Find The Absolute Maximum And Minimum Values Of G ( X ) = E − X 2 G(x)=e^{-x^2} G ( X ) = E − X 2 On The Interval − 4 ≤ X ≤ 2 -4 \leq X \leq 2 − 4 ≤ X ≤ 2 . Then Graph The Function. Label The Points On The Graph Where The Absolute Extrema Occur, And Include Their Coordinates.- Absolute

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Introduction

In this article, we will explore the concept of absolute maximum and minimum values of a function on a given interval. We will focus on the function $g(x)=e^{-x^2}$ and find its absolute maximum and minimum values on the interval $-4 \leq x \leq 2$. Additionally, we will graph the function and label the points on the graph where the absolute extrema occur.

Understanding Absolute Extrema

Absolute extrema are the maximum and minimum values of a function on a given interval. The absolute maximum value is the largest value that the function attains on the interval, while the absolute minimum value is the smallest value that the function attains on the interval.

Finding Critical Points

To find the absolute maximum and minimum values of a function, we need to find the critical points of the function. Critical points are the points on the graph of the function where the derivative of the function is equal to zero or undefined.

Finding the Derivative of $g(x)$

To find the critical points of $g(x)=e^{-x^2}$, we need to find the derivative of the function. Using the chain rule, we can find the derivative of $g(x)$ as follows:

$$g'(x) = -2xe^{-x^2}$$

Finding Critical Points

To find the critical points of $g(x)$, we need to set the derivative of the function equal to zero and solve for $x$. Setting $g'(x) = 0$, we get:

$$-2xe^{-x^2} = 0$$

Since $e^{-x^2}$ is never equal to zero, we can divide both sides of the equation by $e^{-x^2}$ to get:

$$-2x = 0$$

Solving for $x$, we get:

$$x = 0$$

Therefore, the only critical point of $g(x)$ is $x = 0$.

Finding Absolute Extrema

To find the absolute maximum and minimum values of $g(x)$, we need to evaluate the function at the critical points and at the endpoints of the interval. The critical point is $x = 0$, and the endpoints of the interval are $x = -4$ and $x = 2$.

Evaluating $g(x)$ at Critical Points and Endpoints

Evaluating $g(x)$ at the critical point $x = 0$, we get:

$$g(0) = e^{-0^2} = e^0 = 1$$

Evaluating $g(x)$ at the endpoint $x = -4$, we get:

$$g(-4) = e^{-(-4)^2} = e^{-16}$$

Evaluating $g(x)$ at the endpoint $x = 2$, we get:

$$g(2) = e^{-2^2} = e^{-4}$$

Conclusion

In conclusion, we have found the absolute maximum and minimum values of $g(x)=e^{-x^2}$ on the interval $-4 \leq x \leq 2$. The absolute maximum value is $g(0) = 1$, and the absolute minimum value is $g(-4) = e^{-16}$. We have also graphed the function and labeled the points on the graph where the absolute extrema occur.

Graphing the Function

To graph the function $g(x)=e^{-x^2}$, we can use a graphing calculator or a computer algebra system. The graph of the function is a bell-shaped curve that opens upwards.

Labeling Absolute Extrema

To label the points on the graph where the absolute extrema occur, we need to evaluate the function at the critical points and at the endpoints of the interval. The critical point is $x = 0$, and the endpoints of the interval are $x = -4$ and $x = 2$.

Plotting Absolute Extrema

Plotting the points on the graph where the absolute extrema occur, we get:

• The absolute maximum value $g(0) = 1$ occurs at the point $(0, 1)$.
• The absolute minimum value $g(-4) = e^{-16}$ occurs at the point $(-4, e^{-16})$.

Final Thoughts

In this article, we have explored the concept of absolute maximum and minimum values of a function on a given interval. We have focused on the function $g(x)=e^{-x^2}$ and found its absolute maximum and minimum values on the interval $-4 \leq x \leq 2$. We have also graphed the function and labeled the points on the graph where the absolute extrema occur.

References

• [1] Calculus, 3rd edition, by Michael Spivak
• [2] Calculus, 2nd edition, by James Stewart
• [3] Graphing Calculators, by Texas Instruments

Keywords

• Absolute maximum
• Absolute minimum
• Critical points
• Derivative
• Graphing
• Interval
• Maximum
• Minimum
• Points
• Values
  

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Introduction

In our previous article, we explored the concept of absolute maximum and minimum values of a function on a given interval. We focused on the function $g(x)=e^{-x^2}$ and found its absolute maximum and minimum values on the interval $-4 \leq x \leq 2$. In this article, we will answer some frequently asked questions related to the topic.

Q&A

Q1: What is the absolute maximum value of $g(x)=e^{-x^2}$ on the interval $-4 \leq x \leq 2$?

A1: The absolute maximum value of $g(x)=e^{-x^2}$ on the interval $-4 \leq x \leq 2$ is $g(0) = 1$.

Q2: What is the absolute minimum value of $g(x)=e^{-x^2}$ on the interval $-4 \leq x \leq 2$?

A2: The absolute minimum value of $g(x)=e^{-x^2}$ on the interval $-4 \leq x \leq 2$ is $g(-4) = e^{-16}$.

Q3: How do you find the critical points of a function?

A3: To find the critical points of a function, you need to find the derivative of the function and set it equal to zero. Then, solve for the variable.

Q4: What is the derivative of $g(x)=e^{-x^2}$?

A4: The derivative of $g(x)=e^{-x^2}$ is $g'(x) = -2xe^{-x^2}$.

Q5: How do you find the absolute maximum and minimum values of a function on a given interval?

A5: To find the absolute maximum and minimum values of a function on a given interval, you need to evaluate the function at the critical points and at the endpoints of the interval.

Q6: What is the graph of $g(x)=e^{-x^2}$?

A6: The graph of $g(x)=e^{-x^2}$ is a bell-shaped curve that opens upwards.

Q7: How do you label the points on the graph where the absolute extrema occur?

A7: To label the points on the graph where the absolute extrema occur, you need to evaluate the function at the critical points and at the endpoints of the interval.

Q8: What are the coordinates of the absolute maximum and minimum values of $g(x)=e^{-x^2}$ on the interval $-4 \leq x \leq 2$?

A8: The coordinates of the absolute maximum value are $(0, 1)$, and the coordinates of the absolute minimum value are $(-4, e^{-16})$.

Conclusion

In this article, we have answered some frequently asked questions related to the topic of finding the absolute maximum and minimum values of a function on a given interval. We have also provided the answers to some common questions that students may have when working with functions and graphing.

Final Thoughts

In conclusion, finding the absolute maximum and minimum values of a function on a given interval is an important concept in calculus. By understanding how to find these values, you can better analyze and graph functions, and make informed decisions in a variety of real-world applications.

References

• [1] Calculus, 3rd edition, by Michael Spivak
• [2] Calculus, 2nd edition, by James Stewart
• [3] Graphing Calculators, by Texas Instruments

Keywords

• Absolute maximum
• Absolute minimum
• Critical points
• Derivative
• Graphing
• Interval
• Maximum
• Minimum
• Points
• Values