative 4 to an open endpoint below the x-axis at x equals 0. The second branch is a line segment that extends from an open endpoint above the x-axis at x equals 0 to an endpoint below the x-axis at x equals 4. There is a hole in the graph on the x-axis at x equals negative 2. B. -8 8 x y

A coordinate system has a horizontal x-axis labeled from negative 8 to 8 in increments of 2 and an unlabeled vertical y-axis. A graph consists of two branches. The first branch is a line segment that extends from an endpoint below the x-axis at x equals negative 4 to an open endpoint above the x-axis at x equals 0. The second branch is a line segment that extends from an open endpoint that is below the open endpoint of the first branch at x equals 0 and above the x dash axis. The branch extends to an endpoint below the x-axis at x equals 4. There is a hole in the graph on the x-axis at x equals negative 2. C. -8 8 x y

A coordinate system has a horizontal x-axis labeled from negative 8 to 8 in increments of 2 and an unlabeled vertical y-axis. A graph consists of two branches. The first branch is a line segment that extends from an endpoint above the x-axis at x equals negative 4 to an open endpoint below the x-axis at x equals 0. The second branch is a line segment that extends from an open endpoint that is above the open endpoint of the first branch at x equals 0 and below the x dash axis. The branch extends to an endpoint above the x-axis at x equals 4. There is a hole in the graph on the x-axis at x equals negative 2. D. -8 8 x y

Solution

Let's analyze the problem step by step, focusing on the information provided.

Part 1: Continuity

To identify where the function $g(x)$ is not continuous, we look for any discontinuities. The graph describes two parabolic segments with a hole at $x = -2$ . The hole indicates a discontinuity at that point.

Thus, $g(x)$ is not continuous at $x = -2$ in the interval $(-4, 4)$ .

Part 2: Differentiability

Differentiability requires that the function be smooth and have no sharp corners or discontinuities. Since there is a discontinuity at $x = -2$ , $g(x)$ is not differentiable there. Additionally, there is a potential sharp corner or cusp at $x = 0$ where the two parabolas meet, making $g(x)$ non-differentiable at that point as well.

So, $g(x)$ is not differentiable at $x = -2$ and $x = 0$ .

Part 3: Sketching $g'(x)$

To sketch the derivative $g'(x)$ , note that:

The graph is made of two parabolas, which means the derivative will be linear (since the derivative of a parabola is a straight line).
At $x = -2$ , there is a discontinuity, so there will be a hole in the graph of $g'(x)$ at that point.
At $x = 0$ , where the two parabolas meet, there will be a sharp change in slope, indicating a cusp in the graph of the derivative.

Based on the options given, the graph of $g'(x)$ should reflect these observations:

The graph will have two line segments with different slopes.
There will be a hole at $x = -2$ , and the segments will meet at $x = 0$ but with different slopes, indicating non-differentiability at that point.

Conclusion for the graph:

The correct choice for $g'(x)$ is Option A. This graph shows two straight lines with a hole at $x = -2$ and a sharp corner at $x = 0$ , which corresponds to the points where the function is not differentiable.

Do you want further details or clarifications? Here are 5 related questions to deepen understanding:

What is the formal definition of continuity and how is it applied here?
How do we determine if a function is differentiable at a given point?
What are the graphical signs of a non-differentiable function?
Why does a hole in a graph indicate discontinuity?
What is the geometric meaning of the derivative of a function?

Tip: For parabolas, the derivative is always a linear function (a straight line).

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Math Problem Analysis

Mathematical Concepts

Continuity
Differentiability
Graphical Analysis
Derivatives

Formulas

Conditions for continuity
Conditions for differentiability
Derivative of a quadratic function

Theorems

Intermediate Value Theorem
Mean Value Theorem
Definition of Continuity and Differentiability

Suitable Grade Level

Grades 10-12

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