Access the help on integrals (Desmos)Links to an external site. and review the first example in the section about indefinite integrals. This example demonstrates the method of graphing antiderivative of the sample function f(x)=x^2 
. The outcome is the graph of F(x)=x^3/3 
.
In Desmos, graph the first derivative function from your application problem in Discussion 6 y=\frac{x+4}{\left(x-2\right)^{2}+2}+0.5. Then using the integral notation as shown in the above example, graph its antiderivative so that the maximum value of the antiderivative function is twice as great as the maximum value of your function (c) from Discussion 1. 
Using the graph of the antiderivative function, estimate the 
-intercept and the end value of the function (as 
a->infinity). 
Interpret the 
-intercept and the end behavior of the antiderivative function in the context of your application problem.

To tackle this problem, I'll guide you step-by-step on how to approach the graphing of the derivative and antiderivative functions based on your example, as well as how to interpret the results in the context of your application.

### Problem Breakdown:

1. **Step 1: Understand the Function**
   - Your given function is \( f(x) = \frac{x+4}{(x-2)^2 + 2} + 0.5 \). This is the derivative function from your application problem, and we will work with this to find the antiderivative.

2. **Step 2: Graphing the Derivative**
   - In Desmos, start by graphing the derivative function:
   \[
   f(x) = \frac{x+4}{(x-2)^2 + 2} + 0.5
   \]
   This will give you an insight into the behavior of the derivative, including its critical points and overall behavior.

3. **Step 3: Find the Antiderivative**
   - To graph the antiderivative, we need to find the indefinite integral of the function \( f(x) \). In this case, let’s represent it symbolically as \( F(x) = \int f(x) \, dx \).

4. **Step 4: Modify the Constant \( C \)**
   - Since the prompt requires the maximum value of the antiderivative to be **twice as great** as the maximum value of your function from Discussion 1 (denote this value as \( c \)), we need to adjust the constant of integration \( C \) appropriately when graphing the antiderivative.
   - This involves adding a constant \( C \) so that \( \max(F(x)) = 2c \). Once you graph the antiderivative in Desmos, you can adjust the vertical shift (the constant \( C \)) to meet this requirement.

5. **Step 5: Analyze the Graph**
   - **Estimate the y-intercept:** Check where the antiderivative graph intersects the y-axis (where \( x = 0 \)).
   - **End behavior:** As \( x \to \infty \), observe the behavior of the antiderivative. You should notice if the function approaches a horizontal asymptote or grows without bound.

---

### Interpretation in Context:

1. **y-intercept Interpretation**:
   - The y-intercept represents the initial condition of your system in the context of your application problem. It could represent an initial quantity or state before any changes occur (e.g., an initial amount of a resource, position, etc.).

2. **End Behavior Interpretation**:
   - As \( x \to \infty \), the behavior of the antiderivative function reflects the long-term trend of the quantity being measured in your application. For example, if the antiderivative tends towards a constant value, it could mean that the system reaches a steady state or equilibrium as time progresses.

---

### Steps to Take on Desmos:
1. **Graph the derivative function \( f(x) = \frac{x+4}{(x-2)^2 + 2} + 0.5 \).**
2. **Use Desmos's integral feature to graph the antiderivative \( F(x) \).**
3. **Adjust the constant \( C \) to ensure the maximum value of \( F(x) \) is twice the maximum of \( c \).**
4. **Estimate the y-intercept and the behavior as \( x \to \infty \).**

Would you like a step-by-step explanation for setting up the graphs in Desmos? Here are some guiding questions:

1. How do we calculate the constant of integration \( C \) to ensure the correct maximum value?
2. How do you estimate the maximum value of the derivative and antiderivative graphs in Desmos?
3. What role does the constant \( C \) play in adjusting the antiderivative graph?
4. How can we analyze the end behavior of functions based on their graphs?
5. What is the significance of a y-intercept in practical application problems?

**Tip**: In Desmos, use sliders to adjust the constant \( C \) dynamically and observe how the graph shifts

Access the help on integrals (Desmos) and review the first example in the section about indefinite integrals. This example demonstrates the method of graphing antiderivative of the sample function f(x)=x^2. The outcome is the graph of F(x)=x^3/3. In Desmos, graph the first derivative function from your application problem in Discussion 6 y=(x+4)/((x-2)^2+2)+0.5. Then using the integral notation as shown in the above example, graph its antiderivative so that the maximum value of the antiderivative function is twice as great as the maximum value of your function (c) from Discussion 1. Using the graph of the antiderivative function, estimate the y-intercept and the end value of the function (as x->infinity). Interpret the y-intercept and the end behavior of the antiderivative function in the context of your application problem.

Learn how to graph the antiderivative of a complex function using Desmos. This guide demonstrates finding the indefinite integral, adjusting for maximum values, and interpreting results like y-intercepts and end behavior in application contexts. Suitable for college-level calculus students.

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